Found problems: 25757
2007 Greece Junior Math Olympiad, 1
In a triangle $ABC$ with the incentre $I,$ the angle bisector $AD$ meets the circumcircle of triangle $BIC$ at point $N\neq I$.
a) Express the angles of $\triangle BCN$ in terms of the angles of triangle $ABC$.
b) Show that the circumcentre of triangle $BIC$ is at the intersection of $AI$ and the circumcentre of $ABC$.
2018 Yasinsky Geometry Olympiad, 3
Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$.
(Andrey Mostovy)
2004 Croatia National Olympiad, Problem 2
Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.
2021 Novosibirsk Oral Olympiad in Geometry, 2
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.
2000 Iran MO (3rd Round), 3
A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane,
such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and
$N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove
that there is a point $P$ in the plane from which all the segments $MN$ are
visible at a constant angle.
2011 AIME Problems, 6
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2007 AMC 8, 8
In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$.
[asy]
defaultpen(linewidth(0.7));
pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0);
draw(E--B--C--D--A--B);
draw(rightanglemark(A, D, C));
label("$A$", A, NW);
label("$B$", B, NW);
label("$C$", C, SE);
label("$D$", D, SW);
label("$E$", E, NW);
label("$3$", A--D, W);
label("$3$", A--B, N);
label("$6$", E, S);[/asy]
$\textbf{(A)} \: 3\qquad \textbf{(B)} \: 4.5\qquad \textbf{(C)} \: 6\qquad \textbf{(D)} \: 9\qquad \textbf{(E)} \: 18\qquad $
1981 Miklós Schweitzer, 5
Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$.
[i]J. Szucs[/i]
1990 Mexico National Olympiad, 5
Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.
2016 Hong Kong TST, 2
Let $\Gamma$ be a circle and $AB$ be a diameter. Let $l$ be a line outside the circle, and is perpendicular to $AB$. Let $X$, $Y$ be two points on $l$. If $X'$, $Y'$ are two points on $l$ such that $AX$, $BX'$ intersect on $\Gamma$ and such that $AY$, $BY'$ intersect on $\Gamma$. Prove that the circumcircles of triangles $AXY$ and $AX'Y'$ intersect at a point on $\Gamma$ other than $A$, or the three circles are tangent at $A$.
KoMaL A Problems 2019/2020, A. 766
Let $T$ be any triangle such that its side-lengths $a, b$ and $c$ and its circumradius $R$ are positive integers. Show that:
a) the inradius $r$ of $T$ is a positive integer;
b) the perimeter $P$ of $T$ is a multiple of $4$; and
c) all three of $a, b$ and $c$ are even.
2024 Kosovo Team Selection Test, P2
Let $\omega$ be a circle and let $A$ be a point lying outside of $\omega$. The tangents from $A$ to $\omega$ touch $\omega$ at points $B$ and $C$. Let $M$ be the midpoint of $BC$ and let $D$ a point on the side $BC$ different from $M$. The circle with diameter $AD$ intersects $\omega$ at points $X$ and $Y$ and the circumcircle of $\bigtriangleup ABC$ again at $E$. Prove that $AD$, $EM$, and $XY$ are concurrent.
2014 Greece JBMO TST, 2
Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.
2012 Kyoto University Entry Examination, 4
Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example.
$(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3.
$(q)$ In $\triangle{ABC},\ \triangle{A'B'C'}$, if $AB=A'B',\ BC=B'C',\ \angle{A}=\angle{A'}$, then these triangles are congruent.
30 points
2023 Thailand TSTST, 1
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.
2006 Australia National Olympiad, 1
In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.
2021 Bangladeshi National Mathematical Olympiad, 8
Let $ABC$ be an acute-angled triangle. The external bisector of $\angle{BAC}$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle{PMN}=\angle{MQN}=90^{\circ}$. If $PN=5$ and $BC=3$, then the length of $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $(a+b)$?
1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4
Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img]
A. 1/3
B. $ \frac {1}{\sqrt {5}}$
C. $ \sqrt {3} \minus{} \sqrt {2}$
D. $ 6 \minus{} 4 \sqrt {2}$
E. None of these
2012 ELMO Shortlist, 4
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
[i]Ray Li.[/i]
2005 Tournament of Towns, 3
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.
[i](4 points)[/i]
2012 Poland - Second Round, 2
Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.
2020 MMATHS, I11
Let triangle $\triangle ABC$ have side lengths $AB = 7, BC = 8,$ and $CA = 9$, and let $M$ and $D$ be the midpoint of $\overline{BC}$ and the foot of the altitude from $A$ to $\overline{BC}$, respectively. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle{AEM} = m\angle{AFM} = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle{AED}$ and $\angle{AFD}$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
1952 Kurschak Competition, 3
$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.
1987 Mexico National Olympiad, 5
In a right triangle $ABC$, M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.
2005 Junior Balkan Team Selection Tests - Romania, 16
Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle of center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$.