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

2011 IMAC Arhimede, 5
Tags: number theory
Solve in set of integers the following equation $x^5+y^5+z^5+t^5=93$.

2012 Irish Math Olympiad, 4
Tags: number theory
There exists an infinite set of triangles with the following properties: (a) the lengths of the sides are integers with no common factors, and (b) one and only one angle is $60^\circ$. One such triangle has side lengths $5,7,8$. Find two more.

1995 Tuymaada Olympiad, 3
Tags: diophantine equation , algebra , number theory
Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

1986 IMO Longlists, 53
Tags: number theory
For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$-tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$. Prove that \[S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}\] Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$

2022 Germany Team Selection Test, 3
Tags: number theory
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2015 Iran MO (3rd round), 2
Tags: number theory
$M_0 \subset \mathbb{N}$ is a non-empty set with a finite number of elements. Ali produces sets $ M_1,M_2,...,M_n $ in the following order: In step $n$, Ali chooses an element of $M_{n-1} $ like $b_n$ and defines $M_n$ as $$M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \}$$ Prove that at some step Ali reaches a set which no element of it divides another element of it.

2005 AMC 12/AHSME, 24
Tags: analytic geometry , parabola , graphing lines , slope , trigonometry , number theory , conic
All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$? $ \textbf{(A)}\ 14\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 18$

2005 Canada National Olympiad, 5
Tags: number theory
Let's say that an ordered triple of positive integers $(a,b,c)$ is [i]$n$-powerful[/i] if $a\le b\le c,\gcd (a,b,c)=1$ and $a^n+b^n+c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is $5$-powerful. $a)$ Determine all ordered triples (if any) which are $n$-powerful for all $n\ge 1$. $b)$ Determine all ordered triples (if any) which are $2004$-powerful and $2005$-powerful, but not $2007$-powerful.

MOAA Team Rounds, 2019.10
Tags: number theory , team
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

2020 Simon Marais Mathematics Competition, B4
Tags: number theory , geometry
[i]The following problem is open in the sense that no solution is currently known to part (b).[/i] Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices. We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct. (a) show that if $n-1$ is prime then $n$ is taut. (b) Which integers $n\geq 2$ are taut?

2014 Tuymaada Olympiad, 1
Tags: number theory
Given are three different primes. What maximum number of these primes can divide their sum? [i](A. Golovanov)[/i]

2016 AMC 12/AHSME, 19
Tags: probability , number theory , relatively prime
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.) $\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$

2017 Princeton University Math Competition, A1/B3
Tags: number theory
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$.

2016 Bulgaria JBMO TST, 3
Tags: number theory
Let $ M (x,y)=x^2+xy-2y $ , x,y are positive integers a) Solve in positive integers $ x^2+xy-2y=64 $ b) Prove that if M (x,y) is a perfect square, then x+y+2 is composite if x>2.

2019 IMO Shortlist, N4
Tags: number theory , functional equation
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2001 Estonia National Olympiad, 1
Tags: number theory , digit , integer
John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

2007 Purple Comet Problems, 8
Tags: probability , number theory , relatively prime
You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positive integers, find $m+ n$.

2023 Bangladesh Mathematical Olympiad, P1
Tags: factorial , number theory , diophantine equation
Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

2014 Stars Of Mathematics, 2
Tags: number theory
Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. ([i]Dan Schwarz[/i])

1999 All-Russian Olympiad, 2
Tags: induction , number theory
Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]

1998 Estonia National Olympiad, 3
Tags: algebra , number theory , combinatorics
On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?

2022 Iberoamerican, 6
Tags: number theory , functional equation
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.

2001 USA Team Selection Test, 8
Tags: number theory
Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]

2009 Kyrgyzstan National Olympiad, 5
Tags: induction , floor function , number theory
Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$

2009 Austria Beginners' Competition, 3
Tags: number theory
There are any number of stamps with the values ​​$134$, $135$, $...$, $142$ and $143$ cents available. Find the largest integer value (in cents) that cannot be represented by these stamps. (G. Woeginger, TU Eindhoven, The Netherlands) [hide=original wording]Es stehen beliebig viele Briefmarken mit den Werten 134, 135. . .., 142 und 143 Cent zur Verfügung. Man bestimme den größten ganzzahligen Wert (in Cent), der nicht durch diese Briefmarken dargestellt werden kann.[/hide]