Olympiad problems

2013 Estonia Team Selection Test, 1

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

2020-21 IOQM India, 6

Tags: number theory , perfect square

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?

2000 Slovenia National Olympiad, Problem 1

Tags: number theory , bases

Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)

2006 Korea Junior Math Olympiad, 1

Tags: permutation , algebra , number theory , combinatorics , multiple , product

$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$. Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)

2007 Bulgaria Team Selection Test, 2

Tags: inequalities , number theory

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

1987 Brazil National Olympiad, 1

Tags: polynomial , coprime , prime number , number theory , algebra

$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.

2010 Moldova Team Selection Test, 1

Tags: modular arithmetic , number theory

Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.

2022 Purple Comet Problems, 17

Tags: number theory

Find the least positive integer with the property that if its digits are reversed and then $450$ is added to this reversal, the sum is the original number. For example, $621$ is not the answer because it is not true that $621 = 126 + 450$.

Kvant 2021, M2666

Tags: number theory

Let $x{}$ and $y{}$ be natural numbers greater than 1. It turns out that $x^2+y^2-1$ is divisible by $x+y-1$. Prove that $x+y-1$ is composite. [i]From the folklore[/i]

1996 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory , greatest common divisor

Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies: (i) $f(1995) = 1996$ (ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$. Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .

2011 German National Olympiad, 5

Tags: number theory

Prove or disprove: $\exists n\in N$ , s.t. $324 + 455^n$ is prime.

2023 Poland - Second Round, 1

Tags: number theory , divisibility

Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$.

1985 IMO Longlists, 64

Tags: number theory

Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?

2010 Contests, 4

Tags: inequalities , number theory

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

2012 Paraguay Mathematical Olympiad, 4

Tags: number theory

Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$. ($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is).

2009 Kosovo National Mathematical Olympiad, 3

Tags: number theory

Prove that $\sqrt 2$ is irrational.

2023 Switzerland Team Selection Test, 2

Tags: number theory

Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.

2011 Singapore Junior Math Olympiad, 4

Tags: number theory , sequence , odd

Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.

2011 Hanoi Open Mathematics Competitions, 6

Tags: number theory , diophantine equation , diophantine

Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$

2020 IMO, 5

Tags: number theory , arithmetic mean , geometric mean

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

2023 India IMO Training Camp, 3

Tags: number theory , prime factorization , function , prime number , prime

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2021 OMpD, 3

Tags: diophantine equation , number theory

Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

1967 IMO Longlists, 1

Tags: algebra , number theory , decimal representation , perfect cube

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

VI Soros Olympiad 1999 - 2000 (Russia), 10.5

Tags: combinatorics , number theory

For what values of $k\ge2$ can the set of natural numbers be colored in $k$ colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?

2024 Iran MO (3rd Round), 2

Tags: number theory

For all positive integers $n$ Prove that one can find pairwise coprime integers $a,b,c>n$ such that the set of prime divisors of the numbers $a+b+c$ and $ab+bc+ac$ coincides. Proposed by [i]Mohsen Jamali[/i] and [i]Hesam Rajabzadeh[/i]