Olympiad problems

2013 Saudi Arabia IMO TST, 4

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

Mathley 2014-15, 2

Tags: algebra , geometry , integer , triangle , sequence , area of a triangle

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , integer , divisor

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

2017 Singapore MO Open, 3

Tags: integer , sum , sum of squares , number theory

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

Kyiv City MO Seniors 2003+ geometry, 2019.10.3

Tags: integer , triangle , geometry

Call a right triangle $ABC$ [i]special [/i] if the lengths of its sides $AB, BC$ and$ CA$ are integers, and on each of these sides has some point $X$ (different from the vertices of $ \vartriangle ABC$), for which the lengths of the segments $AX, BX$ and $CX$ are integers numbers. Find at least one special triangle. (Maria Rozhkova)

2016 Irish Math Olympiad, 9

Tags: algebra , integer , radical

Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$ is an integer and find its value

2016 Hanoi Open Mathematics Competitions, 15

Tags: prime , algebra , integer , polynomial

Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

2002 Swedish Mathematical Competition, 4

Tags: integer , number theory

For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?

1986 All Soviet Union Mathematical Olympiad, 422

Tags: lattice points , coordinates , integer , geometry

Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.

1952 Moscow Mathematical Olympiad, 213

Tags: algebra , integer , sum , geometric sequence

Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.

2008 Tournament Of Towns, 5

Tags: combinatorics , prime , integer

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?

2017 Peru IMO TST, 10

Tags: polynomial , prime , number theory , squarefree , integer

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

2005 Cuba MO, 8

Tags: geometric inequality , geometry , area , integer

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

2024 VJIMC, 4

Tags: function , sequence , number theory , integer , divisor

Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.) a) Prove that $(b_n)_{n \ge 0}$ is unbounded. b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.

2023 OMpD, 4

Tags: analysis , integer , real number , algebra

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

2016 Singapore Senior Math Olympiad, 4

Tags: geometry , integer , integer area , integer sidelengths , perpendicular

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.

2016 Germany Team Selection Test, 2

Tags: number theory , integer , n-variable inequality , inequalities

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 \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

2019 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , integer , rational

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

2006 Estonia Team Selection Test, 1

Tags: algebra , quadratic , trinomial , integer polynomial , integer

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2003 Korea Junior Math Olympiad, 1

Tags: square , number theory , integer

Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.

2017 India National Olympiad, 6

Tags: algebra , binomial coefficient , geometry , binomial theorem , heron's formula , integer

Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$ Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.

1980 Bundeswettbewerb Mathematik, 4

Tags: number theory , sequence , integer

A sequence of integers $a_1,a_2,\ldots $ is defined by $a_1=1,a_2=2$ and for $n\geq 1$, $$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. $$ (a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms. (b) Prove that no term of the sequence is zero. (c) Show that if $n = 2^k - 1$ for $k\geq 2$, then $a_n$ is divisible by $7$.

2022 Indonesia TST, N

Tags: quadratic , greatest common divisor , diophantine , integer , number theory

For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

2015 India Regional MathematicaI Olympiad, 3

Tags: number theory , integer , fraction

Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$ , for some integers $k, l$.

2004 Mexico National Olympiad, 2

Tags: integer , inequality , number theory

Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$