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.

AND:
OR:
NO:

Found problems: 15460

2024 Israel National Olympiad (Gillis), P5
Tags: number theory , smallest prime divisor , integer sequence
For positive integral $k>1$, we let $p(k)$ be its smallest prime divisor. Given an integer $a_1>2$, we define an infinite sequence $a_n$ by $a_{n+1}=a_n^n-1$ for each $n\geq 1$. For which values of $a_1$ is the sequence $p(a_n)$ bounded?

2013 JBMO Shortlist, 1
Tags: number theory
$\boxed{N1}$ find all positive integers $n$ for which $1^3+2^3+\cdots+{16}^3+{17}^n$ is a perfect square.

2016 Japan Mathematical Olympiad Preliminary, 6
Tags: number theory , combinatorics , algebra
Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

2007 Kyiv Mathematical Festival, 1
Tags: number theory
Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

2006 India IMO Training Camp, 2
Tags: modular arithmetic , geometry , 3d geometry , analytic geometry , pigeonhole principle , number theory
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

2016 PUMaC Individual Finals B, 3
Tags: floor function , number theory
Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

2010 Tournament Of Towns, 1
Tags: number theory
$2010$ ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of $31$.

2013 Macedonian Team Selection Test, Problem 2
Tags: number theory , irrational number
a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational. b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.

2014 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: partition , number theory , inequalities , set
Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

2009 Middle European Mathematical Olympiad, 10
Tags: incenter , number theory , greatest common divisor , cyclic quadrilateral , geometry
Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

2024 IMC, 10
Tags: number theory , fermat primes , primality test
We say that a square-free positive integer $n$ is [i]almost prime[/i] if \[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\] for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$. (An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)

1982 Bundeswettbewerb Mathematik, 4
Tags: prime number , factorization , number theory
Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

2020 Korea National Olympiad, 4
Tags: number theory , diophantine equation
Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

2024 Brazil Cono Sur TST, 3
Tags: number theory , sigma function , tau function , ceiling function , divisor
Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]

PEN Q Problems, 7
Tags: algebra , polynomial , calculus , integration , number theory , relatively prime , rational root theorem
Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2011 Serbia JBMO TST, 1
Tags: combinatorics , algebra , number theory
A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

2017 Bundeswettbewerb Mathematik, 4
Tags: number theory , square , cube
We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.

2015 Peru IMO TST, 11
Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

2003 Iran MO (2nd round), 1
Tags: number theory
We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

1989 Tournament Of Towns, (235) 3
Tags: number theory , perfect square , combinatorics
Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

1993 AIME Problems, 6
Tags: number theory , least common multiple
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

2003 Olympic Revenge, 5
Tags: number theory , prime number , prime , divisibility
Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

2017 Dutch IMO TST, 3
Tags: number theory
Compute the product of all positive integers $n$ for which $3(n!+1)$ is divisible by $2n - 5$.

1998 Vietnam Team Selection Test, 1
Tags: algebra , polynomial , calculus , integration , rational root theorem , number theory
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

1999 Mexico National Olympiad, 2
Tags: arithmetic sequence , number theory , prime
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.