Found problems: 235
2011 JBMO Shortlist, 6
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
1987 IMO Shortlist, 6
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then
\[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \]
[i]Proposed by Greece.[/i]
1980 IMO Shortlist, 15
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
1987 IMO Longlists, 33
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then
\[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \]
[i]Proposed by Greece.[/i]
2018 JBMO TST-Turkey, 8
Let $x, y, z$ be positive real numbers such that
$\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$.
Prove that
$\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
2006 IMO Shortlist, 8
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$.
[i]Proposed by Waldemar Pompe, Poland[/i]
2002 Baltic Way, 12
A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that
$|XY|=|XZ|+|XW|$
Prove that all four points lie on a line.
2003 Brazil National Olympiad, 1
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 $ABCD$.
1996 Spain Mathematical Olympiad, 3
Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)| \leq 2 $ for $ -1 \leq x \leq 1 $.
MathLinks Contest 7th, 4.1
Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that
\[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]
2013 Bogdan Stan, 1
$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $
[b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle.
[b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $
[i]Vasile Pop[/i]
2005 Hong kong National Olympiad, 2
Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.
2008 India Regional Mathematical Olympiad, 6
Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$.
[16 points out of 100 for the 6 problems]
2011 Junior Balkan MO, 4
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
2009 ISI B.Math Entrance Exam, 9
Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.
2004 IMO Shortlist, 1
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
1968 IMO, 1
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
1980 Austrian-Polish Competition, 3
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
2010 Czech-Polish-Slovak Match, 3
Let $ABCD$ be a convex quadrilateral for which \[ AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.\] Prove that $ABCD$ is a parallelogram.
2015 Romania Team Selection Test, 2
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.
1999 Polish MO Finals, 2
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.
2000 India National Olympiad, 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
1983 IMO, 3
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
2011 District Olympiad, 4
Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have:
\[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\]
for all $x,y\in [0,1]$.