Olympiad problems

2007 USA Team Selection Test, 3

Tags: algebra , polynomial , algorithm , greatest common divisor , trigonometry , induction , number theory

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

2018-2019 Fall SDPC, 2

Tags: number theory

Find all pairs of positive integers $(m,n)$ such that $2^m-n^2$ is the square of an integer.

2005 Vietnam National Olympiad, 2

Tags: number theory

Find all triples of natural $ (x,y,n)$ satisfying the condition: \[ \frac {x! \plus{} y!}{n!} \equal{} 3^n \] Define $ 0! \equal{} 1$

2015 Tuymaada Olympiad, 4

Tags: number theory

Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$, $\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point. [i]A.Golovanov [/i]

2001 Argentina National Olympiad, 5

Tags: combinatorics , number theory

All sets of $49$ distinct positive integers less than or equal to $100$ are considered. Leandro assigned each of these sets a positive integer less than or equal to $100$. Prove that there is a set $L$ of $50$ distinct positive integers less than or equal to $100$, such that for each number $x$ of $L$ the number that Leandro assigned to the set of $49$ numbers $L-\{ x\}$ is different from $x$. Clarification: $L-\{x\}$ denotes the set that results from removing the number $x$ from $L$.

2020 LIMIT Category 1, 14

Tags: number theory , limit , algebra

Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.

1979 Chisinau City MO, 169

Tags: number theory , integer , algebra

Prove that the number $x^8+\frac{1}{x^8}$ is an integer if $x+\frac{1}{x }$ is an integer.

2018 IMO Shortlist, N1

Tags: number theory , arithmetic function , number of divisors , divisor

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2021 Princeton University Math Competition, A2 / B4

Tags: number theory

A [i]substring [/i] of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.

1986 Federal Competition For Advanced Students, P2, 4

Tags: number theory

Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.

1936 Moscow Mathematical Olympiad, 024

Tags: representation , number theory

Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs. (P. Dirac)

2016 Dutch IMO TST, 3

Tags: number theory , least common multiple

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

2006 India IMO Training Camp, 2

Tags: analytic geometry , geometry , number theory , 3d geometry , pigeonhole principle , modular arithmetic

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

1999 Mongolian Mathematical Olympiad, Problem 4

Tags: number theory , euler

Maybe well known: $p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: number theory , equation

Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers

1999 APMO, 4

Tags: number theory , inequalities

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.

2012 Tournament of Towns, 3

Tags: polynomial , divide , divisible , number theory

Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.

2012 Czech-Polish-Slovak Match, 1

Tags: relatively prime , number theory

Given a positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$ and $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are relatively prime to $n$. Find all $n$ for which one of the three numbers $n,\tau(n), \varphi(n)$ is the arithmetic mean of the other two.

2006 China Team Selection Test, 3

Tags: number theory

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

2021 BMT, 3

Tags: number theory

How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.)

2022 Saudi Arabia JBMO TST, 1

Tags: number theory , perfect square

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

2000 JBMO ShortLists, 4

Tags: number theory

Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.

2025 Poland - First Round, 5

Tags: number theory , prime number

Positive integers $a, b, n$ are given. Assume that $a$ and $n$ are even, $b$ is odd and the number $ab(a+b)^{n-1}$ is divisible by $a^n+b^n$. Prove that there exist a prime number $p$, such that $p^{n+1}$ divides $a^n+b^n$.

2011 Argentina National Olympiad, 4

Tags: number theory , perfect square

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

1988 Brazil National Olympiad, 1

Tags: prime number , number theory

Find all primes which are sum of two primes and difference of two primes.