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: 288

2013 Saudi Arabia IMO TST, 4
Tags: algebra , polynomial , integer , divisible
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

2005 Mexico National Olympiad, 2
Tags: integer , matrix , combinatorics , number theory
Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

1996 Nordic, 1
Tags: integer , sum of digits , number theory
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

2002 Swedish Mathematical Competition, 4
Tags: number theory , integer
For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?

2020 German National Olympiad, 5
Tags: integer , inequalities
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$. Prove the inequality \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

2020 New Zealand MO, 1
Tags: number theory , integer
What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

Denmark (Mohr) - geometry, 1994.4
Tags: integer , geometry , right triangle
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

2023 OMpD, 4
Tags: analysis , integer , real number , algebra
Are there integers $m, n \geq 2$ such that the following property is always true? $$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$

1995 Nordic, 4
Tags: geometry , integer , area of a triangle
Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying (i) The side lengths of $T $ are consecutive integers. (ii) The area of $T$ is an integer.

2000 Czech and Slovak Match, 6
Tags: integer , number theory , coloring , ramsey theory , combinatorics
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.

2014 Hanoi Open Mathematics Competitions, 5
Tags: sequence , integer , number theory
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is: (A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.

2013 Brazil Team Selection Test, 1
Tags: integer , geometry
Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.

1997 Tournament Of Towns, (562) 3
Tags: number theory , integer , perfect square
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

2015 Costa Rica - Final Round, N4
Tags: integer , number theory , coprime
Show that there are no triples $(a, b, c)$ of positive integers such that a) $a + c, b + c, a + b$ do not have common multiples in pairs. b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.

2016 Germany Team Selection Test, 2
Tags: inequalities , integer , n-variable inequality , number theory
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

2010 Hanoi Open Mathematics Competitions, 4
Tags: number theory , integer
How many real numbers $a \in (1,9)$ such that the corresponding number $a- \frac1a$ is an integer? (A): $0$, (B): $1$, (C): $8$, (D): $9$, (E) None of the above.

2018 India PRMO, 24
Tags: integer , geometry , angle , combinatorics
If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

2013 Grand Duchy of Lithuania, 1
Tags: functional , increasing , integer
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

2018 German National Olympiad, 3
Tags: combinatorics , coloring , integer
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes. Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions: 1) There are no two red boxes in the same column or in the same row. 2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column. [i]Proposed by Christian Reiher[/i]

2022 Turkey EGMO TST, 3
Tags: integer , number theory
Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.

1953 Putnam, B2
Tags: integer , polynomial
Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$

2013 Balkan MO Shortlist, N8
Tags: integer , number theory , divide
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.

2008 Tournament Of Towns, 5
Tags: integer , combinatorics , prime
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?

Ukrainian TYM Qualifying - geometry, 2017.4
Tags: integer , right triangle , geometry
Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

1996 Spain Mathematical Olympiad, 1
Tags: number theory , greatest common divisor , integer
The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.