Found problems: 1269
2010 Contests, 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 $(0,+\infty )$
2011 Czech and Slovak Olympiad III A, 6
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$, we have \[ f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.\]
2012 Centers of Excellency of Suceava, 1
Function ${{f\colon \mathbb[0, +\infty)}\to\mathbb[0, +\infty)}$ satisfies the condition $f(x)+f(y){\ge}2f(x+y)$ for all $x,y{\ge}0$.
Prove that $f(x)+f(y)+f(z){\ge}3f(x+y+z)$ for all $x,y,z{\ge}0$.
Mathematical induction?
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Azerbaijan Land of the Fire :lol:
2009 Kyrgyzstan National Olympiad, 3
For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.
1999 All-Russian Olympiad, 6
Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.
1999 Yugoslav Team Selection Test, Problem 1
For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that:
$P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$.
Prove that $2P(2n + 1) - P(2n + 2) = 1$.
P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)
2011 Polish MO Finals, 1
Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$,
\[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]
1987 USAMO, 5
Given a sequence $(x_1,x_2,\ldots, x_n)$ of 0's and 1's, let $A$ be the number of triples $(x_i,x_j,x_k)$ with $i<j<k$ such that $(x_i,x_j,x_k)$ equals $(0,1,0)$ or $(1,0,1)$. For $1\leq i \leq n$, let $d_i$ denote the number of $j$ for which either $j < i$ and $x_j = x_i$ or else $j > i$ and $x_j\neq x_i$.
(a) Prove that \[A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.\] (Of course, $\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}$.) [5 points]
(b) Given an odd number $n$, what is the maximum possible value of $A$? [15 points]
1989 IMO Longlists, 7
For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.
2001 China Team Selection Test, 2
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
2005 Junior Balkan Team Selection Tests - Romania, 10
Let $k,r \in \mathbb N$ and let $x\in (0,1)$ be a rational number given in decimal representation \[ x = 0.a_1a_2a_3a_4 \ldots . \] Show that if the decimals $a_k, a_{k+r}, a_{k+2r}, \ldots$ are canceled, the new number obtained is still rational.
[i]Dan Schwarz[/i]
1994 Taiwan National Olympiad, 5
Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows:
$+$ $0$ $a$ $b$ $c$
$0$ $0$ $a$ $b$ $c$
$a$ $a$ $0$ $c$ $b$
$b$ $b$ $c$ $0$ $a$
$c$ $c$ $b$ $a$ $0$
a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$.
b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.
1991 IMTS, 2
Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients
\[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\]
form an increasing arithmetic series.
2012 Olympic Revenge, 2
We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$.
Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise.
Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.
2002 Taiwan National Olympiad, 5
Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying
$\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$
$\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$
$...$
$\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$
Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.
2008 Greece National Olympiad, 1
A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.
2006 MOP Homework, 6
Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$.
1988 IMO Longlists, 16
If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right]$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.$
1989 China National Olympiad, 3
Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$).
We define function $f:S\rightarrow S$ as follow: $\forall z\in S$,
$ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$
$f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$
We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy:
$f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$.
Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.
1983 IMO Longlists, 39
If $\alpha $ is the real root of the equation
\[E(x) = x^3 - 5x -50 = 0\]
such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$, where $n$ is a positive integer, prove that:
[b](a)[/b] $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$
[b](b)[/b] $\alpha < x_{n+1} < x_n.$
1987 IberoAmerican, 2
Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$. Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$.
2002 IberoAmerican, 2
The sequence of real numbers $a_1,a_2,\dots$ is defined as follows: $a_1=56$ and $a_{n+1}=a_n-\frac{1}{a_n}$ for $n\ge 1$. Show that there is an integer $1\leq{k}\leq2002$ such that $a_k<0$.
2002 Irish Math Olympiad, 4
Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.
2011 Polish MO Finals, 3
Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy \[f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.\]
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.