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

2017 Latvia Baltic Way TST, 15
Tags: number theory , digit
Let's call the number string $D = d_{n-1}d_{n-2}...d_0$ a [i]stable ending[/i] of a number , if for any natural number $m$ that ends in $D$, any of its natural powers $m^k$ also ends in $D$. Prove that for every natural number $n$ there are exactly four stable endings of a number of length $n$. [hide=original wording]Ciparu virkni $D = d_{n-1}d_{n-2}...d_0$ sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe $m^k$ beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.[/hide]

2023 Hong Kong Team Selection Test, Problem 2
Tags: number theory
Let $n$ be a positive integer. Show that if p is prime dividing $5^{4n}-5^{3n}+5^{2n}-5^{n}+1$, then $p\equiv 1 \;(\bmod\; 4)$.

2024 Iran Team Selection Test, 10
Tags: number theory
Let $\{a_n\}$ be a sequence of natural numbers such that each prime number greater than $1402$ divides a member of that. Prove that the set of prime divisors of members of sequence $\{b_n\}$ which $b_n=a_1a_2...a_n-1$ , is infinite. [i]Proposed by Navid Safaei[/i]

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

1994 Brazil National Olympiad, 5
Tags: number theory
Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$. (Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ ) Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$. Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)

2018 Caucasus Mathematical Olympiad, 3
Tags: number theory
Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.

2008 Switzerland - Final Round, 6
Tags: odd , number theory , divide , divisible
Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

1966 IMO Shortlist, 48
Tags: algebra , equation , number theory , root , parametric equation
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2006 Finnish National High School Mathematics Competition, 1
Tags: number theory
Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.

2014 Peru IMO TST, 9
Tags: diophantine equation , number theory
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$

2022 Kosovo & Albania Mathematical Olympiad, 3
Tags: number theory
Is it possible to partition $\{1, 2, 3, \ldots, 28\}$ into two sets $A$ and $B$ such that both of the following conditions hold simultaneously: (i) the number of odd integers in $A$ is equal to the number of odd integers in $B$; (ii) the difference between the sum of squares of the integers in $A$ and the sum of squares of the integers in $B$ is $16$?

2009 AMC 12/AHSME, 19
Tags: algebra , polynomial , search , vieta , prime number , number theory
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

1985 Bulgaria National Olympiad, Problem 1
Tags: polynomial , number theory , algebra
Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)

2011 Tournament of Towns, 4
Tags: digit , number theory
Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?

2014 ELMO Shortlist, 3
Tags: algebra , polynomial , induction , binomial theorem , number theory
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

2020 CMIMC Algebra & Number Theory, 1
Tags: algebra , number theory
Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.

2018 JBMO TST-Turkey, 5
Tags: number theory
Let $a_1, a_2, ... , a_{1000}$ be a sequence of integers such that $a_1=3, a_2=7$ and for all $n=2, 3, ... , 999$ $a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)$. Find the number of indices $1\leq n\leq 1000$ for which $a_n+2018$ is a perfect square.

2006 Junior Balkan Team Selection Tests - Romania, 4
Tags: digit , number theory , perfect square
For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

1994 Mexico National Olympiad, 1
Tags: number theory , integer sequence
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.

1999 IMO Shortlist, 3
Tags: modular arithmetic , number theory , integer sequence , divisibility , sequence
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

2008 Postal Coaching, 1
Tags: combination , least common multiple , lcm , number theory
Prove that for any $n \ge 1$, $LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$

2014 Contests, 3
Tags: greatest common divisor , least common multiple , inequalities , number theory
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

2023 Serbia JBMO TST, 4
Tags: number theory
Find all triples $(k, m, n)$ of positive integers such that $m$ is a prime and: (1) $kn$ is a perfect square; (2) $\frac{k(k-1)}{2}+n$ is a fourth power of a prime; (3) $k-m^2=p$ where $p$ is a prime; (4) $\frac{n+2}{m^2}=p^4$.

2012 Mathcenter Contest + Longlist, 2
Tags: number theory , prime
Let $p=2^n+1$ and $3^{(p-1)/2}+1\equiv 0 \pmod p$. Show that $p$ is a prime. [i](Zhuge Liang) [/i]

2009 Baltic Way, 9
Tags: modular arithmetic , inequalities , number theory
Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.