Found problems: 15925
2011 Singapore MO Open, 4
Find all polynomials $P(x)$ with real coefficients such that
\[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]
2010 Iran MO (3rd Round), 7
[b]interesting function[/b]
$S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and
$f : P(S) \rightarrow \mathbb N$
is a function with these properties:
for every subset $A$ of $S$ we have $f(A)=f(S-A)$.
for every two subsets of $S$ like $A$ and $B$ we have
$max(f(A),f(B))\ge f(A\cup B)$
prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$.
time allowed for this question was 1 hours and 30 minutes.
2005 Alexandru Myller, 3
Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove:
$$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$
2020 Korean MO winter camp, #7
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $2f(x^2+y^2)=(x+f(y))^2+f(x-f(y))^2$ for all $x,y\in\mathbb{R}$.
2023 Turkey MO (2nd round), 5
Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?
2011 Dutch IMO TST, 2
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.
2018 Canadian Mathematical Olympiad Qualification, 7
Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$.
Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.
2025 Romania National Olympiad, 4
Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$.
a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above.
b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.
2012 Hanoi Open Mathematics Competitions, 6
For every n = 2; 3; : : : , we put
$$A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$
Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.
2023 Bulgarian Autumn Math Competition, 12.1
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.
MOAA Team Rounds, 2018.2
If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.
2011 Saudi Arabia IMO TST, 1
Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that
$$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$
2023 Brazil National Olympiad, 4
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
1952 Moscow Mathematical Olympiad, 213
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.
2013 Romania Team Selection Test, 1
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that
\[
\left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\]
for every positive integer $n$.
2003 Greece National Olympiad, 2
Find all real solutions of the system \[\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}\]
2014 Thailand Mathematical Olympiad, 4
Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$
2017 India PRMO, 3
A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to complete the job?
2013 Greece Junior Math Olympiad, 1
(a) Write $A = k^4 + 4$, where $k$ is a positive integer, as a product of two factors each of them is sum of two squares of integers.
(b) Simplify the expression$$K=\frac{(2^4+\frac14)(4^4+\frac14)...((2n)^4+\frac14)}{(1^4+\frac14)(3^4+\frac14)...((2n-1)^4+\frac14)}$$and write it as sum of squares of two consecutive positive integers
2025 Bulgarian Spring Mathematical Competition, 12.2
Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
2025 All-Russian Olympiad Regional Round, 10.10
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
1901 Eotvos Mathematical Competition, 1
Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5$ if and only if $n$ is not divisible by $4$.
2021 Science ON Seniors, 1
Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy
$$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$
for all $n\in \mathbb{Z}_{\ge 1}$.
[i](Bogdan Blaga)[/i]
2019 Thailand TST, 1
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
2017 China Team Selection Test, 5
A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$