Found problems: 25757
2005 Italy TST, 2
The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.
1990 Romania Team Selection Test, 4
The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.
2022 Argentina National Olympiad Level 2, 3
Let $A$, $X$ and $Y$ be three non-collinear points on the plane. Construct with a straightedge and compass a square $ABCD$ such that $X$ is on the line $BC$ and $Y$ is on the line $CD$.
2015 EGMO, 1
Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$
If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$
2009 Dutch Mathematical Olympiad, 4
Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.) Show that the points $P, Q, C$ and $R$ form a parallelogram.
2009 India IMO Training Camp, 4
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
2007 Princeton University Math Competition, 6
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
1964 Spain Mathematical Olympiad, 4
We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?
2024 China Second Round, 2
\(ABCD\) is a convex quadrilateral, \(AC\) bisects the angle \(\angle BAD\). Points \(E\) and \(F\) are on the sides \(BC\) and \(CD\) respectively such that \(EF \parallel BD\). Extend \(FA\) and \(EA\) to points \(P\) and \(Q\) respectively, such that the circle \(\omega_1\) passing through points \(A\), \(B\), \(P\) and the circle \(\omega_2\) passing through points \(A\), \(D\), \(Q\) are both tangent to line \(AC\). Prove that the points \(B\), \(P\), \(Q\), \(D\) are concyclic.
2022 Junior Balkan Team Selection Tests - Romania, P2
Let $ABC$ be an acute scalene triangle. Let $D$ be the foot of the $A$-bisectrix and $E$ be the foot of the $A$-altitude. The perpendicular bisector of the segment $AD$ intersects the semicircles of diameter $AB$ and $AC$ which lie on the outside of triangle $ABC$ at $X$ and $Y$ respectively. Prove that the points $X,Y,D$ and $E$ lie on a circle.
2019 Taiwan TST Round 1, 2
Given a convex pentagon $ ABCDE. $ Let $ A_1 $ be the intersection of $ BD $ with $ CE $ and define $ B_1, C_1, D_1, E_1 $ similarly, $ A_2 $ be the second intersection of $ \odot (ABD_1),\odot (AEC_1) $ and define $ B_2, C_2, D_2, E_2 $ similarly. Prove that $ AA_2, BB_2, CC_2, DD_2, EE_2 $ are concurrent.
[i]Proposed by Telv Cohl[/i]
2011 ISI B.Stat Entrance Exam, 10
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled.
2022 Kosovo National Mathematical Olympiad, 3
Let $ABCD$ be a parallelogram and $l$ the line parallel to $AC$ which passes through $D$. Let $E$ and $F$ points on $l$ such that $DE=DF=DB$. Show that $EA,FC$ and $BD$ are concurrent.
2011 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
2006 Sharygin Geometry Olympiad, 8.1
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
2021 USEMO, 3
Let $A_1C_2B_1A_2C_1B_2$ be an equilateral hexagon. Let $O_1$ and $H_1$ denote the circumcenter and orthocenter of $\triangle A_1B_1C_1$, and let $O_2$ and $H_2$ denote the circumcenter and orthocenter of $\triangle A_2B_2C_2$. Suppose that $O_1 \ne O_2$ and $H_1 \ne H_2$. Prove that the lines $O_1O_2$ and $H_1H_2$ are either parallel or coincide.
[i]Ankan Bhattacharya[/i]
2019 Latvia Baltic Way TST, 9
Let $ABCD$ be a rhombus with the condition $\angle ABC > 90^o$. The circle $\Gamma_B$ with center at $B$ goes through $C$, and the circle $\Gamma_C$ with center at $C$ goes through $B$. Denote by $E$ one of the intersection points of $\Gamma_B$ and $\Gamma_C$. The line $ED$ intersects intersects $\Gamma_B$ again at $F$. Find the value of $\angle AFB$.
2022 Indonesia Regional, 4
Suppose $ABC$ is a triangle with circumcenter $O$. Point $D$ is the reflection of $A$ with respect to $BC$. Suppose $\ell$ is the line which is parallel to $BC$ and passes through $O$. The line through $B$ and parallel to $CD$ meets $\ell$ at $B_1$. Lines $CB_1$ and $BD$ intersect at point $B_2$. The line through $C$ parallel to $BD$ and $\ell$ meet at $C_1$. Finally, $BC_1$ and $CD$ intersects at point $C_2$. Prove that points $A, B_2, C_2, D$ lie on a circle.
2010 Today's Calculation Of Integral, 522
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
2024 Brazil National Olympiad, 2
Let \( ABC \) be a scalene triangle. Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \), respectively, and let \( D \) be any point on segment \( BC \). The circumcircles of triangles \( BDF \) and \( CDE \) intersect line \( EF \) at points \( K \neq F \), and \( L \neq E \), respectively, and intersect at points \( X \neq D \). The point \( Y \) is on line \( DX \) such that \( AY \) is parallel to \( BC \). Prove that points \( K \), \( L \), \( X \), and \( Y \) lie on the same circle.
2010 National Olympiad First Round, 9
Let $E$ be a point outside of square $ABCD$. If the distance of $E$ to $AC$ is $6$, to $BD$ is $17$, and to the nearest vertex of the square is $10$, what is the area of the square?
$ \textbf{(A)}\ 200
\qquad\textbf{(B)}\ 196
\qquad\textbf{(C)}\ 169
\qquad\textbf{(D)}\ 162
\qquad\textbf{(E)}\ 144
$
1996 Romania National Olympiad, 4
In the triangle $ABC$ the incircle $J$ touches the sides $BC$, $CA$, $AB$ in $D$, $E$, $F$, respectively. The segments $(BE)$ and $(CF)$ intersect $J$ in $G,H$. If $B$ and $C$ are fixed points, find the loci of points $A, D, E, F, G, H$ if $GH \parallel BC$ and the loci of the same points if $BCHG$ is an inscriptible quadrilateral.
2008 Cuba MO, 6
We have an isosceles triangle $ABC$ with base $BC$. Through vertex $A$ draw a line $r$ parallel to $BC$. The points $P, Q$ are located on the perpendicular bisectors of $AB$ and $AC$ respectively, such that $PQ\perp BC$. They are points $M$ and $N$ on the line $r$ such that $\angle APM = \angle AQN = 90^o$. Prove that $$\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}$$
1954 AMC 12/AHSME, 8
The base of a triangle is twice as long as a side of a square and their areas are the same. Then the ratio of the altitude of the triangle to the side of the square is:
$ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{2} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 4$
1999 Croatia National Olympiad, Problem 1
In a triangle $ABC$, the inner and outer angle bisectors at $C$ intersect the line $AB$ at $L$ and $M$, respectively. Prove that if $CL=CM$ then $AC^2+BC^2=4R^2$, where $R$ is the circumradius of $\triangle ABC$.