Found problems: 1311
2005 Kazakhstan National Olympiad, 1
Solve equation
\[2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|\]
2008 ITAMO, 3
Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions:
(i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$;
(ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
1990 Iran MO (2nd round), 2
Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.
2017 Benelux, 1
Find all functions $f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}$ such that $$f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right)
= xyf(\frac{1}{x})f(\frac{1}{y}),$$ for all $x, y \in \Bbb{Q}_{>0,}$ where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b.$
2012 Romanian Master of Mathematics, 3
Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties:
(a) if $x\le y$, then $f(x)\le f(y)$; and
(b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$.
Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$.
[i](United Kingdom) Ben Elliott[/i]
2007 Ukraine Team Selection Test, 4
Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.
2014 Turkey Team Selection Test, 2
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation
\[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\]
holds.
2006 Germany Team Selection Test, 1
Find all real solutions $x$ of the equation
$\cos\cos\cos\cos x=\sin\sin\sin\sin x$.
(Angles are measured in radians.)
1987 India National Olympiad, 7
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
2003 Moldova National Olympiad, 12.5
Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where
$n\in{N^*}$. Find the remainder of the division of polynomial
$P(x^{2n+1})$ by $P(x)$.
2010 Pan African, 3
Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?
2010 Romanian Masters In Mathematics, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
2006 Singapore MO Open, 3
Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$, either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$. Show that any such sequence has a finite number of terms.
2006 Moldova National Olympiad, 11.6
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.
Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
2001 District Olympiad, 2
Two numbers $(z_1,z_2)\in \mathbb{C}^*\times \mathbb{C}^*$ have the property $(P)$ if there is a real number $a\in [-2,2]$ such that $z_1^2-az_1z_2+z_2^2=0$. Prove that if $(z_1,z_2)$ have the property $(P)$, then $(z_1^n,z_2^n)$ satisfy this property, for any positive integer $n$.
[i]Dorin Andrica[/i]
2005 Greece Team Selection Test, 3
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
2013 All-Russian Olympiad, 1
Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution.
1987 IMO Longlists, 7
Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$
2016 Korea Winter Program Practice Test, 1
Find all $\{a_n\}_{n\ge 0}$ that satisfies the following conditions.
(1) $a_n\in \mathbb{Z}$
(2) $a_0=0, a_1=1$
(3) For infinitly many $m$, $a_m=m$
(4) For every $n\ge2$, $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$ $\mod n$
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
1999 Baltic Way, 4
For all positive real numbers $x$ and $y$ let
\[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \]
Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.
1999 Irish Math Olympiad, 2
A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies:
$ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime;
$ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$.
Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$
1989 Federal Competition For Advanced Students, P2, 1
Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times.
$ (a)$ Show that all members of $ S_n$ are real.
$ (b)$ Find the product $ P_n$ of the elements of $ S_n$.
2000 Baltic Way, 17
Find all real solutions to the following system of equations:
\[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]