Found problems: 1269
2014 Contests, 2
Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.
2014 Dutch IMO TST, 1
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.
1994 China National Olympiad, 5
For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
2013 ELMO Shortlist, 7
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
2013 South East Mathematical Olympiad, 3
A sequence $\{a_n\}$ , $a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}$. Show that $a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N$
2005 Italy TST, 1
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and
\[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \]
$(a)$ Prove that $f$ has a fixed point different from $1$.
$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.
2010 Contests, 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
2003 China Team Selection Test, 1
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:
(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and
(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.
Find $|B_n^m|$ and $|B_6^3|$.
1987 Vietnam National Olympiad, 2
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.
2007 Romania National Olympiad, 1
Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2010 Greece Team Selection Test, 4
Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$
and are strictly monotone in $(0,+\infty )$
1984 IMO Longlists, 38
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
2014 Saudi Arabia IMO TST, 2
Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.
1989 IMO Longlists, 3
For each non-zero complex number $ z,$ let $\arg(z)$ be the unique real number $ t$ such that $ \minus{}\pi < t \leq \pi$ and $ z \equal{} |z|(\cos(t) \plus{} \textrm{i} sin(t)).$ Given a real number $ c > 0$ and a complex number $ z \neq 0$ with $\arg z \neq \pi,$ define \[ B(c, z) \equal{} \{b \in \mathbb{R} \ ; \ |w \minus{} z| < b \Rightarrow |\arg(w) \minus{} \arg(z)| < c\}.\] Determine necessary and sufficient conditions, in terms of $ c$ and $ z,$ such that $ B(c, z)$ has a maximum element, and determine what this maximum element is in this case.
2011 Albania National Olympiad, 1
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
2002 China Team Selection Test, 1
Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.
2003 China Team Selection Test, 2
Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.
2001 Federal Competition For Advanced Students, Part 2, 1
Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$
\[f(f(x)^2 + f(y)) = xf(x) + y.\]
2010 India National Olympiad, 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
1985 USAMO, 4
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
1999 Vietnam Team Selection Test, 1
Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$, where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.
2007 Germany Team Selection Test, 2
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
2008 Bundeswettbewerb Mathematik, 2
Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.