Found problems: 4776
PEN M Problems, 21
In the sequence $1, 0, 1, 0, 1, 0, 3, 5, \cdots$, each member after the sixth one is equal to the last digit of the sum of the six members just preceeding it. Prove that in this sequence one cannot find the following group of six consecutive members: \[0, 1, 0, 1, 0, 1\]
2007 AMC 12/AHSME, 17
Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$
$ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$
1968 IMO Shortlist, 26
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
1991 Arnold's Trivium, 6
In the $(x,y)$-plane sketch the curve given parametrically by $x=2t-4t^3$, $y=t^2-3t^4$.
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:
2013 Kazakhstan National Olympiad, 2
Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.
2013 IMC, 4
Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that:
\[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\]
[i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]
1958 November Putnam, A1
Let $f(m,1)=f(1,n)=1$ for $m\geq 1, n\geq 1$ and let $f(m,n)=f(m-1, n)+ f(m, n-1) +f(m-1 ,n-1)$ for $m>1$ and $n>1$. Also let
$$ S(n)= \sum_{a+b=n} f(a,b) \,\,\;\; a\geq 1 \,\, \text{and} \,\; b\geq 1.$$
Prove that
$$S(n+2) =S(n) +2S(n+1) \,\, \; \text{for} \, \, n \geq 2.$$
2011 Romania National Olympiad, 3
Let be three positive real numbers $ a,b,c. $ Show that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , $
$$ f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , $$
is nondecresing on the interval $ \left[ 0,\infty \right) $ and nonincreasing on the interval $ \left( -\infty ,0 \right] . $
1992 IMO Longlists, 48
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity
\[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\]
Where $\alpha,\beta$ are given real numbers.
2019 IMC, 6
Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$.
[i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]
2013 Vietnam National Olympiad, 1
Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(0)=0,f(1)=2013$ and
\[(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))\]
Note: $f^2(x)=(f(x))^2$
1993 USAMO, 1
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
2010 Balkan MO, 4
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
2002 AMC 10, 2
For the nonzero numbers $ a$, $ b$, $ c$, define
\[(a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}.\]
Find $ (2,12,9)$.
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
1992 Poland - First Round, 4
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
2002 China Team Selection Test, 2
Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.
2001 China Team Selection Test, 3
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
2002 Romania National Olympiad, 4
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that:
\[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \]
Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.
2007 China Team Selection Test, 3
Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$
2014 Online Math Open Problems, 25
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i]
1991 Arnold's Trivium, 93
Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.
2009 Costa Rica - Final Round, 1
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
2009 Tuymaada Olympiad, 4
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
2015 Balkan MO Shortlist, A4
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$
(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$
(Albania)