This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

1966 Dutch Mathematical Olympiad, 3
Tags: perfect square , number theory
How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

2025 All-Russian Olympiad, 9.3
Tags: perfect square , number theory
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number \[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \] is a perfect square.

2010 Iran MO (2nd Round), 1
Tags: algorithm , euclidean algorithm , number theory
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

2012 Princeton University Math Competition, A2 / B5
Tags: number theory
How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?

Oliforum Contest V 2017, 4
Tags: prime , number theory
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

2016 Tuymaada Olympiad, 4
Tags: number theory , diophantine equation
For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

2010 Iran MO (3rd Round), 1
Tags: modular arithmetic , number theory
suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)

2014 Israel National Olympiad, 5
Tags: integer polynomial , algebra , polynomial , number theory
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.

2010 Tournament Of Towns, 2
Tags: number theory , ratio
Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio $1 : a$. His goal is to divide the cheese into two piles of equal masses. Can he do it if $(a) a$ is irrational? $(b) a$ is rational, $a \neq 1?$

2006 Tournament of Towns, 2
Tags: number theory
Are there exist some positive integers $n$ and $k$, such that the first decimals of $2^n$ (from left to the right) represent the number $5^k$ while the first decimals of $5^n$ represent the number $2^k$ ? (5)

Russian TST 2021, P2
Tags: nonnegative integers , number theory , functional equation
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

2017 Serbia JBMO TST, 4
Tags: number theory
Positive integer $q$ is the $k{}$-successor of positive integer $n{}$ if there exists a positive integer $p{}$ such that $n+p^2=q^2$. Let $A{}$ be the set of all positive integers $n{}$ that have at least a $k{}$-successor, but every $k{}$-successor does not have $k{}$-successors of its own. Prove that $$A=\{7,12\}\cup\{8m+3\mid m\in\mathbb{N}\}\cup\{16m+4\mid m\in\mathbb{N}\}.$$

2020 Middle European Mathematical Olympiad, 2#
Tags: number theory
We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.

1979 Austrian-Polish Competition, 9
Tags: power of 2 , floor function , number theory
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.

2016 Canadian Mathematical Olympiad Qualification, 6
Tags: greatest common divisor , number theory
Determine all ordered triples of positive integers $(x, y, z)$ such that $\gcd(x+y, y+z, z+x) > \gcd(x, y, z)$.

1993 Taiwan National Olympiad, 3
Tags: number theory
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$. [i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.

2020 International Zhautykov Olympiad, 1
Tags: euler s phi function , order , prime number , zhautykov , number theory
Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

1998 China National Olympiad, 1
Tags: number theory
Find all natural numbers $n>3$, such that $2^{2000}$ is divisible by $1+C^1_n+C^2_n+C^3_n$.

2012-2013 SDML (Middle School), 10
Tags: prime number , number theory
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

2021 Durer Math Competition Finals, 2
Tags: number theory , perfect square
Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.

2019 Baltic Way, 18
Tags: number theory
Let $a,b$, and $c$ be odd positive integers such that $a$ is not a perfect square and $$a^2+a+1 = 3(b^2+b+1)(c^2+c+1).$$ Prove that at least one of the numbers $b^2+b+1$ and $c^2+c+1$ is composite.

2011 IFYM, Sozopol, 8
Tags: permutation , number theory
Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions: (1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$; (2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.

2024 Polish Junior MO Finals, 5
Tags: multiple , divisibility , digit , number theory
Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number \[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\] is a multiple of $19$.

1999 Romania National Olympiad, 2
Tags: prime , number theory
Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$ Prove that $a^2 +b^2 +c^2$ cannot be a prime number.

2010 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: modular arithmetic , number theory
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.