Found problems: 1269
1994 Vietnam Team Selection Test, 2
Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying
\[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\]
for all $x$.
1977 Polish MO Finals, 3
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
2007 All-Russian Olympiad, 2
Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$.
[i]A. Khrabrov [/i]
2010 China Team Selection Test, 2
Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose
\[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\]
holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.
2000 Italy TST, 3
Given positive numbers $a_1$ and $b_1$, consider the sequences defined by
\[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\]
Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.
2011 All-Russian Olympiad, 1
Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?
2010 Contests, 2
Let $\{a_{n}\}$ be a sequence which satisfy
$a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$
[b](a)[/b] Find the general fomular for $a_{n}$
[b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences
2009 Vietnam National Olympiad, 1
[b]Problem 1.[/b]Find all $ (x,y)$ such that:
\[ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\
\sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\;
\]
1987 IMO Longlists, 24
Prove that if the equation $x^4 + ax^3 + bx + c = 0$ has all its roots real, then $ab \leq 0.$
2011 Kyrgyzstan National Olympiad, 7
Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.
1980 IMO, 9
Prove that is $x,y$ are non negative integers then $5x\ge 7y$ if and only if there exist non-negative integers $(a,b,c,d)$ such that
\[\left\{\begin{array}{l}x=a+2b+3c+7d\qquad\\ y=b+2c+5d\qquad\\ \end{array}\right.\]
2011 Estonia Team Selection Test, 3
Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously:
$(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$;
$(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?
2011 China Second Round Olympiad, 10
A sequence $a_n$ satisfies $a_1 =2t-3$ ($t \ne 1,-1$), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$.
[list]
[b][i]i)[/i][/b] Find $a_n$,
[b][i]ii)[/i][/b] If $t>0$, compare $a_{n+1}$ with $a_n$.[/list]
2010 Germany Team Selection Test, 1
A sequence $\left(a_n\right)$ with $a_1 = 1$ satisfies the following recursion: In the decimal expansion of $a_n$ (without trailing zeros) let $k$ be the smallest digest then $a_{n+1} = a_n + 2^k.$ How many digits does $a_{9 \cdot 10^{2010}}$ have in the decimal expansion?
2011 USA Team Selection Test, 4
Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$.
2005 Taiwan TST Round 2, 1
Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$.
This is much harder than the problems we had in the 1st TST...
1970 Canada National Olympiad, 9
Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \] a) Give a formula for $f(n)$.
b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.
2003 IberoAmerican, 3
Pablo copied from the blackboard the problem:
[list]Consider all the sequences of $2004$ real numbers $(x_0,x_1,x_2,\dots, x_{2003})$ such that: $x_0=1, 0\le x_1\le 2x_0,0\le x_2\le 2x_1\ldots ,0\le x_{2003}\le 2x_{2002}$. From all these sequences, determine the sequence which minimizes $S=\cdots$[/list]
As Pablo was copying the expression, it was erased from the board. The only thing that he could remember was that $S$ was of the form $S=\pm x_1\pm x_2\pm\cdots\pm x_{2002}+x_{2003}$. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.
2018 Latvia Baltic Way TST, P3
Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.
2011 Uzbekistan National Olympiad, 1
Find the minimum value of
$|x-y|+\sqrt{(x+2)^2+(y-4)^4}$
2001 Mediterranean Mathematics Olympiad, 2
Find all integers $n$ for which the polynomial $p(x) = x^5 -nx -n -2$ can be represented as a product of two non-constant polynomials with integer coefficients.
1995 Irish Math Olympiad, 5
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
2005 MOP Homework, 3
Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.
1996 Brazil National Olympiad, 6
Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.
2006 China Team Selection Test, 3
Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]