Olympiad problems

2022 JBMO TST - Turkey, 1

For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.

1996 Czech and Slovak Match, 1

Tags: prime , Integers , number theory

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.

2024 Switzerland Team Selection Test, 12

Tags: functional equation , IMO Shortlist , Integers

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

2001 Tuymaada Olympiad, 7

Tags: algebra , fractional part , Integers

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

Denmark (Mohr) - geometry, 1994.4

Tags: geometry , right triangle , Integers

In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

2000 Nordic, 1

Tags: combinatorics , Integers , Sum

In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)

2024 Indonesia MO, 4

Tags: Game Theory , Integers , combinatorics , number theory , Inamo

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner. At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.

2022 Switzerland Team Selection Test, 6

Tags: Integers , number theory , Divisibility

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

2014 JHMMC 7 Contest, 16

Tags: poet s strategy , Integers , JHMMC

The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two integers?

2019 Durer Math Competition Finals, 2

Tags: geometry , sidelengths , Integers , prime , number theory , right triangle

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

2013 Brazil Team Selection Test, 1

Tags: geometry , Integers

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.

2015 JBMO Shortlist, NT1

Tags: number theory , consecutive , divisible , Difference , maximum , Integers

What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

2024 VJIMC, 4

Tags: function , Divisors , Sequences , Integers , number theory

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 $b_n>b_{n+1}$.

2016 Czech And Slovak Olympiad III A, 1

Tags: number theory , Integers , Sum , Fractions

Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.

1993 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , Sum , summand , Integers

Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$

2019 Poland - Second Round, 2

Tags: Integers , algebra

Determine all nonnegative integers $x, y$ satisfying the equation \begin{align*} \sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}. \end{align*}

2024 Indonesia MO, 5

Tags: combinatorics , Integers , number theory , Indonesia , Indonesia MO , Inamo

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties: 1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number. (a) Prove that $0$ and $1$ must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.

1988 Austrian-Polish Competition, 1

Tags: Integer Polynomial , polynomial , Integers , algebra

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.

2024 Irish Math Olympiad, P2

Tags: Integers , irmo

A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.

2023 ISL, N8

Tags: functional equation , IMO Shortlist , Integers

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

1972 Putnam, B4

Tags: Putnam , polynomial , Integers

Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that $$P(x^{n},x^{n+1},x+x^{n+2})=x.$$

2017 India PRMO, 4

Tags: trinomial , roots , Integers , algebra

Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$ ?

2018 India PRMO, 9

Tags: trinomial , Integers , Root , maximum

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

2015 Indonesia MO Shortlist, N2

Tags: trinomial , area of a triangle , algebra , integer root , right triangle , Integers

Suppose that $a, b$ are natural numbers so that all the roots of $x^2 + ax - b$ and $x^2 - ax + b$ are integers. Show that exists a right triangle with integer sides, with $a$ the length of the hypotenuse and $b$ the area .

2012 Romania National Olympiad, 3

Tags: number theory , Integers , Integer

We consider the non-zero natural numbers $(m, n)$ such that the numbers $$\frac{m^2 + 2n}{n^2 - 2m} \,\,\,\, and \,\,\, \frac{n^2 + 2m}{m^2-2n}$$ are integers. a) Show that $|m - n| \le 2$: b) Find all the pairs $(m, n)$ with the property from hypothesis $a$.