Singapore Math Olympiad Past Years Question - Number Theory (Junior)
Join Trial or Access Free ResourcesJoin Trial or Access Free ResourcesThe 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
Let (n) be a positive integer such that (n+11) is a factor of (n^2+121). Find the largest possible value of (n).
Find the largest integer less than or equal to \((3+\sqrt{5})^3\)
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).
If (x) and (y) are real numbers such that (x+y=12) and (x y=10), find the value of (x^4+y^4).
Let (x) be a real number. What is the minimum value of the following expression?
\(\frac{20 x^2+10 x+3}{2 x^2+x+1}\)
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) \(3^6\)
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?
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 ?
If (x) is a real number, how many solutions are there to the equation
\((3 x+2)^{x+5}=1 \text { ? }\)
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).
How many integers (n) are there in \({1,2, \ldots, 2022}\) such that at least one of the digits of \(n\) is ' 2 '?
How many integers (n) are there in \({1,2, \ldots, 2022}\) such that \(\lfloor\sqrt[3]{n}\rfloor\) is a factor of \(n\) ?
If (x) and (y) can take any real values, what is the smallest possible value of the expression
\(2 x^2+4 x y+5 y^2+4 x+10 y+13 ?\)
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) ?
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.
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) \(|a-b|>0\)
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?
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) \(z>x>y\)
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
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 \(x\) ?
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.
ow many distinct triples of positive integers \((a, b, c)\) satisfy \(1 \leqslant a \leqslant b \leqslant c\) and
\(\frac{1}{a_3}+\frac{1}{b_3}+\frac{1}{c_3}=1 \) ?
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 \(z\) ?
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 \(\left\lfloor 10^4 x^2\right\rfloor\) ?
Suppose a positive integer (x) satisfies the following equation
\(\sqrt[5]{x+76638}-\sqrt[5]{x-76637}=5 \).
What is the value of \(x\) ?
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) \(z>x>y\)
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 \(\triangle\), which of the following is true?
(A) \(\square<0<\triangle\)
(B) \(\square<\Delta<0\)
(C) \(\triangle<\square<0\)
(D) \(\triangle<0<\square\)
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
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) ?
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) ?
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) ?
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 ?
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) \(9^{10}\)
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
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 \(m+n\) ?
What is the largest possible two-digit positive integer that is 18 more than the product of its two digits?
A five-digit positive integer \(x\) has the following properties:
(i) \(x\) has distinct digits which are from \({1,2,3,4,5}\);
(ii) \(x>23456\).
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 ?