Found problems: 177
2013 District Olympiad, 2
Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.
2013 Saudi Arabia IMO TST, 4
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
2011 Saudi Arabia Pre-TST, 2.3
Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer
2012 Thailand Mathematical Olympiad, 12
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.
1996 Spain Mathematical Olympiad, 1
The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.
2018 China Girls Math Olympiad, 6
Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have
(i) $168 \in I_i$;
(ii) $\forall x, y \in I_i$, we have $x-y \in I_i$.
Determine the number of $k-chain$ in total.
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
1998 Bosnia and Herzegovina Team Selection Test, 5
Let $a$, $b$ and $c$ be integers such that $$bc+ad=1$$ $$ac+2bd=1$$ Prove that $a^2+c^2=2b^2+2d^2$
2012 Thailand Mathematical Olympiad, 3
Let $m, n > 1$ be coprime odd integers. Show that
$$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$
is an even integer, where $\phi$ is Euler’s totient function.
2016 Irish Math Olympiad, 9
Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$
is an integer and find its value
2018 BAMO, 5
To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
[img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img]
An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.
1997 Tournament Of Towns, (562) 3
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is:
(a) an integer;
(b) the square of an integer.
(A Kanel)
2017 Czech-Polish-Slovak Junior Match, 6
On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.
2009 Junior Balkan Team Selection Tests - Romania, 2
Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.
2003 Estonia National Olympiad, 2
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.
1996 All-Russian Olympiad Regional Round, 9.3
Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.
2000 Czech and Slovak Match, 6
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.
2010 Contests, 3
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure).
Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, O, X, Y, Z;
X = (1,4);
Y = (0,0);
Z = (5,1.5);
O = (1.8,2.2);
A = extension(O, O + Z - X, X, Y);
B = extension(O, O + Y - Z, X, Y);
C = extension(O, O + X - Y, Y, Z);
D = extension(O, O + Z - X, Y, Z);
E = extension(O, O + Y - Z, Z, X);
F = extension(O, O + X - Y, Z, X);
draw(X--Y--Z--cycle);
draw(A--D);
draw(B--E);
draw(C--F);
dot("$A$", A, NW);
dot("$B$", B, NW);
dot("$C$", C, SE);
dot("$D$", D, SE);
dot("$E$", E, NE);
dot("$F$", F, NE);
dot("$O$", O, S);
dot("$X$", X, N);
dot("$Y$", Y, SW);
dot("$Z$", Z, dir(0));
label("$a$", (A + O)/2, SW);
label("$b$", (B + O)/2, SE);
label("$c$", (C + O)/2, SE);
label("$d$", (D + O)/2, SW);
label("$e$", (E + O)/2, SE);
label("$f$", (F + O)/2, NW);
[/asy]
2022 Mexican Girls' Contest, 6
Let $a$ and $b$ be positive integers such that $$\frac{5a^4+a^2}{b^4+3b^2+4}$$ is an integer. Prove that $a$ is not a prime number.
2017 Bundeswettbewerb Mathematik, 4
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$:
(a) The number $a_n$ is a positive integer.
(b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$.
(c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.
1957 Poland - Second Round, 1
Prove that if $ n $ is an integer, then
$$
\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$
is also an integer.
1997 Czech and Slovak Match, 5
The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.
2015 India Regional MathematicaI Olympiad, 6
Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer.
(Here $\{a\}$ denotes the fractional part of $a$.)
2013 Bosnia And Herzegovina - Regional Olympiad, 2
If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer
2001 Dutch Mathematical Olympiad, 4
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