Olympiad problems

1959 Kurschak Competition, 1

$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.

2024 Switzerland Team Selection Test, 12

Tags: integer , functional equation

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

VMEO III 2006 Shortlist, N12

Tags: integer , number theory

Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.

2006 Abels Math Contest (Norwegian MO), 3

Tags: rational , number theory , diophantine , integer

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

2009 Cuba MO, 5

Tags: integer , number theory

Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.

2025 Nepal National Olympiad, 2

Tags: algebra , number theory , integer

(a) Positive rational numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? (b) Positive real numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? [i](Andrew Brahms, USA)[/i]

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 \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

2015 Germany Team Selection Test, 1

Tags: real number , integer , root , algebra , polynomial

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

2006 Korea Junior Math Olympiad, 2

Tags: positive integer , integer , coprime , number theory

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

2014 JHMMC 7 Contest, 16

Tags: poet s strategy , integer

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?

2013 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: integer , number theory

If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer

2019 Hanoi Open Mathematics Competitions, 12

Tags: integer , algebra , combinatorics , polynomial

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.

1998 Mexico National Olympiad, 4

Tags: integer , number theory

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

2015 India Regional MathematicaI Olympiad, 2

Tags: quadratic trinomial , integer , algebra , polynomial

Let $P(x)=x^{2}+ax+b$ be a quadratic polynomial where $a$ is real and $b \neq 2$, is rational. Suppose $P(0)^{2},P(1)^{2},P(2)^{2}$ are integers, prove that $a$ and $b$ are integers.

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.

1996 Czech and Slovak Match, 1

Tags: prime , number theory , integer

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.

2014 Ukraine Team Selection Test, 5

Tags: number theory , modulo , integer

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

2016 Czech And Slovak Olympiad III A, 1

Tags: number theory , integer , sum , fraction

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.

2015 India PRMO, 3

Tags: integer , number theory , algebra

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

1978 Putnam, B4

Tags: equation , integer

Prove that for every real number $N$ the equation $$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$ has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$

2001 Estonia National Olympiad, 1

Tags: number theory , digit , integer

John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: perfect square , integer , number theory

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

2008 Austria Beginners' Competition, 1

Tags: number theory , integer

Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.

1987 Tournament Of Towns, (161) 5

Tags: combinatorics , coloring , integer

Consider the set of all pairs of positive integers $(A , B)$ in which $A < B$ . Some of these pairs are to $be$ designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form $A, A + D, A + 2D$, in which $D > 0$, the associated pairs $(A, A + D )$ , $(A , A + 2D)$ and $(A + D, A + 2D)$ would include at least one pair of each colour?

Brazil L2 Finals (OBM) - geometry, 2010.6

Tags: geometric inequality , minimum , area of a triangle , integer , geometry

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?