Singapore Math Olympiad Past Years Question - Number Theory (Junior)

Problem 03: SMO Year-2023

The median and mean of five distinct numbers, (4,7,10,11, N), are equal. Find the sum of all possible values of (N).
(A) 18
(B) 21
(C) 26
(D) 29
(E) 35

Problem 07:SMO Year-2023

Let (n) be a positive integer such that (n+11) is a factor of (n^2+121). Find the largest possible value of (n).

Problem 08:SMO Year-2023

Find the largest integer less than or equal to \((3+\sqrt{5})^3\)

Problem 09:SMO Year-2023

The product of the two-digit number \(\overline{x 4}\) and the three-digit number \(\overline{3 y z}\) is 7656 . Find the value of (x+y+z).

Problem 10:SMO Year-2023

If (x) and (y) are real numbers such that (x+y=12) and (x y=10), find the value of (x^4+y^4).

Problem 14:SMO Year-2023

Let (x) be a real number. What is the minimum value of the following expression?
\(\frac{20 x^2+10 x+3}{2 x^2+x+1}\)

Problem 01: SMO Year-2022

Which of the five numbers, \(30^2, 10^3, 5^4, 4^5\) or \(3^6\), is the largest?

(A) \(30^2\)
(B) \(10^3\)
(C) \(5^4\)
(D) \(4^5\)
(E) \(3^6\)

Problem 03: SMO Year-2022

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use \(\square<\Delta\) to represent \(\square\) is lighter than \(\Delta\), which of the following is true?

Problem 07:SMO Year-2022

The digits (1,2,3,4,5) and 6 are arranged to form two positive integers with each digit appearing exactly once. How many ways can this be done if the sum of the two integers is 570 ?

Problem 12:SMO Year-2022

If (x) is a real number, how many solutions are there to the equation

\((3 x+2)^{x+5}=1 \text { ? }\)

Problem 10:SMO Year-2023

The lengths of the sides of a triangle are \( \log_{10} 12\), \(\log_{10} 75\) and \(\log _{10}n\) where (n) is a positive integer. Find the number of possible values for (n).

Problem 17:SMO Year-2022

How many integers (n) are there in \({1,2, \ldots, 2022}\) such that at least one of the digits of \(n\) is ' 2 '?

Problem 23:SMO Year-2022

How many integers (n) are there in \({1,2, \ldots, 2022}\) such that \(\lfloor\sqrt[3]{n}\rfloor\) is a factor of \(n\) ?

Problem 24:SMO Year-2022

If (x) and (y) can take any real values, what is the smallest possible value of the expression
\(2 x^2+4 x y+5 y^2+4 x+10 y+13 ?\)

Problem 08:SMO Year-2022

Consider the following product of two mixed fractions
\(m\frac{6}{7} \times n \frac{1}{3}=23\),
where (m) and (n) are positive integers. What is the value of (m+n) ?

Problem 09:SMO Year-2022

What are the last four digits of the sum
\(222+2022+20022+\cdots+2 \underbrace{0000000000}_{\text {ten } 0 s} 22\) ?

Give your answer as a 4-digit number.

Problem 01:SMO Year-2021

Let \(a\) and \(b\) be real numbers satisfying \(a<0<b\). Which of the following is not true?

(A)\(a^2 b<0\)
(B) \(a b^2<0\) (C) \(\frac{a}{b}>0\)
(D) \(b-a>0\)
(E) \(|a-b|>0\)

Problem 02:SMO Year-2021

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use \(\square<\Delta\) to represent is lighter than \(\triangle\), which of the following is true?

Problem 03:SMO Year-2021

Let \(x=2^{20} \cdot 3^5, y=2^5 \cdot 5^{10}\) and \(z=7^{10}\). Which of the following is true?
(A) \(x>y>z\)
(B) \(x>z>y\)
(C) \(y>z>x\)
(D) \(y>x>z\)
(E) \(z>x>y\)

Problem 05:SMO Year-2021

Which of the following is closest to the value of
\(\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{2021}+\sqrt{2020}}\) ?
(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

Problem 06:SMO Year-2021

Let \(x\) be a positive integer. Suppose that the lowest common multiple of \(x\) and \(14\) is \(42\) and the lowest common multiple of \(x\) and \(33\) is \(66\) . What is the value of \(x\) ?

Problem 07:SMO Year-2021

What are the last four digits of the sum
\(1+22+333+4444+\cdots+\underbrace{999999999}_{\text {nine } 9 \mathrm{~s}} \text { ? A205 }\). Give your answer as a 4-digit number.

Problem 08:SMO Year-2021

ow many distinct triples of positive integers \((a, b, c)\) satisfy \(1 \leqslant a \leqslant b \leqslant c\) and
\(\frac{1}{a_3}+\frac{1}{b_3}+\frac{1}{c_3}=1 \) ?

Problem 14:SMO Year-2021

If \(x\) is a 3-digit number, we define \(M(x)\) and \(m(x)\) respectively as the largest and smallest positive number that can be formed by rearranging the three digits of (x). For example, if \(x=123\), then \(M(123)=321\) and \(m(123)=123\). If \(y=898\), then \(M(898)=988\) and \(m(898)=889\).
Given that \(z\) is a 3-digit number that satisfies \(z=M(z)-m(z)\), what is the value of \(z\) ?

Problem 19:SMO Year-2021

Let (x) be the positive real number that satisfies \(\sqrt{x^2-4 x+5}+\sqrt{x^2+4 x+5}=3 x \).

What is the value of \(\left\lfloor 10^4 x^2\right\rfloor\) ?

Problem 25:SMO Year-2021

Suppose a positive integer (x) satisfies the following equation
\(\sqrt[5]{x+76638}-\sqrt[5]{x-76637}=5 \).

What is the value of \(x\) ?

Problem 01:SMO Year-2020

Let \(x=2^{300}, y=3^{200}\) and \(z=6^{100}\). Which of the following is true?
(A) \(x>y>z\)
(B) \(x>z>y\)
(C) \(y>z>x\)
(D) \(y>x>z\)
(E) \(z>x>y\)

Problem 03:SMO Year-2020

(A) \(\square<0<\triangle\)
(B) \(\square<\Delta<0\)
(C) \(\triangle<\square<0\)
(D) \(\triangle<0<\square\)

Problem 04:SMO Year-2020

The integer 6 has exactly four positive factors, namely (1,2,3) and 6 . Likewise, the integer 8 has exactly four positive factors, (1,2,4) and 8 . How many integers from 9 to 50 (inclusive) have exactly four positive factors?
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

Problem 06:SMO Year-2020

Let (n) be a positive integer. Suppose the lowest common multiple of 4,5 and (n) is 2020 . What is the sum of the smallest possible value of (n) and the largest possible value of (n) ?

Problem 07:SMO Year-2020

When the five-digit integer \(\overline{2 x 6 y x}\) is divided by the four-digit integer \(\overline{5 y 27}\), the quotient is 4 and remainder is \(\overline{x 106}\), which is a four-digit integer. What is the value of the digit (x) ?

Problem 09:SMO Year-2020

A quadruple ((a, b, c, d)) of positive integers is skewed if the median and mode of (a, b, c, d) are equal, but strictly greater than the mean of (a, b, c, d). How many skewed ((a, b, c, d)) of positive integers are there that satisfy \(a \leq b \leq c \leq d\) and (a+b+c+d=40) ?

Problem 19:SMO Year-2020

Let \(X=1234 \cdots 78798081\) be the integer that consists of all the integers from 1 to 81 written from left to right. What is the remainder of (X) when divided by 2020 ?

Problem 01:SMO Year-2019

Which of the five numbers \(2^{30}, 3^{19}, 4^{14}, 6^{12}, 9^{10}\)

has the largest value?

(A) \(2^{30}\)
(B) \(3^{19}\)
(C) \(4^{14}\)
(D) \(6^{12}\)
(E) \(9^{10}\)

Problem 04:SMO Year-2019

Let \(x, y\) and \(z\) be positive integers satisfying
\(x^2+y^2+z^2=2(x y+1) \quad \text { and } \quad x+y+z=2022\) .

If \(x_1\) and \(x_2\) are two distinct solutions for \(x\), what is the value of \(x_1+x_2\) ?

(A) 2019
(B) 2020
(C) 2021
(D) 2022
(E) 2023

Problem 08:SMO Year-2019

Suppose that \(m\) and \(n\) are positive integers where \(\frac{100 m}{n}\) is a perfect cube greater than 1 . What is the minimum value of \(m+n\) ?

Problem 09:SMO Year-2019

What is the largest possible two-digit positive integer that is 18 more than the product of its two digits?

Problem 18:SMO Year-2019

A five-digit positive integer \(x\) has the following properties:
(i) \(x\) has distinct digits which are from \({1,2,3,4,5}\);
(ii) \(x>23456\).

Problem 21:SMO Year-2019

A positive integer is said to be "twelvish" if the sum of digits in its decimal representation is equal to 12. For example, the first four twelvish integers are 39,48,57 and 66 . What is the total number of twelvish integers between 1 and 999 ?

Singapore Math Olympiad Past Years Question - Algebra (Junior)

Problem 06:SMO Year 2023

The product of the ages of three adults is 26390 . Find the sum of their ages. (A person is an adult if he or she is at least 21 years old.)

Problem 09:SMO Year 2023

The product of the two-digit number \(\overline{x 4}\) and the three-digit number \(\overline{3 y z}\) is 7656 . Find the value of (x+y+z).

Problem 10:SMO Year 2023

If (x) and (y) are real numbers such that (x+y=12) and (x y=10), find the value of (x^4+y^4).

Problem 12:SMO Year 2023

Find the value of the integer (n) such that the following equation holds:
\([
\frac{\sqrt{5}+n \sqrt{3}-2 \sqrt{2}}{(\sqrt{5}+\sqrt{3})(\sqrt{3}-\sqrt{2})}=\sqrt{5}+\sqrt{2} .
]\)

Problem 14:SMO Year 2023

Let (x) be a real number. What is the minimum value of the following expression?
\([
\frac{20 x^2+10 x+3}{2 x^2+x+1}
]\)

Problem 19:SMO Year 2023

If \(\sqrt{19-8 \sqrt{3}}\) is a root of the equation (x^2-a x+b=0) where (a) and (b) are rational numbers, find the value of (a+b).

Problem 24:SMO Year 2023

Four positive integers (x, y, z) and (w) satisfy the following equations:
\([
\begin{aligned}
& x y+x+y=104 \
& y z+y+z=146 \
& z w+z+w=524
\end{aligned}
]\)

Problem 08:SMO Year 2021

Consider the following product of two mixed fractions
\(m\frac{6}{7} \times n \frac{1}{3}=23\),
where (m) and (n) are positive integers. What is the value of (m+n) ?

Problem 10:SMO Year 2021

If (a) and (b) are distinct solutions to the equation
\(x^2+10 x+20=0\),
what is the value of \(a^4+b^4\) ?

Problem 22:SMO Year 2021

If we have
\(\frac{\sqrt{15}+\sqrt{35}+\sqrt{21}+5}{\sqrt{3}+2 \sqrt{5}+\sqrt{7}}=\frac{a \sqrt{7}+b \sqrt{5}+c \sqrt{3}}{2}\)
for some integers (a, b, c). What is the value of (a+b+c) ?

Problem 15:SMO Year 2021

How many integers (k) are there such that the quadratic equation \(k x^2+20 x+20-k=0\) has only integer solutions?

Problem 19:SMO Year 2021

Let (x) be the positive real number that satisfies \(\sqrt{x^2-4 x+5}+\sqrt{x^2+4 x+5}=3 x \).

What is the value of \(\left\lfloor 10^4 x^2\right\rfloor\) ?

Problem 20:SMO Year 2021

What is the number of positive integers (c) such that the equation \(x^2-2021 x+100 c=0\) has real roots?

Problem 20:SMO Year 2020

Let \(A=\frac{1}{7} \times 3.14 \mathrm{i} \dot{5}), where (3.14 \mathrm{i} \dot{5}\) is the rational number with recurring digits 15 . In other words,
\([
3.14 \dot{1} \dot{5}=3.14+0.0015+0.000015+0.00000015+\cdots .
]\) Suppose that \(A=\frac{m}{n}\), where (m) and (n) are positive integers with no common factors larger than 1 . What is the value of (m+n) ?

Problem 22:SMO Year 2020

If we have
\([
(1-3 x)+(1-3 x)^2+\cdots+(1-3 x)^{100}=a_0+a_1 x+a_2 x^2+\cdots+a_{100} x^{100},
]\)
for some integers (a_0, a_1, \ldots, a_{100}), what is the value of
\([
\left|\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{100}}{3^{100}}\right| ?
]\)

Problem 25:SMO Year 2020

What is the value of
\([
\begin{aligned}
\left(\frac{1}{2}+\frac{1}{3}\right. & \left.+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{37}\right)+\left(\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+\cdots+\frac{2}{37}\right) \
& +\left(\frac{3}{4}+\frac{3}{5}+\frac{3}{6}+\cdots+\frac{3}{37}\right)+\cdots+\left(\frac{35}{36}+\frac{35}{37}\right)+\frac{36}{37} ?
\end{aligned}
]\)

Problem 04:SMO Year 2019

Let \(x, y\) and \(z\) be positive integers satisfying
\(x^2+y^2+z^2=2(x y+1) \quad \text { and } \quad x+y+z=2022\) .

If \(x_1\) and \(x_2\) are two distinct solutions for \(x\), what is the value of \(x_1+x_2\) ?

Problem 8:SMO Year 2019

Suppose that \(m\) and \(n\) are positive integers where \(\frac{100 m}{n}\) is a perfect cube greater than 1 . What is the minimum value of \(m+n\) ?

Problem 16:SMO Year 2019

If the equation \(\frac{x-1}{x-5}=\frac{m}{10-2 x}\) has no solutions in \(x\), what is the value of \(|m| ?\)

Problem 20:SMO Year 2019

If (x) is a nonnegative real number, find the minimum value of
\[
\sqrt{x^2+4}+\sqrt{x^2-24 x+153} .
\]

Singapore Math Olympiad Past ears Questions- Combinatorics (Junior)

Problem 04 - SMO Year 2022

A shop sells two types of buns, with either cream or jam filling, which are indistinguishable until someone bites into the buns. Four mathematicians visited the shop and ordered (not necessarily in that sequence): three cream buns, two cream buns and one jam bun, one cream and two jam buns, and three jam buns. Each knew precisely what the others had ordered. Unfortunately, the shop owner mixed up the orders and gave each mathematician the wrong order!
The mathematicians started eating, all still unaware of the mixup, until the shop owner ran over to inform them of the mistake. Mathematician A said: "I ate two buns and both had cream filling. So, if my order was wrong, I now know what type my third bun is." Mathematician B then said: "I only ate one bun and it had cream filling. Based on what A said and since I remember A's order, I now know what type my other two buns are." Finally, Mathematician C said: "I have not started eating but I must have received three jam buns." Which of the following statements about Mathematician D is correct?

(A) D ordered two cream and one jam but received three jam buns.
(B) D ordered one cream and two jam but received two cream and one jam buns.
(C) D ordered three cream but received one cream and two jam buns.
(D) D ordered three jam but received three cream buns.
(E) None of the above

Problem 07- SMO Year 2022

The digits (1,2,3,4,5) and 6 are arranged to form two positive integers with each digit appearing exactly once. How many ways can this be done if the sum of the two integers is 570 ?

Problem 13 - SMO Year 2022

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If \(\mathrm{W}\) represents 5 , what number does TROOP represent?

Problem 16 - SMO Year 2022

Eggs in a certain supermarket are sold only in trays containing exactly 10,12 or 30 eggs per tray. It is thus impossible to buy exactly 14 eggs or any odd number of eggs. However, it is possible to buy exactly 78 eggs using four trays of 12 and one tray of 30 . What is the largest even number of eggs that is impossible to be bought from this supermarket?

Problem 11 - SMO Year 2021:

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If \(S\) represents \(6 \)and \(E\) represents \(8\),-what number does SIX represent?

Problem 21 - SMO Year 2021

In chess, two queens are said to be attacking each other if they are positioned in the same row, column or diagonal on a chessboard. How many ways are there to place two identical queens in a (4 \times 4) chessboard such that they do not attack each other?
\(\frac{1}{2} \times \frac{1}{4} \times 401 \times 403 x \times 801=\)

Problem 23 - SMO Year 2021:

A \(3 \times 3\) grid is filled with the integers 1 to 9 . An arrangement is nicely ordered if the integers in each horizontal row is increasing from left to right and the integers in each vertical column is increasing from top to bottom. Two examples of nicely ordered arrangements are given in the diagram below. What is the total number of distinct nicely ordered arrangements?

Problem 24 - SMO Year 2021:

A class has exactly 50 students and it is known that 40 students scored (A) in English, 45 scored (A) in Mathematics and 42 scored (A) in Science. What is the minimum number students who scored (A) in all three subjects?

Problem 02 - SMO Year 2020:

An expensive painting was stolen and the police rounded up five suspects Alfred, Boris, Chucky, Dan and Eddie. These were the statements that were recorded.
Alfred: "Either Boris or Dan stole the painting."
Boris: "I think Dan or Eddie is the guilty party."
Chucky: "It must be Dan."
Dan: "Boris or Eddie did it!"
Eddie: "I am absolutely sure the thief is Alfred."
The police knew that only one of the five suspects stole the painting and that all five were lying. Who stole the painting?
(A) Alfred
(B) Boris
(C) Chucky
(D) Dan
(E) Eddie

Problem 12 - SMO Year 2020:

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If the letter K represents 6 , what number does SHAKE represent?

Problem 21- SMO Year 2020:

Ali and Barry went running on a standard 400 metre track. They started simultaneously at the same location on the track but ran in opposite directions. Coincidentally, after 24 minutes, they ended at the same location where they started. Ali completed 12 rounds of the track in those 24 minutes while Barry completed 10 rounds. How many times did Ali and Barry pass each other during the run? (Exclude from your answer the times that they met at the start of the of run and when they completed the run after 24 minutes.)

Problem 2 - SMO Year 2019

In a strange island. there are only two types of inhabitants: truth-tellers who only tell the truth and liars who only tell lies. One day, you meet two such inhabitants \(A\) and \(B\). \(A\) said "Exactly one of us is a truth teller." \(B\) kept silent. Which of the following must be true?

(A) Both \(A\) and \(B\) are truth-tellers
(B) Both \(A\) and \(B\) are liars
(C) \(A\) is a truth-teller and \(B\) is a liar
(D) \(A\) is a liar and \(B\) is a truth-teller
(E) Not enough information to decide

Problem 19 - SMO Year 2019

In the figure below, each distinct letter represents a unique digit such that the arithmetic holds. What digit does the letter \(\mathrm{L}\) represent?

Problem 22 - SMo Year 2019

Two secondary one and \(m\) secondary two students took part in a round-robin chess tournament. In other words, each student played with every other student exactly once. For each match, the winner receives 3 points and the loser 0 points. If a match ends in a draw, both contestants receive 1 point each. If the total number of points received by all students was 130 , and the number of matches that ended in a draw was less than half of the total number of matches played, what is the value of \(m\) ?