Olympiad problems

2024 Indonesia MO, 5

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.

2019 Poland - Second Round, 2

Tags: integer , algebra

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

1987 Tournament Of Towns, (161) 5

Tags: coloring , integer , combinatorics

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?

2012 Thailand Mathematical Olympiad, 12

Tags: number theory , integer , perfect cube

Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.

2024 Indonesia MO, 4

Tags: integer , game theory , combinatorics , number theory

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.

2010 Ukraine Team Selection Test, 6

Tags: number theory , integer , divisibility

Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.

1974 Putnam, A1

Tags: relatively prime , integer

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

2019 China Girls Math Olympiad, 2

Tags: integer , algebra , number theory

Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.

2018 Junior Regional Olympiad - FBH, 5

Tags: integer , exponential , exponent

Find all integers $x$ and $y$ such that $2^x+1=y^2$

2012 Romania National Olympiad, 3

Tags: number theory , 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 $a$.

1997 Czech and Slovak Match, 5

Tags: number theory , integer , sum of powers

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

2004 Mexico National Olympiad, 2

Tags: integer , number theory , inequality

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} .$

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 $n$.

2004 German National Olympiad, 4

Tags: integer , square root , algebra , sum

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

2010 Hanoi Open Mathematics Competitions, 6

Tags: number theory , quadratic , 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 $a, b, c, d$.

2009 Cuba MO, 5

Tags: number theory , integer

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

2001 Dutch Mathematical Olympiad, 4

Tags: number theory , integer

The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$. Determine all positive integers $x$ for which $f(x)$ is an integer

2022 SAFEST Olympiad, 2

Tags: integer , 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.

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.

2015 JBMO Shortlist, NT1

Tags: integer , number theory , consecutive , divisible , difference , maximum

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 Irish Math Olympiad, P2

Tags: integer

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 $p \leq n^2 < q$.

2003 Korea Junior Math Olympiad, 2

Tags: quadratic equation , integer , number theory , quadratic , algebra

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

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 $7$.

2013 Singapore Junior Math Olympiad, 4

Tags: integer , number theory

Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer

2004 Peru MO (ONEM), 4

Tags: geometry , integer , sides of a triangle , area of a triangle , minimum

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.