Found problems: 6530
2004 Uzbekistan National Olympiad, 3
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
2002 Poland - Second Round, 3
Find all positive integers $n$ such that for all real numbers $x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n$ the following inequality holds:
\[ x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot \]
2008 IberoAmerican Olympiad For University Students, 7
Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$).
Prove that each subgroup of $A$ of finite index is isomorphic to $A$.
1986 China National Olympiad, 1
We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.
Brazil L2 Finals (OBM) - geometry, 2003.5
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
2010 China Team Selection Test, 3
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
1981 USAMO, 3
If $A,B,C$ are the angles of a triangle, prove that
\[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\]
and determine when equality holds.
2009 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that
$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$
2025 All-Russian Olympiad Regional Round, 10.5
The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$.
[i]A. Kuznetsov[/i]
1985 Austrian-Polish Competition, 1
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
2000 Turkey MO (2nd round), 3
Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.
2004 Putnam, A1
Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N),$ of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than 80% of $N,$ but by the end of the season, $S(N)$ was more than 80% of $N.$ Was there necessarily a moment in between when $S(N)$ was exactly 80% of $N$?
2010 USA Team Selection Test, 2
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]
2010 Contests, 2
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$
\[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]
1990 Baltic Way, 4
Prove that, for any real numbers $a_1, a_2, \dots , a_n$,
\[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]
1996 Bundeswettbewerb Mathematik, 4
Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.
2005 Junior Balkan Team Selection Tests - Romania, 14
Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \]
[i]Robert Szasz[/i]
2011 Postal Coaching, 5
Let $P$ be a point inside a triangle $ABC$ such that
\[\angle P AB = \angle P BC = \angle P CA\]
Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that
\[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]
2007 Postal Coaching, 2
Let $a_1, a_2, a_3$ be three distinct real numbers. Define
$$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\
b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\
b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$
Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$
When does equality hold?
2012 Macedonia National Olympiad, 3
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:
$f(x+y) < f(x) + f(y)$
$f(f(x)) = \lfloor {x} \rfloor + 2$
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2005 Romania Team Selection Test, 2
Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality
\[ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r \] is true for all $r\geq \rho (n)$.
2004 German National Olympiad, 5
Prove that for four positive real numbers $a,b,c,d$ the following inequality holds and find all equality cases:
$$a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.$$
2006 Italy TST, 2
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that
\[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\]
if and only if $ABC$ is acute-angled.
1957 Polish MO Finals, 4
Prove that if $ a \geq 0 $ and $ b \geq 0 $, then
$$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$