Found problems: 25757
2003 Poland - Second Round, 5
Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.
2018 JHMT, 1
Let $m$ be the area and let $n$ be the perimeter of a regular octagon. The ratio $\frac{m^2}{n}$ can be expressed as $p \tan (q \pi)$ where $p$ is a positive integer. Find $pq$.
1974 Yugoslav Team Selection Test, Problem 2
Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.
Denmark (Mohr) - geometry, 2010.1
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
2023 Belarusian National Olympiad, 9.8
On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different.
Find the smallest possible amount of the sum of all written numbers.
2003 China Western Mathematical Olympiad, 2
A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.
Swiss NMO - geometry, 2008.5
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
2014 Iran Geometry Olympiad (senior), 3:
Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $.
Author:Ali zo'alam , Iran
2003 May Olympiad, 4
Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$.
Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.
2000 Irish Math Olympiad, 2
In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that:
$ R^2\equal{}\frac{(ab\plus{}cd)(ac\plus{}bd)(ad\plus{}bc)}{16Q^2}$.
Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.
2017 Sharygin Geometry Olympiad, P9
Let $C_0$ be the midpoint of hypotenuse $AB$ of triangle $ABC$; $AA_1, BB_1$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_0I$ and $A_1B_1$ meet on the altitude from $C$.
[i]Proposed by A.Zaslavsky[/i]
1994 Portugal MO, 5
Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?
1988 Irish Math Olympiad, 3
$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.
1998 Harvard-MIT Mathematics Tournament, 5
A man named Juan has three rectangular solids, each having volume $128$. Two of the faces of one solid have areas $4$ and $32$. Two faces of another solid have areas $64$ and $16$. Finally, two faces of the last solid have areas $8$ and $32$. What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed.)
2023 All-Russian Olympiad Regional Round, 11.5
Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.
2014 Federal Competition For Advanced Students, P2, 6
Let $U$ be the center of the circumcircle of the acute-angled triangle $ABC$. Let $M_A, M_B$ and $M_C$ be the circumcenters of triangles $UBC, UAC$ and $UAB$ respecrively. For which triangles $ABC$ is the triangle $M_AM_BM_C$ similar to the starting triangle (with a suitable order of the vertices)?
Ukrainian From Tasks to Tasks - geometry, 2013.13
In the quadrilateral $ABCD$ it is known that $ABC + DBC = 180^o$ and $ADC + BDC = 180^o$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the diagonal $AC$.
VII Soros Olympiad 2000 - 01, 8.6
Three cyclists started simultaneously on three parallel straight paths (at the time of the start, the athletes were on the same straight line). Cyclists travel at constant speeds. $1$ second after the start, the triangle formed by the cyclists had an area of $5$ m$^2$. What area will such a triangle have in $10$ seconds after the start?
2017 Taiwan TST Round 1, 4
Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$.\\
Prove that $I$, $A$, $M$ are collinear.
2008 South East Mathematical Olympiad, 2
Circle $I$ is the incircle of $\triangle ABC$. Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$. Show that $M,N,D$ are collinear.
2009 Indonesia TST, 3
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.
2009 Middle European Mathematical Olympiad, 9
Let $ ABCD$ be a parallelogram with $ \angle BAD \equal{} 60$ and denote by $ E$ the intersection of its diagonals. The circumcircle of triangle $ ACD$ meets the line $ BA$ at $ K \ne A$, the line $ BD$ at $ P \ne D$ and the line $ BC$ at $ L\ne C$. The line $ EP$ intersects the circumcircle of triangle $ CEL$ at points $ E$ and $ M$. Prove that triangles $ KLM$ and $ CAP$ are congruent.
2022 Yasinsky Geometry Olympiad, 5
Let $X$ be an arbitrary point on side $BC$ of triangle ABC. Triangle $T$ formed by the bisectors of the angles $ABC$, $ACB$ and $AXC$. Prove that:
a) the circumscribed circle of the triangle $T$ passes through the vertex $A$.
b) the orthocenter of triangle $T$ lies on line $BC$.
(Dmytro Prokopenko)
2016 Dutch BxMO TST, 3
Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.
(a) Prove that $\vartriangle ABC \sim \vartriangle DEF$.
(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.
2018 Germany Team Selection Test, 3
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.