Olympiad problems

2022 Bulgarian Autumn Math Competition, Problem 10.3

Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$

2003 Bosnia and Herzegovina Junior BMO TST, 3

Tags: divide , divisible , number theory

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

2011 Bundeswettbewerb Mathematik, 4

Tags: diophantine equation , diophantine , euclidean algorithm , number theory

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

2018 Saint Petersburg Mathematical Olympiad, 1

Tags: analytic geometry , geometry , number theory

Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$

2018 Bosnia and Herzegovina EGMO TST, 2

Tags: positive integer , divide , number theory

Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2008 Postal Coaching, 3

Tags: number theory , divide , divisible

Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

1955 Moscow Mathematical Olympiad, 309

Tags: combinatorial geometry , combinatorics , number theory , convex polygon , enumeration

A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered. a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$. b) Prove that for $n = 10$ there is no such numeration.

2024/2025 TOURNAMENT OF TOWNS, P4

Tags: number theory , geometry

Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)

2018 Federal Competition For Advanced Students, P2, 5

Tags: combinatorics , number theory

On a circle $2018$ points are marked. Each of these points is labeled with an integer. Let each number be larger than the sum of the preceding two numbers in clockwise order. Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers. [i](Proposed by Walther Janous)[/i]

Kvant 2021, M2671

Tags: algebra , number theory

Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]

2009 Bundeswettbewerb Mathematik, 4

Tags: palindrome , number theory , 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.

1993 Mexico National Olympiad, 4

Tags: number theory , function , recursive

$f(n,k)$ is defined by (1) $f(n,0) = f(n,n) = 1$ and (2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$. How many times do we need to use (2) to find $f(3991,1993)$?

2004 Chile National Olympiad, 1

Tags: number theory , combinatorics

A company with $2004$ workers celebrated its anniversary by inviting everyone to a lunch served at a round table. When the $2004$ workers sat around this table, they formed a circle of people and soon discovered that they all had salaries. different and also that the difference between the salaries of any two neighbors, at the round table, was $2000$ or $3000$ pesos. Calculate the maximum difference that can exist between the wages of these workers.

2012 Bosnia Herzegovina Team Selection Test, 4

Tags: function , induction , algebra , polynomial , modular arithmetic , number theory

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

2020 ELMO Problems, P2

Tags: number theory , fibonacci

Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number? [i]Proposed by Fedir Yudin.[/i]

2015 China Northern MO, 4

Tags: combinatorics , number theory

It is known that $a_1, a_2,...a_{108}$ are $108$ different positive integers not exceeding $2015$. Prove that there is a positive integer $k$ such that there are at least four different pairs $(i, j) $satisfying $a_i-a_j =k$.

2024 Malaysian IMO Team Selection Test, 5

Tags: number theory

Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$. [i]Proposed by Anzo Teh Zhao Yang[/i]

1971 IMO Longlists, 2

Tags: inequalities , ratio , number theory

Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ($1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.

2010 Regional Olympiad of Mexico Center Zone, 5

Tags: number theory , diophantine equation , diophantine

Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions: $\bullet$ $r$ is a positive integer with exactly $8$ positive divisors. $\bullet$ $p$ and $q$ are prime numbers.

PEN H Problems, 81

Tags: diophantine equation , modular arithmetic , number theory , relatively prime

Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.

2019 Durer Math Competition Finals, 3

Tags: greatest common divisor , relatively prime , totient function , number theory , function

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2008 ITest, 44

Tags: probability , floor function , arithmetic series , number theory , relatively prime

Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

2020 Durer Math Competition Finals, 3

Tags: number theory , sum of digits

Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?

2016 Saudi Arabia IMO TST, 1

Tags: number theory , sum , coprime

Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.

2023 Malaysian IMO Training Camp, 1

Tags: number theory

Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]