Olympiad problems

2020 Argentina National Olympiad, 4

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

2015 Saudi Arabia IMO TST, 3

Tags: number theory , Integer

Let $n$ and $k$ be two positive integers. Prove that if $n$ is relatively prime with $30$, then there exist two integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2 - b^2 + k}{n}$ is an integer. Malik Talbi

2022 SG Originals, Q2

Tags: functional equation , set , Integer , algebra , function

Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that $$f(A+B)=f(A)+f(B)$$ for all non-empty sets of integers $A$ and $B$. $A+B$ is defined as $\{a+b: a \in A, b \in B\}$.

2016 Germany Team Selection Test, 3

Tags: Game Theory , game , Integer , number theory , combinatorics

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

2019 Nigerian Senior MO Round 3, 4

Tags: combinatorial geometry , Integer , rectangle , grid , combinatorics

A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

2023 OMpD, 4

Tags: analysis , Integer , real number , algebra

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

2009 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , Integer

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

1989 Nordic, 4

Tags: Integer , combinatorics

For which positive integers $n$ is the following statement true: if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$ then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that the indices $i_1, i_2,... , i_j$ are different numbers, and $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?

2014 Irish Math Olympiad, 8

Tags: algebra , polynomial , Integer

(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ . Assume that $P(-1), P(0)$ and $P(1)$ are integers. Prove that $P(n)$ is an integer for all integers $n$. (b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 $. Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers. Prove that $Q(n)$ is an integer for all integers $n$.

2011 Tournament of Towns, 2

Tags: geometry , rectangle , perimeter , Integer , combinatorial geometry

A rectangle is divided by $10$ horizontal and $10$ vertical lines into $121$ rectangular cells. If $111$ of them have integer perimeters, prove that they all have integer perimeters.

2015 Hanoi Open Mathematics Competitions, 12

Tags: altitudes , Integer , geometry

Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.

1993 Italy TST, 2

Tags: Integer , number theory , primes

Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer. Show that $p = q$.

2019 District Olympiad, 3

Tags: number theory , Integer

Consider the sets $M = \{0,1,2,, 2019\}$ and $$A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} $$ a) How many elements does the set $A$ have? b) Determine the smallest natural number $n$, $n \ge 2$, which has the property that any $n$-element subset of the set $A $contains two distinct elements whose difference is divisible by $40$.

1952 Moscow Mathematical Olympiad, 213

Tags: geometric sequence , Integer , Sum , algebra

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.

2015 India Regional MathematicaI Olympiad, 3

Tags: number theory , Integer , Fractions

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 $k, l$.

2012 Dutch IMO TST, 3

Tags: number theory , Integer

Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

2015 Germany Team Selection Test, 2

Tags: Integer , positive , consecutive , form , number theory

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

2001 Austrian-Polish Competition, 6

Tags: floor function , Sequence , Integer , algebra , number theory

Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

2008 Cuba MO, 3

Tags: number theory , Integer

Prove that there are infinitely many ordered pairs of positive integers $(m, n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer.

2013 Tournament of Towns, 4

Tags: Sum , perfect cube , number theory , Integer

Is it true that every integer is a sum of finite number of cubes of distinct integers?

2019 Singapore Junior Math Olympiad, 3

Tags: number theory , Integer

Find all positive integers $m, n$ such that $\frac{2m-1}{n}$ and $\frac{2n-1}{m}$ are both integers.

2010 Hanoi Open Mathematics Competitions, 6

Tags: number theory , quadratics , Integer

Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.

2019 Dürer Math Competition (First Round), P4

Tags: combinatorics , Integer , coordinates

Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

2002 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: Integer , number theory , positive integers

Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

2024 ITAMO, 6

Tags: inequalities , inequalities proposed , Integer , extermal , Fractions , Maximal

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\] for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.