Olympiad problems

2003 Korea Junior Math Olympiad, 4

When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

2016 India Regional Mathematical Olympiad, 8

Tags: algebra , polynomial , Integer

At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , Divisors , Integer

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.

2001 Estonia National Olympiad, 1

Tags: number theory , Digits , 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

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 $x \in R$.

2015 Germany Team Selection Test, 1

Tags: algebra , polynomial , real number , Root , Integer

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 $P(\xi)=0$.

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]

1976 Vietnam National Olympiad, 4

Tags: number theory , Integer , integer equation , factorial equation

Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$

2019 Durer Math Competition Finals, 7

Tags: number theory , Integer

Find the smallest positive integer $n$ with the following property: if we write down all positive integers from $1$ to $10^n$ and add together the reciprocals of every non-zero digit written down, we obtain an integer.

2011 Saudi Arabia Pre-TST, 3

Tags: number theory , radical , Integer

Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

1991 Spain Mathematical Olympiad, 2

Tags: linear algebra , matrix , Integer , Subsets , algebra

Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ . Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.

2007 Switzerland - Final Round, 9

Tags: number theory , Integer , divides , divisible

Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

2012 Belarus Team Selection Test, 2

Tags: floor function , Integer , odd , algebra

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

1994 Mexico National Olympiad, 2

Tags: combinatorics , Integer , number theory

The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

2017 Singapore MO Open, 3

Tags: Integer , number theory , Sum , Sum of Squares

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

2006 JBMO ShortLists, 15

Tags: geometry , areas , minimum , prime numbers , Integer

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

2018 Junior Regional Olympiad - FBH, 2

Tags: Integer , frac integer

Find all integers $m$ such that $\frac{2m^2+7m-9}{m^2+m+1}$ is integer

2016 Hanoi Open Mathematics Competitions, 15

Tags: polynomial , Integer , algebra , prime

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.

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.