Found problems: 1311
2007 Romania Team Selection Test, 2
Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that \[|f(x)-f(y)|\leq (x-y)^{2}\] for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.
2012 ELMO Shortlist, 10
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.
[i]David Yang.[/i]
2006 Taiwan TST Round 1, 1
Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$:
(i) $f((xy)^p)=(f(x)f(y))^p$
(ii)$f((x+y)^q)=(f(x)+f(y))^q$
Find the function $f$.
2005 IberoAmerican Olympiad For University Students, 3
Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$,
$(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$.
Find $\lim_{n\to \infty}(x_n,y_n)$.
2007 Tuymaada Olympiad, 2
Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?
2008 All-Russian Olympiad, 2
Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$
1999 Czech and Slovak Match, 5
Find all functions $f: (1,\infty)\text{to R}$ satisfying
$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.
[hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution.
I have also solved by using this method.By finding $f(x^5)$ in two ways
I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]
1998 Baltic Way, 8
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that
\[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \]
for every real number $x$ and every positive integer $n$.
1984 Canada National Olympiad, 5
Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.
2006 All-Russian Olympiad, 3
Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?
2009 Paraguay Mathematical Olympiad, 4
Let $a_1, a_2, ..., a_n $ be a sequence such that the arithmetic mean of the $n$ terms is $n$. Consider $n = 2009$. Determine the sum of the $2009$ terms of the sequence.
2014 Bundeswettbewerb Mathematik, 2
For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$.
Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation
\[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]
2003 Vietnam National Olympiad, 2
Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.
2007 South africa National Olympiad, 5
Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively.
Let $ f: Z \rightarrow R$ be a function satisfying:
(i) $ f(n) \ge 0$ for all $ n \in Z$
(ii) $ f(mn)\equal{}f(m)f(n)$ for all $ m,n \in Z$
(iii) $ f(m\plus{}n) \le max(f(m),f(n))$ for all $ m,n \in Z$
(a) Prove that $ f(n) \le 1$ for all $ n \in Z$
(b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) \equal{} 1$
2009 Indonesia TST, 1
Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]
2010 Poland - Second Round, 1
Solve in the real numbers $x, y, z$ a system of the equations:
\[
\begin{cases}
x^2 - (y+z+yz)x + (y+z)yz = 0 \\
y^2 - (z + x + zx)y + (z+x)zx = 0 \\
z^2 - (x+y+xy)z + (x+y)xy = 0. \\
\end{cases}
\]
2005 District Olympiad, 1
Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that
i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing;
ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.
1986 India National Olympiad, 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
2004 CentroAmerican, 2
Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$.
Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.
2000 Romania Team Selection Test, 2
Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$.
[i]Marius Cavachi[/i]
2008 District Olympiad, 4
Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$.
a) Prove that $ A$ is non-empty.
b) Find, with proof, whether $ A$ is infinite.
c) Prove that all the elements of $ A$ are bijective functions.
(Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)
2012 ELMO Shortlist, 10
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.
[i]David Yang.[/i]
2005 Georgia Team Selection Test, 4
Find all polynomials with real coefficients, for which the equality
\[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\]
holds for any real number $ x$.
1985 IMO Longlists, 51
Let $f_1 = (a_1, a_2, \dots , a_n) , n > 2$, be a sequence of integers. From $f_1$ one constructs a sequence $f_k$ of sequences as follows: if $f_k = (c_1, c_2, \dots, cn)$, then $f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1)$, where $(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}})$ is a permutation of $(c_1, c_2, \dots, c_n)$. Give a necessary and sufficient condition for $f_1$ under which it is possible for $f_k$ to be a constant sequence $(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n$, for some $k.$
2007 International Zhautykov Olympiad, 1
Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?