Found problems: 85335
2019 IMC, 9
Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$.
[i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]
1900 Eotvos Mathematical Competition, 2
Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r$ of its inscribed circle, and the radius $r_c$ of its ex-circle tangent to the side $AB$ and the extensions of $BC$ and $CA$.
2005 MOP Homework, 4
Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.
2016 Bosnia And Herzegovina - Regional Olympiad, 2
Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$
1996 South africa National Olympiad, 3
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.
2011 QEDMO 9th, 9
In a very long corridor there is an infinite number of cabinets, which start with $1,2,3,...$ numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set $A \subseteq \{1, 2,3,...\}$ . In ascending order, the QED people now cause chaos: the person with number $a \in A$ visits the cabinet with the numbers $a,2a,3a,...$ opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from $A$ open.
2005 Oral Moscow Geometry Olympiad, 4
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.
(M. Volchkevich)
1975 Vietnam National Olympiad, 2
Solve this equation
$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
2010 Math Prize For Girls Problems, 15
Compute the value of the sum
\begin{align*}
\frac{1}{1 + \tan^3 0^\circ}
&+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ}
+ \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\
&+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ}
+ \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ}
\, .
\end{align*}
2006 MOP Homework, 1
There are $2005$ players in a chess tournament played a game. Every pair of players played a game against each other. At the end of the tournament, it turned out that if two players $A$ and $B$ drew the game between them, then every other player either lost to $A$ or to $B$. Suppose that there are at least two draws in the tournament. Prove that all players can be lined up in a single file from left to right in the such a way that every play won the game against the person immediately to his right.
2022 Novosibirsk Oral Olympiad in Geometry, 3
Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are $50^o$, $60^o$ and $70^o$. Find the angles of the original triangle.
2002 India IMO Training Camp, 17
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
2015 Indonesia MO, 6
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
2006 Sharygin Geometry Olympiad, 11
In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $ A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.
2008 IberoAmerican, 3
Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.
2001 China Team Selection Test, 2
In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.
1978 Romania Team Selection Test, 6
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.
2024 Malaysian Squad Selection Test, 8
Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$.
Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent.
[i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]
2011 China Northern MO, 8
It is known that $n$ is a positive integer, and the real number $x$ satisfies $$|1-|2-...|(n-1)-|n-x||...||=x.$$ Find the value of $x$.
2007 Dutch Mathematical Olympiad, 4
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.)
2021 Azerbaijan Junior NMO, 2
Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.
2014 Iran MO (3rd Round), 4
$2 \leq d$ is a natural number.
$B_{a,b}$={$a,a+b,a+2b,...,a+db$}
$A_{c,q}$={$cq^n \vert n \in\mathbb{N}$}
Prove that there are finite prime numbers like $p$ such exists $a,b,c,q$ from natural numbers :
$i$ ) $ p \nmid abcq $
$ ii$ ) $A_{c,q} \equiv B_{a,b} (mod p ) $
(15 points )
1979 IMO Longlists, 76
Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$. Find the sides of the triangle.
1972 Miklós Schweitzer, 2
Let $ \leq$ be a reflexive, antisymmetric relation on a finite set $ A$. Show that this relation can be extended to an appropriate finite superset $ B$ of $ A$ such that $ \leq$ on $ B$ remains reflexive, antisymmetric, and any two elements of $ B$ have a least upper bound as well as a greatest lower bound. (The relation $ \leq$ is extended to $ B$ if for $ x,y \in A , x \leq y$ holds in $ A$ if and only if it holds in $ B$.)
[i]E. Freid[/i]