Olimpiade Math SD

Pecahan, Persen dan Rasio

1. Compute the sum of a, b and c given that ${\frac {a}{2}}={\frac {b}{3}}={\frac {c}{5}}$ and the product of a, b and c is 1920. (PMWC 9th Team)

2. Find the fraction with the smallest denominator between ${\frac {97}{36}}$ and ${\frac {96}{35}}$. (PMWC 7th Individual)

3. On what letter in the following figure the number ${\frac {1}{5}}$ lies exactly? (OSN SD Uraian 2012)

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4. There are nine fractions between ${\frac {1}{5}}$ and ${\frac {1}{2}}$ such that the difference between any two successive fractions is constant. Find the sum of these eleven fractions. (PMWC 8th Team)

5. Evaluate $2014\times ({\frac {1}{962}}+{\frac {2}{45}})+1969\times ({\frac {1}{2014}}-{\frac {2}{45}}+45\times ({\frac {1}{2014}}-{\frac {2}{962}}$ . (NMOS 2014)

6. Given that x and y are whole numbers such that 1/x-1/y=1/6, find the largest value of x+y. (NMOS 2014 Special Round)

Find the value of (101 + 103 + 105 + … + 197 + 199)/(1 + 3 + 5 + … + 97 + 99). (NMOS 2011 Special Round) A sequence of 20 numbers is given as follows. (4/5+5/25),(4/6+6/25),(4/7+7/25),…,(4/23+23/25),(4/24+24/25) It is known that each of the numbers is of the form (4/k+k/25), where k=5,6,…,24. Given that m/n is the smallest value among the 20 numbers in the sequence and that m/n is in its simplest form, determine the value of m+n. (NMOS 2014 Special Round)

The decimal form of 3/32 is 0, 09375. The decimal form of 29/32 is … (OSN SD Isian Singkat 2012)

Given that x/9900=0.201414141414…, a repeated decimal where ‘14’ keeps repeating, find the value of the whole number x. (NMOS 2014 Special Round)

Let a = 0.363636… and b = 0.515. Find the digit on the 2011th decimal place of the product ab. (IMSO 2011 Essay Problem)

What is the largest value of positive integer n such that n/666=0.2y17y17y17⋯, where y is a single digit and y17 is repeating? (PMWC 20th Individual)

The positive integers a and b are such that 5/7<a/b<9/11. Find the value of a+b when b takes the smallest possible value. (IMSO 2012 Short Answer)

The extended fraction 1/(3+1/(3+1/(3+1/3))) Can be expressed as a simple fraction 33/A. Find the value of A. (NMOS 2015)

Let a,b dan c are natural numbers such that a/2+b/5+c/7=69/70. The result of 2a+5b+7c is … (OSN SD Isian Singkat 2018)

Find the value of

(2009/2010+2010/2011+6/7)(1/2+2009/2010+2010/2011+2/5)-(1/2+2009/2010+2010/2011+6/7)(2009/2010+2010/2011+2/5)

(IMSO 2011 Short Answer)

Study the following pattern.

1/(1×2)=1/2,1/(1×2)+1/(2×3)=2/3,1/(1×2)+1/(2×3)+1/(3×4)=3/4.

Given that 1/(1 × 2)+1/(2 × 3)+1/(3 × 4)+⋯+1/(2013 × 2014)=(a + 2)/(a + 3), where a is a positive integer. Find the value of a. (IMSO 2013 Short Answer)

Find x if x/(1×2×3)+x/(2×3×4)+x/(3×4×5)+⋯+x/(8×9×10)=11.

(PMWC 16th Team)

Replace the letters a, b, c and d in the following expression with the numbers 1, 2, 3 and 4, without repetition: a+1/(b+1/(c+1/d))

Find the difference between the maximum value and the minimum value of the expression. (PMWC 9th Team)

Find the greatest value of a+1/(b + 1/c)+d+1/(e + 1/f)+g+1/(h + 1/i) where each letter represents a different non-zero digit. (PMWC 12th Individual)

Note that 1+ 1/(1+1/(2+1/4))=1+1/(1+4/9)=1+9/13=22/13. It is known that the fraction 13/10 can be written as 1+1/(a+1/(b+1/c)), where a,b and c are whole numbers. Find the value of a+2b+3c. (NMOS 2013 Special Round)

AXXX and XXXB are two four-digit numbers, where A,B and X are all distinct. If AXXX/XXXB=2/5, find A,B and X. (PMWC 3rd Individual)

The fraction 44/37 can be written in the form of 1+1/(x + 1/(y + 1/z)), where x,y and z are distinct integers. Find the value of x+y+z. (PMWC 3rd Team)

Let A=x+1/(y+1/z), B= y+1/(z+1/x), C=z+1/(x+1/y), where x,y and z are positive integers. If A=37/16, calculate the value of A×B×C. (PMWC 15th Team)

What is the 2013th term in the sequence 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4,… ? (PMWC 16th Individual)

What is the largest integer less than or equal to the expression (IMSO 2015 Essay Problem)

1/(1/1985+1/1986+1/1987+⋯+1/2015)

Find the last digit of N, where N=(1+2+3+4)+(1^2+2^2+3^2+4^2 )+⋯+(1^2012+2^2012+3^2012+4^2012 ). (PMWC 15th Individual)

There are positive integers k,n,m such that 19/20<1/k+1/n+1/m<1. What is the smallest possible value of k+n+m? (PMWC 4th Team)

Let a,b and c be different positive integers such that 1=1/2+1/3+1/7+1/a+1/b+1/c. What is the smallest possible value of a+b+c? (IMSO 2015 Essay Problem) ?

Diketahui lima pecahan 1/3,1/6,1/9,1/12 dan 1/15. Empat dari lima pecahan tersebut dilambangkan dengan huruf a,b,c dan d. Jika a+b+c=7/12 dan c×d=1/45 maka nilai d= … (OSN SD Isian Singkat 2016) Whenever Sam reads a date like 20/11/2016, he incorrectly interprets it as two divisions, with the second one evaluated before the first one: 20÷(11÷2016)=40320/11=3665 5/11

For some dates, like this one, he does not get an integer, while for others, like 20/8/2016, he gets 20÷(8÷2016)=5040 , an integer. How many dates this year (day/month/year) give him a non-integer? (IMSO 2016 Short Answer)

Penjumlahan pecahan di antara empat pecahan 1/3, 1/9,1/27, dan 1/81 menghasilkan berbagai bilangan. Sebagai contoh bilangan 4/9 dapat diperoleh dengan menjumlahkan dua pecahan, yaitu 1/3+1/9, atau empat pecahan 1/9+1/9+1/9+1/9 . Bilangan 70/81 dapat diperoleh dengan menjumlahkan paling sedikit . . . pecahan. (OSN SD Isian Singkat 2015)

We insert +,-,× and ÷ each exactly once into the following four boxes, (1/2□1/9), (1/3□1/8),(1/4□1/7), (1/5□1/6) so that the sum of these four terms is the largest. In this case, if the second largest among these four terms is written as A/B in its simplest form, find A+B. (NMOS 2012 Special Round)

In the expression a/b+ c/d+ e/f, each letter is replaced by a different digit among 1, 2, 3, 4, 5, and 6. What is the smallest possible value of this expression? (IMSO 2007 Short Answer)

Let ∎ and ∆ be two distinct positive integers such that ∎-∆=2013, ∎/∆=(∎ - ∆ - 669)/(∎ - ∆ - 2011). What is the value of ∎? (IMSO 2012 Short Answer)

The sum of the reciprocals of four positive integers a,b,c and d (not necessarily different) is 7/10, i.e. 1/a+1/b+1/c+1/d=7/10. What is the smallest possible sum of these four integers? (PMWC 17th Individual)

Given that 1+1/2^2 +1/3^2 +⋯=M, and 1+1/3^2 +1/5^2 +⋯=K, find the ratio of M∶K. (PMWC 16th Individual)

In an examination, students can obtain 4 possible grades: A,B,C and D. 1/7 of the students got A, 1/3 of them got B, and 1/2 of them got C. If there are less than 50 students taking the examination, how many students got D? (NMOS 2006)

A huge bowl contains many sweets. On the first day, Kenneth ate 1/7 of the number of sweets. On the second day, he ate 1/6 of the remaining number of sweets. On the third day, he ate 1/5 of the remaining number of sweets. On the fourth day, he ate 1/4 of the remaining number of sweets. How many sweets were there in the bowl initially if there are 6 sweets remaining after the fourth day? (NMOS 2007)

Wayne has 7/13 of the number of stamps that Monica has. After Peter gave an equal number of stamps to Wayne and Monica, Wayne has 9/10 of the number of stamps that Monica has. If the number of stamps Peter gave to Wayne is between 500 and 550, what is the number of stamps that Peter gave to Monica? (NMOS 2009 Special Round)

The amount of money David has is 1/4 of the amount of money Samuel has. If Samuel gives David $500, Samuel will have 2/3 of the amount of money David has. How much money do they have altogether? (NMOS 2010)

At a wedding dinner, all the men wear pants. 1/4 of the number of women wear pants while the rest of the women dresses. If the number of women is 2/3 of the number of men, what percentage of the people at the wedding dresses? (NMOS 2010)

Di kotak terdapat sejumlah bola merah, hijau, dan biru. Banyaknya bola merah dan biru di kotak tersebut berturut-turut adalah 1/4 bagian dan 2/5 bagian. Banyaknya bola hijau sama dengan dua kali banyaknya bola merah dikurangi 9. Banyaknya bola hijau di kotak tersebut adalah . . . . (OSN SD Isian Singkat 2011)

Sejumlah salak yang berbobot sama dimasukkan ke dalam sebuah keranjang. Apabila diambil 1/4 bagian dari isi keranjang, berat keranjang beserta salak yang tersisa turun menjadi 19, 5 kg. Namun apabila yang diambil hanya 1/6 bagian dari isi keranjang, berat keranjang beserta salak yang tersisa hanya turun menjadi 21, 5 kg. Berapa berat keranjang beserta salak sisanya apabila diambil 2/3 bagian dari salak? (OSN SD Uraian 2011)

Pada hari lebaran, Pak Samsul ingin membagikan sejumlah uang kepada cucu-cucunya. Pak Samsul membagi cucu-cucunya ke dalam beberapa kategori, yaitu usia TK, SD, SMP dan SMA. Dari sejumlah uang yang disiapkan, 2/5 -nya untuk usia TK, 1/5 -nya untuk usia SD, dan 1/3-nya untuk usia SMP, dan sisanya untuk usia SMA. Bagian untuk usia SMA adalah . . . bagian. (OSN SD Isian Singkat 2012)

Linda has 2 containers, Container A and Container B. The amount of water in Container A is 4/5 of the amount in Container B at first. When 200 ml of water from Container A is poured into Container B, the amount of water in Container A is 1/2 the amount in Container B. How much water was in Container A at first? (NMOS 2006)

Melvin, Nelson and Oliver were playing cards. At first, Melvin started with 40 cards and then he gained 5/8 more than his original number of cards from Nelson. In the next round, Melvin and Nelson each lost 1/4 of Melvin’s original number of cards to Oliver. In the final round, Melvin gained 1/8 of his original number of cards from Nelson. After the final round, Melvin, Nelson and Oliver all have the same number of cards. What was the total number of cards Melvin, Nelson and Oliver had at first? (NMOS 2007)

A piece of square paper, B, is cut from a big piece of square paper , A, such that its side is 1/2 that of the side of A. Another smaller square piece of paper, C, is cut from B such that its side 2/3 that of the side of B. The area of C is Δ/ of the area of A, where Δ and  are whole numbers. Find the smallest value of Δ+ . (NMOS 2007)

In a class of 45 students, 6 girls and 1/7 of the boys took part in a Mathematics competition. There are an equal number of girls and boys who did not take part in this competition. Find the number of girls in the class. (NMOS 2008 Special Round)

Aaron, Betty, Calvin, Diana and Edward ate 21 cakes altogether. They took turns to eat their share. After each of them finished their share, they told the group how much they had eaten. You are not told who ate the cakes first, but you are told what they had said. Aaron said: “I have eaten 2/3 of the remaining.” Betty said: “I have eaten half of the remaining.” Calvin said: “I have eaten half of the remaining.” Diana said: “I have eaten all of the remaining.” Edward said: “The numbers of cake eaten by us are all different whole numbers.” What is the number of cakes Edward has eaten? (NMOS 2014)

In a specific math Olympiad training class, the proportion of female students is more than 4/15, but less than 3/10, what is the smallest possible number of pupils in the class? (NMOS 2015 Special Round)

Jika 15% dari 4/5 uang Rani adalah Rp 45.000.00, maka 5/6 uang Rani adalah… (OSN SD Isian Singkat 2016)

Banyaknya bilangan pecahan a/b yang nilainya kurang dari 4 dengan pembilang dan penyebut bilangan 1, 2, 3, 4, 5 adalah (OSN SD Isian Singkat 2017)

Misalkan a,b,c dan d adalah bilangan berbeda yang dapat diganti dengan 1, 2, 3, 4 dan 5, sehingga hasil operasi a/1+b/10+c/100+d/1000 dapat diurutkan dari terkecil sampai terbesar. Berapakah hasil operasi bilangan tersebut pada urutan ke-32? (OSN SD Uraian 2018)

In the fraction a/b,a and b are positive integers. Amanda multiplied the numerator of this fraction by 340. Which positive integer closest to 2017 should Amanda add to the denominator to obtain the fraction 2a/3b  ? (PMWC 20th Individual)

The fraction 221/210 is obtained as a sum of three positive fractions each less than 1 with single digit denominators. Find the largest (greatest) of these fractions in simplest form. (PMWC 21th Individual)

In a classroom, 1/4 of the pupils are girls. After 10 boys leave the room, the portion of the girls in the room becomes 1/3. What is the total number of pupils before the 10 boys leave? (IMSO 2004 Essay Problem)

The income of a taxi driver is the sum of the regular salary and some tips. The tips are 5/4 of his salary. What is the fraction of his income which comes from his tips? (IMSO 2014 Short Answer)

Unit fractions are those fractions whose numerator is 1 and denominator is any positive integer. Express the number 1 as the sum of seven different unit fractions, given five of them are 1/3,1/5,1/9,1/15 and 1/30. Find the product of the two remaining unit fractions. (IMSO 2015 Short Answer)

Class A and Class B have the same number of students. The number of students in class A who took part in a mathematics competition is 1/3 of the students in class B who did not take part. The number of students in class B who took part in a mathematics competition is 1/5 of the students in class A who did not take part. Find the ratio of the number of students in class A who did not take part in this competition to the number of students in class B who did not take part. (IMSO 2016 Short Answer)

What number should be subtracted from the numerator of the fraction 537/463 and added to the denominator so that the resulting fraction is equal to 1/9? (IMSO 2017 Short Answer)

The digits 0 to 9 without repetition form two 5-digit numbers M and N. Given that M/N equals to 1/2, find the largest possible sum of M and N. (PMWC 4th Team)

Three persons together own a pile of about 200 gold coins. They originally posses 1/2 , 1/3, and 1/6 of the coins, respectively. Now each person is going to take out some coins from the pile until there is nothing left. Then the first person is to return 1/2 of what he has taken out, the second person 1/3 of what he had taken, and third person 1/6 of what he had taken. If the returned coins are equally distributed to the three persons, than each person will get back the same number of coins which he originally possessed. How many gold coins were there originally? (PMWC 5th Team)

Let 1/a+1/b+1/c=1/2005, where a and b are different four-digit positive integers (natural numbers) and c is a five-digit positive integer (natural number). What is the number c  ? (PMWC 9th Individual)

Let x be a fraction between 36/35 and 183/91. If the denominator of x is 455 and the numerator and denominator have no common factor except 1, how many possible values are there for x? (PMWC 9th Individual)

Elvis has many identical pizzas. He gives 1/3 of his pizzas plus 2/3 of a pizza to Adam. He then gives 1/4 of the remaining pizzas plus half a pizza to Benny. He gives half of the remaining pizzas to Clinton. Lastly, he gives half of the remaining pizzas plus half a pizza to Deon. In the end, Elvis is left with 5 pizzas. How many pizzas does Elvis have originally? (PMWC 13th Individual)

There are three boxes of marbles. Each box contains a different number of marbles. From the first box, I remove 1/3 of the number of marbles, from the second box, I remove 1/4 of the number of marbles and from the third box, I remove 1/5 of the number of marbles. Finally, there is an equal number of marbles remaining in all the three boxes. What is the smallest possible number of marbles which I may have removed in total? (PMWC 14th Individual) There are only white and red balls in a bag. Tom takes one ball, looks in the bag and says “ 5/7 of the remaining balls are white”. After that he puts the ball back into the bag. Then Masha takes one of the balls, looks in the bag and says “ 12/17 of the remaining balls are white”. How many balls in total were there in the bag initially? (PMWC 15th Individual)

Frank had a total of 424 local and foreign stamps. He gave away 40% of the local stamps and 5/6 of the foreign stamps to his friends. Then he bought 28 foreign stamps. As a result, the number of foreign stamps he had was 16 2/3% of the number of local stamps left. How many local stamps did have at first? (NMOS 2016 Special Round)

Pak Abun menjual dua buah rumah yang masing-masing harganya Rp52.000.000,00. Ia memperoleh keuntungan 30% dari rumah pertama, tetapi menderita kerugian 20% dari rumah kedua. Ternyata secara keseluruhan Pak Abun mengalami kerugian. Berapa rupiahkah kerugiannya? (OSN SD Isian Singkat 2003)

Benedict spent $600 of his monthly salary and saved the rest. When he increased his spending by 35%, his savings decreased by 10%. How much, in $, was his monthly salary? (NMOS 2015)

There are some marbles in a box. 40% of the marbles are red. There are 12 more yellow marbles than red marbles in the box and the rest of the marbles are green. If there is a total of 132 red and yellow marbles, what is the percentage of the green marbles in the box? (NMOS 2015)

The price of a shirt is reduced from Rp. 24.000,00 to Rp.18.000,00. If normally the profit is 60%, how many percent is the profit or loss after the price reduction? (OSN SD Uraian 2004)

Suatu kegiatan ekstrakurikuler yang diikuti oleh 100 anak menempati tiga ruangan A, B, dan C. Setelah satu bulan, 50% anak dari ruang A pindah ke ruang B, 20% anak dari ruang B pindah ke ruang C, dan sepertiga anak dari ruang C pindah ke ruang A. Setelah perpindahan terjadi, ternyata banyak anak di setiap ruangan tidak berubah. Berapakah banyak anak di ruang A? (OSN SD Uraian 2007)

Sebuah kotak berisi bola merah dan bola putih dengan 80% di antaranya adalah bola merah. Mula-mula diambil 35 bola merah dan 5 bola putih dari kotak tersebut. Sisanya dibagi menjadi beberapa kelompok, masing-masing terdiri atas 7 bola. Pada setiap kelompok terdapat 5 bola merah. Pada awalnya paling sedikit terdapat … bola dalam kotak tersebut. (OSN SD Isian Singkat 2010)

Seorang karyawan baru di sebuah perusahaan memulai kerja pada tanggal 1 januari dengan gaji per bulan sebesar 5 juta rupiah. Setelah bekerja 6 bulan gajinya naik sebesar 20%. Di akhir tahun dia harus menghitung pajak penghasilannya yang dihitung dari jumlah gajinya selama satu tahun. 18 juta rupiah dari gajinya setahun tidak kena pajak, dan sisanya kena pajak sebesar 5%. Berapa rata-rata pajaknya per bulan di tahun tersebut? (OSN SD Uraian 2015)

Harga satu buah baju di Toko A adalah Rp 5.000, 00 lebih mahal dibanding harga satu buah baju di toko B. Toko B memberikan diskon 10% untuk setiap baju sedangkan toko A memberi harga khusus jika seseorang membeli baju lebih dari satu buah, seseorang akan memperoleh diskon 40% untuk baju kedua yang dia beli. Dengan kondisi seperti itu ternyata harga dua buah baju di Toko A sama dengan harga dua buah baju di Toko B. Berapa harga satu buah baju di Toko A? (OSN SD Uraian 2011)

Perbandingan banyak siswa kelas A,B dan C adalah 2:3:4. Perbandingan nilai rata-rata ujian matematika kelas A,B dan C adalah 4:3:2. Rentang nilai ujian matematika 0 sampai dengan 100. Tentukan nilai rata-rata terbesar seluruh siswa yang mungkin. (OSN SD Uraian 2016)

Ali, Beni, dan Cepi masing-masing memilih satu bilangan positif. Mereka lalu membanding bandingkan yang mereka pilih sepasang-sepasang. Ada tiga rasio yang mereka dapatkan, ketiganya lebih kecil dari 1. Dua rasio adalah 2/5 and 5/7, sedangkan rasio ketiga adalah R, nilai R terbesar yang mungkin adalah … (OSN SD Isian Singkat 2008)

In a class, the ratio between the number of girls and boys is 4 : 5. If four boys go out, then the ratio becomes 1 : 1. How many girls are there in the class? (OSN SD Uraian 2009)

Alif memiliki 4 kotak kelereng yang masing-masing berisi 10, 15, 20 dan 28 butir kelereng. Alif mengambil sejumlah kelereng dari masing- masing kotak sehingga perbandingan banyak kelereng pada kotak tersebut menjadi 1:2:3:4. Banyaknya kelereng maksimal seluruhnya yang tersisa adalah ⋯ . (OSN SD Isian Singkat 2017)

Joko dan Badrun berdiri pada suatu antrian. Pada antrian tersebut, perbandingan antara banyaknya orang di depan dan di belakang Joko adalah 1:3. Sedangkan perbandingan antara banyaknya orang di depan dan di belakang Badrun adalah 2:5. Paling sedikit banyaknya orang pada antrian tersebut adalah (OSN SD Isian Singkat 2010)

Pada tahun 2009, perbandingan banyaknya rusa jantan dan rusa betina di suatu kebun binatang adalah 2:3. Pada tahun 2010 banyaknya rusa jantan bertambah 9 ekor dan banyaknya rusa betina berkurang 4 ekor, sehingga perbandingannya menjadi 3:2. Banyaknya rusa jantan pada tahun 2010 di kebun binatang tersebut adalah (OSN SD Isian Singkat 2010)

Pak Karto memiliki dua bidang tanah. Perbandingan luas tanah pertama dengan luas tanah kedua adalah 2 : 3. Ia menanam jagung dan kedelai pada tanah pertama maupun pada tanah kedua. Pada tanah pertama, perbandingan luas tanaman jagung dan luas tanaman kedelai adalah 1 : 3, sedangkan secara keseluruhan perbandingannya adalah 3 : 7. Berapakah perbandingan luas tanah yang ditanami tanaman jagung dan tanaman kedelai pada tanah ke- dua? (OSN SD Uraian 2010)

Alice and John had some sweets in the ratio 5∶7. After Alice gave John some sweets, the ratio of the number of sweets Alice had to that of John was 17∶31. Express the number of sweets that Alice gave John as a percentage of the number of sweets she had first. If your answer is m%, shade “m” as your answer. For example, if the answer 50%, shade “50”. (NMOS 2012)

Ali bought a camera from Betty, and then sold it to Cathy at a profit. If Ali had given Cathy a 10% discount, Ali would have made a $140 profit. If Ali had given Cathy a 5% discount, then Ali would have made a $200 profit. What was the price of the camera that Ali bought from Betty? (NMOS 2010)

The ratio of male students to female students in a primary school is 5:4. The ratio of students who wear spectacles to students who do not is 11:1. Given that 64% of the students who do not wear spectacles are female, find the percentage of female students who do not wear spectacles. (NMOS 2010 Special Round)

Kadar garam dalam enam liter air laut adalah 4%. Setelah air laut tersebut menguap sebanyak 1 liter, kadar garam menjadi ... persen. (OSN SD Isian Singkat 2007)

Wira mempunyai dua buah botol yaitu botol A dan botol B. Botol A berisi (3 )/4 air putih, dan botol B berisi (3 )/4 susu. Wira kemudian menuangkan isi botol A ke botol B sampai botol B terisi penuh, lalu botol B dikocok sehingga air dan susu tercampur rata. Setelah itu, campuran di botol B dituangkan ke dalam botol A sampai botol A terisi penuh. Perbandingan air dan susu yang ada dibotol A sekarang adalah . . . : . . . . (OSN SD Isian Singkat 2014)

Toko kopi Aroma mempunyai persediaan kopi arabika dan kopi robusta. Harga tiap kilo- gram kopi arabika adalah Rp.70.000, 00 sedangkan harga tiap kilogram kopi robusta adalah Rp.50.000, 00. Karena banyak konsumen yang menyukai kopi campuran antara arabika dan robusta, mereka mencampur kopi arabika senilai Rp.280.000, 00 dengan kopi robusta senilai Rp.300.000, 00. Berapa harga 1 kg kopi campuran tersebut? (OSN SD Uraian 2012)

A watermelon, with 92% of its weight being water, was left to stand in the sun. Some of the water evaporated so that now only 91% of its weight is water. The weight of the watermelon is now 4048 grams. What was the weight of the watermelon (in grams) before the water evaporated? (NMOS 2013)

A kind of drink contains 5% pure chocolate. If 5 liters of milk are added to 20 liters of this drink, find the percentage of chocolate in the mixture. (IMSO 2006 Essay Problem)

Fatimah wants to make a drink that contains 40% pure orange juice. This drink is called 40% orange juice. Her mom gives her 100ml of 20% orange juice, and a large bottle of drink that contains 80% pure orange juice. Fatimah needs . . . ml the drink in the bottle to produce the 40% orange juice. (IMSO 2008 Short Answer)

A big container is filled with 150 litres of syrup and 50 litres of water. Then 40 litres of the mixture is removed and the container is filled with water again to obtain the original volume. What is the percentage of the syrup in the final mixture? (NMOS 2008 Special Round)

A painter mixed different colours of paint in a pail. He mixed 8 litres of blue paint and 12 liters of yellow paint together in the pail to obtain green paint. Not satisfied with the colour of the mixture, he poured away 5 litres of the mixture and added blue paint to obtain the original volume. What percentage of the final mixture was made up to blue paint? (NMOS 2009)

Bilangan Bulat

Find the value of 1-2+3-4+5-6+7-…+2001-2002+2003-2004+2005. (NMOS 2006)

Dengan mengganti tanda ± dengan tanda + atau - saja mungkinkah kamu mendapatkah persamaan ini menjadi benar? Beri penjelasan dan berikan beberapa contoh! ± 1 ±2 ±3 ± . . . ± 12 = 1

Dengan mengganti tanda ± dengan tanda + atau - saja mungkinkah kamu mendapatkah persamaan ini menjadi benar? Beri penjelasan dan beberapa contoh! ± 1 ±2 ±3 ± . . . ± 19± 20±21=1

On a certain planet, the following equations are true. D + A + R + T = 11 T + A + R + T = 12 C + A + R + T = 13 Each letter represents a different whole number. No letter takes the value 0. Find the largest possible value of A + R + T. (NMOS 2007)

Bilangan tiga digit ABC mempunyai sifat : A + B + C = 18 B − C = 1 CBA − ABC = 396 Bilangan ABC tersebut adalah . . . (OSN SD Isian Singkat 2013)

If the same 4-digit number is subtracted from 2010, 2000 and 1990, three different prime numbers will be obtained. What is the 4-digit number? (PMWC 13th Individual)

Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012, what number did Maria erase? (PMWC 15th Individual)

Using only brackets, +,-,× and/or ÷, we can form the number 48 from single-digits of 4 as follows: (4+4)×4+4×4=48

What is the minimum number of single-digit of 4 required to form the number 400? (NMOS 2009 Special Round)

When a two-digit number is divided by the sum of its digits, what is the largest possible remainder? (IMSO 2012 Short Answer)

Pak Amat memiliki suatu neraca dan tiga buah anak timbangan yang terdiri atas anak timbangan dengan berat 1 kg, 3 kg, dan 9 kg. Dengan alat tersebut, Pak Amat dapat menimbang benda A yang beratnya 2 kg dengan cara berikut.

Berat benda paling ringan (dalam bentuk bilangan bulat) yang tidak dapat ditimbang Pak Amat adalah . . . kg. (OSN SD Isian Singkat 2013) A large number “123456789101112131415 … 454647484950” s formed by writing 1, 2, …, 49, 50 in increasing order. Find the maximum possible value of the number formed when 80 digits are removed from the original number while the remaining digits stay in the same order. (PMWC 6th Team)

The sum of the 13 distinct positive integers is 2013. What is the maximum value of the smallest integer? (PMWC 16th Individual)

Bilangan-bilangan 14, 21, 28, 42, 49, 63, 84, 91, 105 dipisah menjadi dua kelompok, sedemikian hingga kita mendapatkan nilai terkecil dari selisih jumlah masing-masing kelompok yang mungkin. Selisih yang terkecil tersebut adalah … (OSN SD Isian Singkat 2011)

Pada jam digital, waktu 23 : 57 adalah salah satu contoh waktu yang semua angka digitnya menunjukkan bilangan-bilangan prima berbeda (2, 3, 5, 7). Dalam sehari semalam, jam digital tersebut menunjukkan waktu yang semua angka digitnya menunjukkan bilangan-bilangan prima berbeda sebanyak . . . kali. (OSN SD Isian Singkat 2013)

The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are to be filled into the boxes below and each digit can be used only once.

So that the expression produce the smallest possible product. What is the 4-digit number? (NMOS 2012 Special Round)

The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are to be filled into the boxes below and each digit can be used only once.

So that the expression produce the largest possible product. What is the 4-digit number? By using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the following boxes ◻◻◻◻◻×◻◻◻×◻, write down the expression that will produce the largest product. (Each digit is used only once.) (PMWC 5th Team)

p is the product of two 4-digit numbers formed by the digits 1, 2, 3, 4, 5, 6, 7, 8 without any repetition. Find the largest value of p in the form of ….. × ….. (You are not required to multiply the numbers). (PMWC 4th Individual)

A 2013-digit number is formed by writing the whole numbers starting from 1, in a connected way, as follows: 123456789101112….. What are the last of four digits of this number? (NMOS 2013 Special Round)

How many ways can we select six consecutive positive integers from 1 to 999 so that the tailing of the product of these six consecutive positive integers end with exactly four 0’s? (IMSO 2011 Essay Problems)

It is known that 5×6×7×8×9×10=1512 × 102 and that 2 is the largest value of n when 5×6×7×8×9×10 is written as A × 10^n (where A is a whole number). What is the largest value of m if 5×8×11×14×17×20×…×95×98×101 is written as B × 10m (where B is a whole number)? (NMOS 2014)

Suppose 1×2×3×…×2010×2011=14^n×A, where n and A are both positive integers. What is the maximum value of n? PMWC 14th Individual)

The product of 1×4×7×10×…×1999 can be expressed in the form of 7^a×10^b×A where a,b and A are positive integers. Find a+b if a and b are as large as possible. (PMWC 3rd Team)

Find the value of 100×99-99×98+98×97-97×96+⋯+4×3-3×2+2×1. (PMWC 14th Individual)

Find the last two digits of 1!+2!+3!+⋯+2010!+2011!. Note that n!=1×2×3×…×(n-1)×n. (PMWC 14th Individual)

Total selisih 2 dengan bilangan-bilangan 1, 4, 7, 8 dan 10 yaitu

(2-1)+(4-2)+(7-2)+(8-2)+(10-2)=22.
Jika total selisih a dengan bilangan-bilangan 1, 4, 7, 8 dan 10 nilainya minimal, maka nilai a adalah … (OSN SD Isian Singkat 2017)

Perhatikan petak-petak di bawah ini Petak A,B,C dan D diisi bilangan-bilangan 2, 0, 1 dan 7 yang semua petaknya memuat bilangan yang berbeda. Bila hasil operasinya berupa bilangan bulat positif yang kemungkinan hasilnya adalah 1 sampai dengan 10, maka banyaknya bilangan yang tidak mungkin diperoleh dan hasil operasinya tersebut adalah … (OSN SD Isian Singkat 2017)

Let A,B and C be three difference positive integers such that A+B+C=10. The maximum value A×B×C+A×B+B×C+C×A is (OSN SD Isian Singkat 2017)

Find the last digit of N, where N=(1+2+3+4) + (1^2+2^2+3^2+4^2 )+ (1^2012+2^2012+3^2012+4^2012 ). (PMWC 15th Individual)

The number of zeros in the end digits in the product of 1×5×10×15×20×25×30×35×40×45×50×55×60×65×70×75×80×85×90×95 is . . . . (IMSO 2010 Short Answer)

The last 4 digit of the sum 1+11+111+1111+11111=12345 is 2345. What are the last 4 digits of the following sum

1+11+111+1111+⋯+⏟(111…1)┬(2015 '1's)? (NMOS 2015)

An integer N is formed by writing the integers from 1 to 50 in order. That is N=123456789101112…484950. Some digits are removed from N to form a new integer such that the sum of the digits of the new number is 200. If M is the largest number that can be formed in this way, what are the first ten digits of M? (PMWC 17th Team)

Alex must take the sum of 11 consecutive positive integers. But he is very careless and misses two consecutive numbers and gets a total of 9832. What total would he get if he counted correctly? (PMWC 17th Individual)

The prime numbers  : 11, 13, 17, 19, 23, 29, 31 and 37 , are divided into two groups A and B. Suppose the sum of numbers in A is x and the sum of numbers in B is y. Given that x is a multiple of y, what is the smallest value of y? (NMOS 2015)

Bilangan 3461 mempunyai sifat jumlah dua angka pertama sama dengan jumlah dua angka terakhir. Berapa banyak bilangan di antara 1000 sampai 2000 yang mempunyai sifat seperti itu? (OSN SD Isian Singkat 2006) The first digit of the smallest number, which the sum of its digits is 2013, is . . . (OSN SD Isian Singkat 2013)

Andi memberitahu Budi bahwa hasil kali tiga bilangan bulat positif A, B, dan C adalah 36.Andi juga memberitahu Budi berapa jumlah tiga bilangan tersebut. Ternyata Budi tetap tidak tahu persis berapa saja bilangan-bilangan tersebut. Berapakah jumlah ketiga bilangan tersebut? (OSN SD Uraian 2013)

Anto sedang bermain-main dengan bilangan asli dari 1 sampai 100. Ia memilih dua bilangan (keduanya boleh sama) lalu menjumlahkannya. Anto akan memilih dua bilangan yang jumlahnya lebih besar atau sama dengan 190. Banyaknya cara yang dapat dilakukan Anto adalah … (OSN SD Isian Singkat 2014)

Adi mengalikan 18 bilangan asli berurutan yaitu 1×2×3×… ×17×18. Ia melakukannya lima kali dan selalu mendapat hasil yang berbeda yaitu: 6.402.373.705.727.800, 6.402.373.705.728.000, 6.402.373.705.730.000, 6.402.373.705.800.000, dan 6.402.373.706.000.000. Ternyata dari lima hasil itu ada satu yang benar. Hasil perhitungan yang benar adalah . . . (OSN SD Isian Singkat 2014)

Tanggal 21 mei 2015 dapat juga ditulis 21 05 2015. Jumlah empat angka pertama (yang menyatakan tanggal dan bulan), yaitu (2+1+0+5)=8 sama dengan jumlah empat angka terakhir (yang menyatakan tahun) (2+0+1+5)=8. Banyak tanggal di tahun 2015 yang memiliki sifat seperti itu adalah . . . buah. (OSN SD Isian Singkat 2015)

Arrange the following 5 numbers in descending order: 2^847,3^539,5^363,7^308,11^242. (PMWC 19th Individual) If the five numbers 2^147×6^49, 3^98×5^49, 5^98×2^49, 7^98 and 128^7×23^49 , are arranged in ascending order, which number will be in the middle? (PMWC 17th Individual)

How many digits does the product  have ? (PMWC 11th Individual)

Banyak digit hasil operasi bilangan 3217 × 580 adalah……… (OSN SD Isian Singkat 2016)

Bilangan 30 dapat dinyatakan sebagai penjumlahan tiga bilangan cacah, misalkan 1+1+28 atau 5+0+25. Banyaknya cara menuliskan 15 sebagai penjumlahan tiga bilangan cacah tanpa memperhatikan urutan adalah …………. (OSN SD Isian Singkat 2017)

Jumlah angka-angka pada bilangan 4400 adalah 8, yaitu 4 + 4 + 0 + 0 = 8. Banyaknya bilangan antara 4.000 dan 5.000 yang jumlah angka-angkanya 8 adalah . . . ……….. (OSN SD Isian Singkat 2011)

You are asked to choose three different numbers from 1 to 10. The sum of the three numbers must be 12. How many choices do you have altogether? (IMSO 2008 Essay Problems)

Diberikan tiga bilangan bulat positif a, b, and c sedemikian sehingga a∶b=b∶c=c∶a. Nilai dari ((150 × a) + (200 × b) + (250 × c))/(a + (3 × b) - (2 × c)) adalah … ……… (OSN SD Isian Singkat 2016)

Bilangan 2016 dapat ditulis sebagai jumlah dari bilangan asli berurutan (contoh untuk jumlah tiga bilangan: 2016 = 671 + 672 + 673). Berapa cara untuk menuliskannya, bila banyak bilangan yang dijumlahkan empat bilangan atau lebih? (OSN SD Isian Singkat 2016 )

Tabel operasi * dan # untuk bilangan 1, 2, dan 3 masing-masing sebagai berikut.

• 1 2 3

1 2 3 1 2 3 1 2 3 1 2 3

1. 1 2 3

1 3 2 1 2 2 1 3 3 1 3 2

Hasil dari ((3*2)#(1*2)*((3#3)*(2#1) = ……… (OSN SD Isian Singkat 2018)

A box of size 4×2×1 is divided into eight compartments. Sixteen identical balls are put inside the compartments. Each compartment must be filled with at least one ball. The numbers of balls put into rows of compartments are printed on the box, see figure.

For example, there must be 7 balls in the four front compartments, 9 balls in the four back compartments, and 6 balls in the two rightmost compartments. We can put the sixteen balls into the eight compartments in ……ways. (IMSO 2008 Short Answer)

Alan, Billy, Candy, and David are queuing (lining up) in alphabetical order. Alan is in the 7th position from the front while David is in the 9th position from the back. The number of persons between Alan and Billy is the same as those between Candy and David. In total, there are 48 persons in the queue, and six of them are between Billy and Candy. How many persons are there between Alan and Candy? (IMSO 2009 Essay Problems)

Six bags of marbles contain 18, 19, 21, 23, 25 and 34 marbles, respectively. One bag contains red marbles only. The other five bags contain no red marbles. Jane takes three of the bags and George takes two of the others. Only the bag of red marbles remains. If Jane gets twice as many marbles as George, how many red marbles are there? (IMSO 2010 Essay Problems)

Untuk mengurangi abrasi pantai suatu pulau, sekelompok remaja menanam pohon mangrove dalam satu baris di sepanjang pantai sejauh 1 1/4 kilometer. Jarak antar dua mangrove dibuat sama. Jika jarak antara mangrove urutan ke-7 dan ke-23 adalah 32 meter, berapakah banyaknya pohon mangrove yang ditanam? (OSN SD Uraian 2011)

Pak Amir membeli sejumlah mangga untuk dijual di toko buah miliknya. Jika dia menjual mangga tersebut Rp.2000 per buah, dia akan rugi sebanyak Rp.8.000, 00. Jika dia menjual Rp.3.000, 00, per buah, dia akan untung sebanyak Rp.32.000, 00. Berapa harga pembelian mangga tersebut per buahnya? (OSN SD Uraian 2012)

Jumlah 25 bilangan asli berbeda sama dengan 2013. Jika x adalah bilangan terkecil di antara 25 bilangan tersebut, maka nilai x terbesar yang mungkin adalah . . . (OSN SD Isian Singkat 2013)

Toko Roti “Puri Indah” menjual roti isi coklat seharga Rp2.000,00 per potong, roti isi kelapa seharga Rp1.500,00 per potong dan roti isi keju seharga Rp2.500,00 per potong. Ibu Ayu membeli semua jenis roti tersebut sebanyak 10 potong dengan harga Rp20.000,00. Tentukan semua kemungkinan roti yang dibeli oleh Ibu Ayu. (OSN SD Uraian 2016)

Divide 108 students into four groups such that two times the number of students in group 1 is half of the number of students in group 2, 2 less than the number of students in group 3. 2 more than the number of students in group 4. Find the number of students in group 1. (IMSO 2016 Short Answer)

Mahatma memiliki 10 butir kelereng. Kemudian Mahatma bergabung dalam sekelompok anak untuk bermain kelereng. Pemenang akan mendapatkan 1 kelereng dari tiap-tiap anak yang bermain. Jika setelah bermain lima kali, kelereng Mahatma menjadi 31 butir, maka berapakah banyak anak yang bermain bersama Mahatma? (OSN SD Uraian 2016)

Siti had 60 green apples and 90 red apples for sale. The cost of every three green apples was $10, and every five red apples was $8. She mixed the apples and sold them all. If the selling price of every five mixed apples was $15, how much profit did Siti get from selling all the apples? (IMSO 2010 Essay Problems)

There are 111 apples to be distributed to 9 children. The number of apples received by each child is different. The child receiving the smallest number of apples is kid #1 while the child receiving the most number of apples is kid #9. What is the least number of apples received by kid #9? (IMSO 2011 Essay Problems)

Andy bought three packages of goods, each worth $ 35, $ 30, and $ 40. The first package contains 2 books, 1 pencil, and 1 eraser. The second package contains 1 book, 1 pencil, 2 erasers. The third package contains 3 books and 2 erasers. Andy wants to buy the fourth package containing 2 books, 1 pencil, and 3 erasers. What is the price of the fourth package? (IMSO 2009 Essay Problems)

99 apples are divided among a number of children so that each child gets at least one apple and all children get a different number of apples. How many children are there at the most? (PMWC 3rd Team)

276 pupils are seated in a school hall. There are 22 rows of seats and each row has 15 seats. At least how many rows have an equal number of pupils? (IMSO 2011 Essay Problems)

Tempat duduk penonton pada suatu gedung pertunjukan terdiri dari tujuh baris. Banyak kursi penonton pada masing-masing baris membentuk pola sebagai berikut, baris pertama 20 kursi, baris kedua 30 kursi, baris ketiga 50 kursi, baris keempat 80 kursi, dan seterusnya. Harga tiket termahal pada baris pertama dan harga tiket pada baris berikutnya lebih murah Rp5.000,00 dari baris di depannya. Jika pada satu kali pertunjukan seluruh tempat terisi maka pengelola memperoleh pendapatan Rp33.600.000,00. Tentukan harga tiket termahal. (OSN SD Uraian 2018)

In a target,

The center is 10 points worth, the inner rim 8 points and the outer rim 5 points. Tia hits the center as many as the inner rim, and misses the target altogether one quarter of all her shots. Her total score is 99 points. The total number of shots is . . . . (OSN SD Isian Singkat 2014)

Sekelompok anak berencana membeli bola. Masing-masing anak harus iuran sebesar Rp 10.000, 00. Ternyata pada saat mereka mau membeli bola ada lima anak yang tidak jadi ikut iuran. Agar mereka tetap dapat membeli bola tiap anak yang tersisa harus menambah iurannya sebesar Rp 2.000, 00. Berapakah harga bola tersebut? (OSN SD Uraian 2014)

Misalkan sekarang uang koin yang beredar adalah koin Rp100-an, Rp200-an, Rp500-an, dan Rp1.000-an. Harga barang-barang jajanan di sekolah Elisa merupakan kelipatan Rp100, dengan harga jajanan termahalnya sebesar Rp1.000. Suatu hari Elisa membawa sejumlah uang koin ke sekolah. Dengan uang yang dibawa, dia dapat membeli sebuah jajanan yang manapun juga tanpa memerlukan kembalian. Paling sedikit banyak uang koin yang di bawa Elisa adalah . . . buah. (OSN SD Isian Singkat 2015) Marshella membutuhkan 30 kantong permen. Harga setiap kantong permen Rp5.000,00. Di setiap kantong permen tersebut terdapat satu lembar kupon. Setiap tiga lembar kupon dapat ditukarkan dengan satu kantong permen yang sama. Paling sedikit, uang yang dibutuhkan untuk mendapatkan 30 kantong permen tersebut adalah Rp . . . (OSN SD Isian Singkat 2015)

Harry, Larry and Parry were each given some marbles on Monday. On Tuesday, Harry gave Larry and Parry some marbles so that Larry and Parry each ended up with 4 times their original number of marbles. On Wednesday, Larry gave Harry and Parry some marbles so that Harry and Parry each ended up with 3 times the number of marbles they had on Tuesday. On Thursday, Parry gave Harry and Larry some marbles so that Harry and Larry each ended up with twice the number of marbles they had on Wednesday. If Harry, Larry and Parry ended up with 48 marbles each on Thursday, how many marbles did Parry have on Monday? (IMSO 2017 Short Answer)

Terdapat lima macam uang kertas yang dimiliki Pak Firman, yaitu pecahan bernilai Rp1.000, Rp5.000, Rp10.000, Rp50.000 dan Rp100.000. Pada suatu hari, Pak firman mengambil uangnya senilai Rp128.000 sebanyak 41 lembar. Ia masih ingat dari lima macam uang kertas yang dimilikinya, ada pecahan yang tidak di ambilnya dan ia mengambil 5 lembar uang pecahan Rp5.000. Banyaknya lembaran uang bernilai Rp1.000 yang diambil Pak Firman adalah . …. ….. lembar. (OSN SD Isian Singkat 2015)

There are four kinds of dollar-note (or dollar-bills) of value $1, $5, $10 and $50 respectively. There is a total of nine dollar-notes, with at least one dollar-note of each kind. If the total value of these dollar notes is $177, how many $10 dollar-notes are there? (PMWC 7th Individual)

Five boxes have different weights, each less than 100 kilograms. The boxes are weighed in pairs from all possible combinations. (that means each box is weighed with every other box) The weights of all possible pairs are 113, 116, 110, 117, 112, 118, 114, 121, 120, 115. What is the weight of the heaviest box ? (PMWC 8th Individual) Saat ini, usia Hasan sama dengan 7 kali usia Yenny. Dua tahun lalu, usia Hasan sama dengan 9 kali usia Yenny. Usia Hasan saat ini adalah (OSN SD Isian Singkat 2010)

Two years ago, Steve was three times as old as Bill, and in three years he will be twice as old as Bill. Find the sum of their ages. (IMSO 2012 Essay Problems)

Grandfather says to Ann: “My age is 7 times your age now. In a few years, my age will be 6 times your age. Subsequently, my age will be 5 times, 4 times and 3 times your age.” If we know that the age of Grandfather is a two-digit integer and the age of Ann is an integer, how old is Grandfather now? (PMWC 13th Individual)

Budi bekerja pada suatu perusahaan. Ia memutuskan untuk berhenti bekerja pada perusahaan itu apabila jumlah usia dan masa kerjanya sama dengan 75. Saat ini ia berusia 40 tahun dan telah bekerja selama 13 tahun. Usia Budi ketika ia berhenti bekerja pada perusahaan itu adalah ……. tahun (OSN SD Isian Singkat 2012)

The sum of the two digits of Emma’s age this year is 5. Seven years from now, her age will be 2 less than the reverse of the digits of her age this year. How old is Emma now ? (IMSO 2011 Short Answer)

Pak Budi berkata pada anaknya yang besok berulang tahun: “Lucu ya, hari ini kalau digit pada umur saya dibalik, kita dapatkan umur kamu. Tetapi besok umur kamu setengah umur saya.” Umur Pak Budi adalah . . . tahun. (OSN SD Isian Singkat 2015)

In 2017, Alex’s age is equal to the sum of all the digits of his year of birth and his mother’s age is also equal to the sum of all the digits of her year of birth. How old was Alex’s mother when she gave birth to Alex? (IMSO 2017 Essay Problems)

There are two brothers. The younger brother is between 30 and 40 years of age and the other is between 40 and 50 years of age. The product of their ages is a perfect cube. Find the sum of their ages. (PMWC 13th Individual)

On her 40th birthday, Mrs. Sharma makes gifts to her two sons whose ages are prime numbers. She gives to one son a number of dollars equal to the square of his age, and to the other son a number of dollars equal to his age. She gives 300 dollars in total. Find the sum of the ages of Mrs. Sharma’s two sons. (IMSO 2016 Short Answer)

Ami merayakan hari ulang tahunnya pada tanggal 9 Agustus 2008. Pada hari tersebut usia Ami merupakan jumlah dari angka-angka tahun ia dilahirkan. Ami lahir pada tahun ... . (OSN SD Isian Singkat 2008)

There are 2001 pupils standing in a line. From the beginning to the end, they count off numbers from 1 to 3 (1, 2, 3, 1, 2, 3, …). Then, from the end to the beginning, they count off from 1 to 4 (1, 2, 3, 4, 1, 2, 3, 4, …). How many pupils say 1 twice? (PMWC 5th Individual)

In the two arithmetic problems below, the four different shapes

represent exactly one of the numbers 1, 2, 4 or 6 but not necessary in that order. The symbol 0 is zero. What number does each shape represent so that both problems work? (PMWC 12th Individual)

The diagram below shows six distinct positive integers in a ring and the sum of any two neighboring numbers is a perfect square.

The below diagram is to be filled with six different positive integers such that it has the same property. If X≤20 , find all possible values of X. (IMSO 2012 Essay Problems)

Fill in all the numbers 0,1,2,3,4,5,6,7,8,9 on the ten squares below, so that the sum of numbers located on each arrowed line is 20. Two numbers are already filled in. The number on the square with a question mark (“?”) is .... (IMSO 2009 Short Answer)

The diagram below shows the multiplication of two three-digit numbers, yielding a six-digit product. Some of the digits are replaced by boxes. What is the value of this six-digit product? (IMSO 2013 Essay Problems)

Replace the asterisks with digits so that the multiplication below is correct: The product is ………. (IMSO 2009 Short Answer)

In the following division problem, the blanks represent missing digits. If A and B represent the digits of the quotient, find the value of A×B. (NMOS 2015)

Each of the letters E,I,N,S,T,V,W,X and Y represents a different one of the digits 0, 1, 2, 3, 4, 5, 6, 7, and 8 such that

Given that S=6 and E=8, find the 4-digit number “TIVY”. (NMOS 2015 Special Round)

In the following equation, each letter represents a distinct digit.

5×(ABCDEF) ̅ = 6×(EFABCD) ̅ Given that B=2 and D=0, find the 4-digit number “FACE”. (NMOS 2015 Special Round)

The numbers 5, 6, 7, 8, 9, 10 are to be filled in the squares so that the sum of the numbers in the row is equal to the sum of the numbers in the column. How many different possible values of A are there ? (IMSO 2016 Short Answer) A

Pada gambar di samping, tiga lingkaran yang terletak pada titik-titik sudut suatu segitiga disebut lingkaran-lingkaran serumpun. Setiap kali Fikry menambahkan bilangan 1 pada suatu lingkaran, ia harus menambahkan pula bilangan 1 pada dua lingkaran lain yang serumpun dengan lingkaran tersebut. Setelah beberapa kali penambahan bilangan 1, diperoleh susunan bilangan seperti tampak pada gambar tersebut. Jika pada awalnya semua lingkaran tersebut berisi bilangan 0, maka nilai a adalah …. (OSN SD Isian Singkat 2010)

Place all the prime numbers between 4 and 25 into the circles below such that each row and diagonal adds up to the same prime number. (PMWC 19th Team)

In the circles below place six different numbers from 1, 2, 3, 4, 5, 6 and 7 such that in each of the four small triangles of the same size two of the numbers add up to the third number. Which number will not be used ? (PMWC 19th Team)

Put the numbers from 2000 to 2010 into the boxes in the figure below. Each box consists of one number and each number must be used exactly once. The sum of the three numbers along each of the ten segments must be the same. (PMWC 13th Team)

Andika memiliki lima kartu yang masing-masing bertuliskan bilangan satu digit a,b,c,d and e. Andika mengambil dua kartu secara acak, kemudian mencatat selisih dari kedua bilangan yang tertulis di kartu tersebut. Beberapa hasil operasi bilangan yang muncul pada catatan Andika adalah 1, 2, 4, 6, 7. Nilai terbesar yang mungkin dari a+b+c+d+e adalah …….. (OSN SD Isian Singkat 2016)

In a multi-digit positive integer multiple of 7, every digit except the units digit is 6. What are the possible values of units digits? (IMSO 2013 Essay Problems) Amir akan mendesain bendera dengan 59 bintang merah pada dasar kuning. Ketentuan yang harus ia patuhi adalah: Banyaknya bintang pada baris bernomor ganjil (baris ke-1, ke-3, dan seterusnya) adalah sama. Banyaknya bintang pada baris bernomor genap adalah sama. Banyaknya bintang pada setiap baris bernomor ganjil adalah satu lebihnya atau satu kurangnya dari banyaknya bintang pada baris bernomor genap. Banyaknya baris adalah tujuh. Berapa banyak bintang pada baris keempat? (OSN SD Isian Singkat 2006)

Saya mempunyai empat buah bilangan asli yang berbeda. Hasil kali tiga bilangan pertama adalah 1200, sedangkan jumlah ketiga bilangan pertama adalah 10 kurangnya dari bilangan keempat. Dari semua kemungkinan susunan empat bilangan tersebut, bilangan keempat terbesar adalah... . (OSN SD Isian Singkat 2007)

Randi membuat buku kecil dari 10 lembar kertas A4 dengan cara melipat bagian tengah dan membundelnya.

Ia menomori halaman buku tersebut secara berurutan mulai dari 1 pada halaman paling depan sampai dengan 40 pada halaman paling belakang. Randi melepas bundel bukunya. Lembar kertas yang memuat nomor halaman 15, juga memuat tiga nomor halaman lainnya. Tiga nomor halaman tersebut adalah . . . . …. (OSN SD Isian Singkat 2009) A palindromic number is a whole number that is the same when written forwards or backwards (for example, 11511, 22222, 10001). Find the ratio, in proper fraction form, of the number of all five-digit palindromic numbers which are multiples of eleven to the number of all five-digit palindromic numbers. (PMWC 11th Team)

A palindrome is a number that can be read the same forwards and backwards. For example, 246642, 131 and 5005 are palindromic numbers. Find the smallest even palindrome that is larger than 56789 which is also divisible by 7. (IMSO 2011 Essay Problems)

The display of a digital clock is of the form MM : DD : HH : mm, that is, Month : Day : Hour : minute. The display ranges are

Month (MM) from 01 to 12 Day (DD) from 01 to 31 Hour (HH) from 00 to 23 Minute (mm) from 00 to 59 How many times in the year 2005 does the display show a palindrome? (A palindrome is a number which is read the same forward as backward. Examples: 12 : 31 : 13 : 21 and 01 : 02 : 20 : 10.) (IMSO 2005 Short Answer)

Palimage bilangan asli adalah bilangan yang mempunyai digit yang sama dengan bilangan yang diketahui tetapi urutannya terbalik. Misalnya, 123 adalah palimage dari 321 dan sebaliknya, begitu juga 1327 adalah palimage dari 7231. Misalkan X adalah hasil penjumlahan bilangan AB4C dengan palimage-nya. Kemudian X dijumlahkan kembali dengan palimage-nya, akan menghasilkan bilangan PQ4R. Jika A,B,C,P,Q dan R menunjukan angka-angka yang berbeda dan tidak sama dengan 4, maka tentukan bilangan AB4C dan PQ4R? (OSN SD Uraian 2017)

What is the smallest value of positive integer A so that 7560 × A is a square number? (OSN SD Uraian 2013)

Terdapat dua bilangan, bilangan prima dikali 456 ditambah bilangan kedua kali 654 hasilnya 2325. Apabila bilangan pertama dikali 654 dan bilangan kedua dikali 456 hasilnya 3225. Kuadrat jumlah bilangan tersebut adalah … … (OSN SD Isian Singkat 2017)

The sum of the digits of a two-digit number (ab) ̅ is 6. By reversing the digits, one obtained another two-digit number (ba) ̅. If (ab) ̅-(ba) ̅=18, find the original two-digit number. (IMSO 2013 Short Answer)

If (abcdef) ̅ represents a 6-digit number, where a, b, c, d, e and f denote different digits, what is the largest possible value of (abc) ̅ + (bcd) ̅ + (cde) ̅ + (def) ̅? (IMSO 2017 Short Answer)

A, B and C are positive integers. The sum of 160 and the square of A is equal to the sum of 5 and the square of B. The sum of 320 and the square of A is equal to the sum of 5 and the square of C. Find the positive integer A. (IMSO 2017 Short Answer)

A 4035-digit number is a multiple of 13. Its first 2017 digits are all 5s, and its last 2017 digits are all 6s. What is the middle digit? (IMSO 2017 Essay Problems) ⏟(555⋯55)┬(2017 digit)  ? ⏟(666⋯66)┬(2017 digit)

All positive integers from 1 to 10,000 are arranged according to the following rules: The first row contains all the integers whose sum of their digits is 1. The second row contains all the integers whose sum of their digits is 2. The third row contains all the integers whose sum of their digits is 3. Integers in each row are arranged in increasing order. Each of these rows are then rearranged in a line such that the first number in the second row comes after the last number of the first row; the first number in the third row comes after the last number of the second row;… and so on. Which term is the number 9799 in the final arrangement? (IMSO 2017 Essay Problems)

How many integers can be expressed as a sum of three distinct integers chosen from the set {4,7,10,13,…,46}? (PMWC 3rd Individual)

A six digit integer 1◻◻◻◻◻ has 1 as its first digit. If the first digit is transferred to the end of the number, it becomes ◻◻◻◻◻1. It is found that ◻◻◻◻◻1=3×1◻◻◻◻◻. Find the original number 1◻◻◻◻◻. (PMWC 5th Individual)

In the sequence of natural numbers 1, 2, 3, 4, 5, 6, …, if a number cannot be expressed as the sum of two composite numbers, it will be eliminated. For example, 1 should be eliminated, 12 can be written as the sum of 4 and 8, so it shouldn’t be eliminated. Putting the remaining natural numbers in ascending order, what is the 2001st number? (PMWC 5th Team)

Consider the following conditions on the positive integer (natural number) a: 3a + 5 > 40 49a ≥ 301 20a ≤ 999 101a + 53 ≥ 2332 15a – 7 ≥ 144 If only three of these conditions are true, what is the value of a? (PMWC 9th Individual)

A group of 100 people consists of men, women and children (at least one of each). Exactly 200 apples are distributed in such a way that each man gets 6 apples, each woman gets 4 apples and each child gets 1 apple. In how many possible ways can this be done? (PMWC 9th Individual)

Mr. Wong has a 7-digit phone number ABCDEFG. The sum of the number formed by the first 4 digits ABCD and the number formed by the last 3 digits EFG is 9063. The sum of the number formed by the first 3 digits ABC and the number formed by the last 4 digits DEFG is 2529. What is Mr. Wong’s phone number  ? (PMWC 9th Team)

Find the largest 12-digit number for which every two consecutive digits form a distinct 2-digit prime number. (PMWC 9th Team)

a,b and c are two-digit numbers. The unit digit of a is 7, the unit digit of b is 5 and the tens digit of c is 1. If a×b+c=2006, find the value of a+b+c. (PMWC 10th Individual)

A class of students bought and equally distributed a certain number of notebooks. If the notebooks are distributed to girls only, each girl will receive 15 notebooks. If the notebooks are distributed to boys only, each boy will receive 10 notebooks. If the notebooks are equally distributed to everyone in the class, how many notebooks will each student receive? (PMWC 10th Individual)

On a true/false test of 100 items, every item number that is a multiple of 4 is true, and all others are false. If a student marks every item that is a multiple of 3 false and all others true, how many of the 100 items will be correctly answered? (PMWC 10th Individual)

In a group of ten people, each person is asked to write the sum of the ages of all the other nine people. The ten sums are 82, 83, 84, 85, 87, 89, 90, 90, 91 and 92. Find the age of the youngest person. Assume that the ages are all whole numbers. (PMWC 10th Team)

There are 10 hats. Each hat is a different colour. Two hats are cotton ($30 each), five are leather ($50 each) and three are wool ($10 each). How many ways are there to buy 5 hats such that the total cost is more than $101 but less than $149  ? (PMWC 11th Team)

The digits of a 3-digit number, which are all different, are rearranged to form new numbers. The greatest such number and the smallest such number are still 3-digit numbers. The difference between the greatest number formed and the smallest number formed is the original 3-digit number. What is the original number? (PMWC 12th Team)

Find a number which satisfies the following conditions: The number is between 8500 and 8700. The sum of its digits is 21. The number is divisible by 4. The number contains different digits. (IMSO 2004 Short Answer)

The number N has the following properties: It consists of 4 digits, each digit is a number less than 7. It is a square of a certain number. If 3 is added to each digit, the resulting number is also a square of a number. Find N (Imso 2005 Short Answer)

Barbara writes numbers consisting of four digits: 3, 5, 7 and 9, according to the following rules: Digit 7 does not appear in the first nor the last positions. Digit 7 should be to the right of the digit 5. (For example, digit 5 in the number 7395 appears to the right of digits 7, 3 and 9). Find all such possible numbers. (IMSO 2005 Essay Problems)

How many positive whole numbers less than 2005 can be found, if the number is equal to the sum of two consecutive whole numbers and also equal to the sum of three consecutive whole numbers? (For example, 21=10+11=6+7+8.) (IMSO 2005 Essay Problems)

Euis has a list of all 4-digit natural numbers. Each number satisfies the following four conditions:

All digits are different. No digit is 0, 5, 7 or 9. The sum of the four digits is 20. It is divisible by 4. How many numbers are there in the list? (IMSO 2007 Short Answer)

The integer 8 has two properties: If the number 1 is added, we get the number 9, which is a square number, i.e., 9 = 3×3. Half of it is 4, which is also a square number, i.e., 4 = 2×2. The next natural number which has the same properties is … .…. (IMSO 2009 Short Answer) The number N consists of three different digits and is greater than 200. The digits are greater than 1. For any two digits, one digit is a multiple of the other or the difference is 3. For example, 258 is one of such numbers. There are at most … possible N. (IMSO 2009 Short Answer)

A two-digit odd number is a multiple of 9. The product of its digits is also a multiple of 9. What is this number? (IMSO 2012 Short Answer)

The first four digits of an eight-digit perfect square are 2012. Find its square root. (IMSO 2012 Short Answer)

A three-digit number is multiplied by a two-digit number whose tens’ digit is 9. The product is a four-digit number whose hundreds digit is 2. How many three-digit numbers satisfy this condition? (IMSO 2012 Essay Problems)

Two numbers are written using the digits 1, 2, 3, 4, 5, 6, 7 and 8 so that each digit is used exactly once. One of the two numbers is a perfect square and the other number is a perfect cube of the same positive integer. Find these two numbers. (PMWC 13th Team)

A “PLK” number is a counting number which has the following features. When 1 is added to it, the sum is a perfect square. When 1 is added to its half, the sum is another perfect square. For example, 48 is a “PLK” number since 48 + 1 = 49 and 48/2+1= 25, and both 49 and 25 are perfect squares. Find the next “PLK” number 2 which is greater than 48. (PMWC 13th Team)

A “PoLeungKuk” number is 9-digit number with the following properties: It is the product of the square of 4 different prime numbers; each prime number is less than 50. The number formed by the first 3 digits and the number formed by the last 3 digits is the same. The number formed by the middle 3 digits is twice the number formed by the first 3 digits. The “PoLeungKuk” number is greater than 300,000,000. What is this “PoLeungKuk” number? (PMWC 15th Individual)

If 4n+1 and 6n+1 are both perfect squares, what is the minimum value of the positive integer n? (PMWC 15th Team)

The ‘4’ button on my calculator is defective, so I cannot enter numbers which contain the digit 4. Moreover, my calculator does not display the digit 4 if 4 is part of an answer. Thus I cannot enter the calculation 2×14 and do not attempt to do so. Also, the result of multiplying 3 by 18 is displayed as 5 instead of 54 and the result of multiplying 2 by 71 is displayed as 12 instead of 142. If I multiply a positive one-digit number by a positive two-digit number on my calculator and it displays 26, list all possibilities which I could have multiplied? (IMSO 2013 Essay Problems)

There are six three-digit numbers (abc) ̅,(acb) ̅,(bac,) ̅ (bca) ̅,(cab) ̅,(cba) ̅. One of these numbers is removed and the sum of the remaining five numbers is 1990. What is the numerical value of the number that was removed? (PMWC 15th Team)

There is a secret code where every integer has a unique icon and every icon represents only one integer. The eight icons below represent the first eight positive integers (1 to 8) according to a specific rule.

Draw circles into the figure below to create the icon for 60. (PMWC 15th Team)

My twelve-hour clock has four digits, two for the hours and two for the minutes. The minutes are shown with leading 0s, but the hours are not. Upside down, the digits 0, 1, 2, 5, 6, 8 and 9 read 0, 1, 2, 5, 9, 8 and 6 respectively. One day, I made a phone call on the appointed time, and learned that it was too early. Then I realized that I had read the clock upside down. If you know by how much I was early, you will know what time I made the call. What time should I have made the call? (IMSO 2013 Essay Problems)

INTERNUTS company offers internet service with an initial payment of 300000 rupiahs and a monthly fee of 72000 rupiahs. Another company, VIDIOTS, offers internet service with no initial payment but a monthly fee of 90000 rupiahs. Johnny prefers INTERNUTS company. What is the minimum number of months he should subscribe in order to pay less than the subscription with VIDIOTS company? (IMSO 2014 Short Answer)