Olympiad problems

1983 IMO Longlists, 26

Tags: number theory

Let $a, b, c$ be positive integers satisfying $\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1$. Show that $2abc-ab-bc-ca$ cannot be represented as $bcx+cay +abz$ with nonnegative integers $x, y, z.$

2002 China Western Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities , number theory

Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$ $ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$

2011 Denmark MO - Mohr Contest, 3

Tags: diophantine , number theory

Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.

1962 Poland - Second Round, 6

Tags: number theory , digit

Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.

1991 Bundeswettbewerb Mathematik, 2

Tags: number theory , divide

Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$. Prove that for every positive integer $n$ we have: a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$ b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$

2007 Hanoi Open Mathematics Competitions, 7

Tags: number theory , sequence

Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$.

1999 Israel Grosman Mathematical Olympiad, 1

Tags: divisible , number theory

For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$. Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.

1998 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , power of 2

Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$, not necessarily distinct, such that $a + b$ is a power of $2$.

2024 JHMT HS, 13

Tags: number theory , quadratic

Compute the largest nonnegative integer $T \leq 30$ that is the remainder when $T^2 + 4$ is divided by $31$.

2023 Puerto Rico Team Selection Test, 4

Tags: number theory

Find all positive integers $n$ such that: $$n = a^2 + b^2 + c^2 + d^2,$$ where $a < b < c < d$ are the smallest divisors of $n$.

2018 China Team Selection Test, 2

Tags: number theory , arithmetic series , number of divisors

A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2001 Taiwan National Olympiad, 1

Tags: algebra , polynomial , number theory

Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.

2023 New Zealand MO, 7

Tags: inequalities , combinatorics , number theory

Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

2018 Taiwan APMO Preliminary, 7

Tags: combinatorics , number theory

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

2019 Regional Olympiad of Mexico Center Zone, 5

Tags: number theory

A serie of positive integers $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is $auto-delimited$ if for every index $i$ that holds $1\leq i\leq n$, there exist at least $a_{i}$ terms of the serie such that they are all less or equal to $i$. Find the maximum value of the sum $a_{1}+a_{2}+\cdot \cdot \cdot+a_{n}$, where $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is an $auto-delimited$ serie.

2018-IMOC, N6

Tags: algebra , polynomial , number theory

If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that $$a_1f(1)+\ldots+a_tf(t)=k.$$

2013 IMAC Arhimede, 2

Tags: number theory , perfect cube , divisible , exponential

For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

2001 Estonia National Olympiad, 2

Tags: sum , divisible , number theory

Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.

2016 Macedonia JBMO TST, 1

Tags: number theory

Solve the following equation in the set of integers $x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1$.

2011 Puerto Rico Team Selection Test, 1

Tags: number theory

The product of 22 integers is 1. Show that their sum can not be 0.

2021 Brazil National Olympiad, 3

Tags: perfect square , power , number theory , floor function , irrational number

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

2010 China National Olympiad, 2

Tags: number theory

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

2012 Dutch BxMO/EGMO TST, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

2012 USAMTS Problems, 3

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

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

2023 Harvard-MIT Mathematics Tournament, 8

Tags: floor function , number theory

Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]