Olympiad problems

1945 Moscow Mathematical Olympiad, 099

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

2013 Nordic, 3

Tags: number theory , sequence , rational

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

2001 Tournament Of Towns, 7

Tags: number theory

It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4?

2001 China Team Selection Test, 3

Tags: number theory

Consider the problem of expressing $42$ as $42 = x^3 + y^3 + z^3 - w^2$, where $x, y, z, w$ are integers. Determine the number of ways to represent $42$ in this form and prove your conclusion.

2016 IFYM, Sozopol, 4

Tags: number theory , algebra , sequence

$a$ and $b$ are fixed real numbers. With $x_n$ we denote the sum of the digits of $an+b$ in the decimal number system. Prove that the sequence $x_n$ contains an infinite constant subsequence.

2022 CMIMC, 2.6 1.3

Tags: number theory , algebra

Find the smallest positive integer $N$ such that each of the $101$ intervals $$[N^2, (N+1)^2), [(N+1)^2, (N+2)^2), \cdots, [(N+100)^2, (N+101)^2)$$ contains at least one multiple of $1001.$ [i]Proposed by Kyle Lee[/i]

2021 Princeton University Math Competition, A7

Tags: number theory

We say that a polynomial $p$ is respectful if $\forall x, y \in Z$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in Z$, $p(x) \in Z$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $ 1$, exclusive. Determine $\sum deg(f)\cdot f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.

2009 Bundeswettbewerb Mathematik, 4

Tags: number theory , palindrome , multiple

A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.

2016 Turkey Team Selection Test, 5

Tags: number theory , function , functional equation , divisibility

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .

2012 Regional Olympiad of Mexico Center Zone, 2

Tags: number theory , modular arithmetic

Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

1983 All Soviet Union Mathematical Olympiad, 360

Tags: divisible , exponential , number theory

Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

2008 Cuba MO, 3

Tags: number theory , integer

Prove that there are infinitely many ordered pairs of positive integers $(m, n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer.

2020 Stars of Mathematics, 2

Tags: number theory , combinatorics

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

2023 HMNT, 9

Tags: number theory

Let $r_k$ denote the remainder when ${127 \choose k}$ is divided by $8$. Compute$ r_1 + 2r_2 + 3r_3 + · · · + 63r_{63}.$

2018 Iran Team Selection Test, 4

Tags: number theory

Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$. Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number) [i]Proposed by Mohsen Jamali[/i]

2013 Singapore Junior Math Olympiad, 3

Tags: prime number , number theory

Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.

2005 Morocco TST, 3

Tags: number theory , quadratic , calculus , integration

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

2007 Indonesia TST, 3

Tags: number theory , floor function , diophantine equation , inequalities

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

2011 Saudi Arabia Pre-TST, 2.1

Tags: number theory , integer

Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.

2023 Brazil Team Selection Test, 2

Tags: sequence , number theory

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

2022 Moscow Mathematical Olympiad, 5

Tags: number theory

Tanya wrote numbers in forms $n^7-1$ for $n=2,3,...$ and noticed that for $n=8$ she got number divisible by $337$. For what minimal $n$ did she get number divisible by $2022$?

2020 Iran Team Selection Test, 1

Tags: polynomial , number theory

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

2011 May Olympiad, 5

Tags: number theory , digit , divisible

We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

2002 India IMO Training Camp, 9

Tags: induction , algorithm , combinatorics , greatest common divisor , number theory

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

2013 Iran Team Selection Test, 8

Tags: modular arithmetic , greatest common divisor , number theory

Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$