Found problems: 1581
2014 Iran Team Selection Test, 6
$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively .
Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$.
Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$.
Prove that the circumcenter of $AIT$ is on the $BC$.
2012 All-Russian Olympiad, 4
The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.
2014 Baltic Way, 11
Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$
2010 ELMO Shortlist, 1
Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent.
[i]Carl Lian.[/i]
2006 China Team Selection Test, 3
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
2005 Turkey Team Selection Test, 2
Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .
1974 IMO Longlists, 51
There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.
2007 South East Mathematical Olympiad, 2
$AB$ is the diameter of semicircle $O$. $C$,$D$ are two arbitrary points on semicircle $O$. Point $P$ lies on line $CD$ such that line $PB$ is tangent to semicircle $O$ at $B$. Line $PO$ intersects line $CA$, $AD$ at point $E$, $F$ respectively. Prove that $OE$=$OF$.
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$
2006 China Team Selection Test, 1
The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively.
Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.
1997 Hungary-Israel Binational, 3
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
2016 CCA Math Bonanza, I6
Let $a,b,c$ be non-zero real numbers. The lines $ax + by = c$ and $bx + cy = a$ are perpendicular and intersect at a point $P$ such that $P$ also lies on the line $y=2x$. Compute the coordinates of point $P$.
[i]2016 CCA Math Bonanza Individual #6[/i]
2023 Switzerland - Final Round, 1
Let $ABC$ be an acute triangle with incenter $I$. On its circumcircle, let $M_A$, $M_B$ and $M_C$ be the midpoints of minor arcs $BC, CA$ and $AB$, respectively. Prove that the reflection $M_A$ over the line $IM_B$ lies on the circumcircle of the triangle $IM_BM_C$.
2004 Tournament Of Towns, 1
Let us call a triangle rational if each of its angles is a rational number when measured in degrees. Let us call a point inside triangle rational if joining it to the three vertices of the triangle we get three rational triangles. Show that any acute rational triangle contains at least three distinct rational points.
2005 Georgia Team Selection Test, 2
In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.
2009 USA Team Selection Test, 3
For each positive integer $ n$, let $ c(n)$ be the largest real number such that
\[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\]
for all triples $ (f, a, b)$ such that
--$ f$ is a polynomial of degree $ n$ taking integers to integers, and
--$ a, b$ are integers with $ f(a) \neq f(b)$.
Find $ c(n)$.
[i]Shaunak Kishore.[/i]
1967 AMC 12/AHSME, 40
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:
$\textbf{(A)}\ 159\qquad
\textbf{(B)}\ 131\qquad
\textbf{(C)}\ 95\qquad
\textbf{(D)}\ 79\qquad
\textbf{(E)}\ 50$
2011 China Team Selection Test, 1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
2014 Middle European Mathematical Olympiad, 6
Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.
Prove that the points $B, C, N,$ and $L$ are concyclic.
1994 Moldova Team Selection Test, 7
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:
$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
2014 PUMaC Algebra B, 3
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]
2007 Mongolian Mathematical Olympiad, Problem 3
Let $P$ be a point outside of the triangle $ABC$ in the plane of $ABC$. Prove that by using reflections $S_{AB}$, $S_{AC}$, and $S_{BC}$ across the lines $AB$, $AC$, and $BC$ one can shift point $P$ inside the triangle $ABC$.
2008 Sharygin Geometry Olympiad, 7
(A.Zaslavsky) The circumradius of triangle $ ABC$ is equal to $ R$. Another circle with the same radius passes through the orthocenter $ H$ of this triangle and intersect its circumcirle in points $ X$, $ Y$. Point $ Z$ is the fourth vertex of parallelogram $ CXZY$. Find the circumradius of triangle $ ABZ$.
2013 NIMO Problems, 3
Let $ABC$ be a triangle. Prove that there exists a unique point $P$ for which one can find points $D$, $E$ and $F$ such that the quadrilaterals $APBF$, $BPCD$, $CPAE$, $EPFA$, $FPDB$, and $DPEC$ are all parallelograms.
[i]Proposed by Lewis Chen[/i]
2015 Iran MO (3rd round), 3
Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.