Found problems: 2023
2008 Czech-Polish-Slovak Match, 2
$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and $|AB| =|BC|,$ $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines $AD$, $BE$ and $CF$ are concurrent.
2007 ISI B.Math Entrance Exam, 4
Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.
2024 Baltic Way, 14
Let $ABC$ be an acute triangle with circumcircle $\omega$. The altitudes $AD$, $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$. A point $K$ is chosen on the line $EF$ such that $KH\parallel BC$. Prove that the reflection of $H$ in $KD$ lies on $\omega$.
2013 All-Russian Olympiad, 2
The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.
1990 IMO Longlists, 94
Given integer $n > 1$ and real number $t \geq 1$. $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$. Here, ${F_n}$ is the $n$-th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$. Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$, and $M$ be the area of $P$, which is $t^2F_{2n+1}.$
[b][i]i)[/i][/b] Prove that for any integral point $(a, b)$, there exists a unique pair of integers $(j, k)$ such that$ j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$, that is,$ jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$
[i][b]ii)[/b][/i] Using [i][b]i)[/b][/i] or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$
Cono Sur Shortlist - geometry, 2012.G4.2
2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.
2014 Contests, 3
From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
2000 Turkey Team Selection Test, 2
In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$
2006 Baltic Way, 13
In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if
$\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$
then the points $A,D,F,E$ lie on a circle.
2008 Romania Team Selection Test, 1
Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.
2013 Uzbekistan National Olympiad, 5
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
2011 Bosnia Herzegovina Team Selection Test, 3
In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.
2004 China National Olympiad, 1
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.
Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.
[i]Xiong Bin[/i]
2013 Korea National Olympiad, 6
Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.
2018 Greece Team Selection Test, 2
A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ .
We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.
2010 Bosnia Herzegovina Team Selection Test, 4
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
1997 Romania Team Selection Test, 1
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.
[i]Mircea Becheanu[/i]
1993 Iran MO (3rd Round), 2
In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles.
[asy]
import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16;
pair A=(0,0), F=(9,0), B=(4,0), C=(3.5,2), D=(1.94,2.59), O=(2.75,1.57);
draw(A--(3,4),linewidth(1.2)); draw((3,4)--F,linewidth(1.2)); draw(A--F,linewidth(1.2)); draw((3,4)--B,linewidth(1.2)); draw(A--C,linewidth(1.2)); draw(B--D,linewidth(1.2)); draw((3,4)--O,linewidth(1.2)); draw(C--F,linewidth(1.2)); draw(F--O,linewidth(1.2));
dot(A,ds); label("$A$",(-0.28,-0.23),NE*lsf); dot(F,ds); label("$F$",(8.79,-0.4),NE*lsf); dot((3,4),ds); label("$E$",(3.05,4.08),NE*lsf); dot(B,ds); label("$B$",(4.05,0.09),NE*lsf); dot(C,ds); label("$C$",(3.55,2.08),NE*lsf); dot(D,ds); label("$D$",(1.76,2.71),NE*lsf); dot(O,ds); label("$O$",(2.57,1.17),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
2001 Balkan MO, 2
A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.
2022 German National Olympiad, 3
Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$.
Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$.
Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.
2015 Saint Petersburg Mathematical Olympiad, 7
Let $BL$ be angle bisector of acute triangle $ABC$.Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$.point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$.Point $T$ is diametrically opposite the point $K$ on the circumcircle of $\triangle AKC$.Prove that $ST$ passes through midpoint of arc $ABC$.(S. Berlov)
[hide] :trampoline: my 100th post :trampoline: [/hide]
1990 Federal Competition For Advanced Students, P2, 3
In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that:
$ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$
2007 Polish MO Finals, 1
1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$
2007 Bulgaria Team Selection Test, 1
In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$
2008 Kyiv Mathematical Festival, 4
Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?