Found problems: 85335
1990 Polish MO Finals, 1
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy
\[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]
1988 All Soviet Union Mathematical Olympiad, 470
There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?
2024 Middle European Mathematical Olympiad, 1
Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.
2019 Saudi Arabia JBMO TST, 4
Let $a, b, c$ be positive reals. Prove that
$a/b+b/c+c/a=>(c+a)/(c+b) + (a+b)/(a+c) + (b+c)/(b+a)$
2014 Costa Rica - Final Round, 2
Let $p_1,p_2, p_3$ be positive numbers such that $p_1 + p_2 + p_3 = 1$. If $a_1 <a_2 <a_3$ and $b_1 <b_2 <b_3$ prove that
$$(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)$$
1971 Dutch Mathematical Olympiad, 1
Given a trapezoid $ABCD$, where sides $AB$ and $CD$ are parallel; the points $P$ on $AD$ and $Q$ on $BC$ lie such that the lines $AQ$ and $CP$ are parallel. Prove that lines $PB$ and $DQ$ are parallel.
2004 Regional Olympiad - Republic of Srpska, 3
Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$
and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.
1993 China Team Selection Test, 1
For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$
2014 Singapore Senior Math Olympiad, 29
Find the number of ordered triples of real numbers $(x,y,z)$ that satisfy the following systems of equations:
$x^2=4y-4,y^2=4z-4,z^2=4x-4$
2001 South africa National Olympiad, 1
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)
2010 Today's Calculation Of Integral, 536
Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.
1969 IMO Longlists, 38
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
2010 Vietnam National Olympiad, 5
Let a positive integer $n$.Consider square table $3*3$.One use $n$
colors to color all cell of table such that
each cell is colored by exactly one color.
Two colored table is same if we can receive them from other by a rotation
through center of $3*3$ table
How many way to color this square table satifies above conditions.
2005 Canada National Olympiad, 2
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$.
$a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$.
$b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.
2010 239 Open Mathematical Olympiad, 7
In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.
1966 IMO Longlists, 46
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
1997 Slovenia National Olympiad, Problem 3
Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.
2016 AIME Problems, 9
Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac15$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.
1997 Israel National Olympiad, 7
A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.
1979 Chisinau City MO, 170
The numbers $a_1,a_2,...,a_n$ ( $n\ge 3$) satisfy the relations $$a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)$$ Prove that the numbers $a_1,a_2,...,a_n$ are non-negative.
1967 AMC 12/AHSME, 23
If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{no finite number}$
2012 Vietnam National Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct.
(a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$
(b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.
1999 Polish MO Finals, 1
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
2022-IMOC, C5
Define a "ternary sequence" is a sequence that every number is $0,1$ or $2$. ternary sequence $(x_1,x_2,x_3,\cdots,x_n)$, define its difference to be $$(|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|)$$ A difference will make the length of the sequence decrease by $1$, so we define the "feature value" of a ternary sequence with length $n$ is the number left after $n-1$ differences. How many ternary sequences has length $2023$ and feature value $0$?
[i]Proposed by CSJL[/i]
2008 Gheorghe Vranceanu, 2
Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.