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: 387

2016 Costa Rica - Final Round, N2
Tags: divide , divisible , number theory
Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

2019 Durer Math Competition Finals, 11
Tags: sum , divide , divisible , number theory
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?

2008 Postal Coaching, 3
Tags: product , divide , divisible , number theory
Prove that there exists an in nite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2007 IMAC Arhimede, 4
Tags: divide , sum , sums and products , number theory , pigeonhole principle , combinatorics
Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

2021 Saudi Arabia Training Tests, 34
Tags: number theory , divide , polynomial
Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

2019 Saudi Arabia IMO TST, 1
Tags: divide , number theory , divisible
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$

2002 Singapore Team Selection Test, 3
Tags: divide , number theory
For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).

2013 Balkan MO Shortlist, A5
Tags: algebra , divide , integer polynomial , polynomial
Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

1925 Eotvos Mathematical Competition, 1
Tags: divisible , divide , number theory
Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

2018 Istmo Centroamericano MO, 1
Tags: number theory , divide , recurrence relation , divisible
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

2018 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: coprime numbers , sum of powers , divide , number theory , coprime
Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2013 Thailand Mathematical Olympiad, 1
Tags: prime , divide , number theory
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2010 Thailand Mathematical Olympiad, 3
Tags: divide , divisible , number theory
Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

2001 Estonia National Olympiad, 2
Tags: number theory , least common multiple , divide , divisor
A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

1982 Poland - Second Round, 5
Tags: divide , divisible , number theory
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible , functional
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

2018 Thailand Mathematical Olympiad, 5
Tags: minimum , divide , number theory
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?

2022 IFYM, Sozopol, 1
Tags: divide , number theory
Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

2015 Saudi Arabia GMO TST, 4
Tags: divide , number theory , divisible
Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

2009 Kyiv Mathematical Festival, 5
Tags: number theory with sequences , divide , sequence , number theory
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?

2010 Thailand Mathematical Olympiad, 6
Tags: prime , divide , number theory
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

2003 Singapore Senior Math Olympiad, 1
Tags: number theory , perfect square , divide , divisible
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

2018 District Olympiad, 2
Tags: divide , number theory
Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.

2015 Balkan MO Shortlist, N5
Tags: maximum , power of 2 , divide , product , number theory
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2019 Ecuador NMO (OMEC), 3
Tags: power of 2 , divide , divisible , number theory
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$