Found problems: 25757
2009 Purple Comet Problems, 14
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD, AB$ has length $1,$ and $CD$ has length $41.$ Let points $X$ and $Y$ lie on sides $AD$ and $BC,$ respectively, such that $XY$ is parallel to $AB$ and $CD,$ and $XY$ has length $31.$ Let $m$ and $n$ be two relatively prime positive integers such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $\tfrac{m}{n}.$ Find $m+2n.$
2011 China Second Round Olympiad, 1
Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.
Cono Sur Shortlist - geometry, 1993.6
Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.
2023 Sharygin Geometry Olympiad, 8.6
For which $n$ the plane may be paved by congruent figures bounded by $n$ arcs of circles?
2019 Belarusian National Olympiad, 10.2
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$.
Prove that the lines $BC$ and $B_1C_1$ are parallel.
[i](A. Voidelevich)[/i]
1990 Tournament Of Towns, (270) 4
The sides $AB$, $BC$, $CD$ and $DA$ of the quadrilateral $ABCD$ are respectively equal to the sides $A'B'$, $B'C'$, $C'D' $ and $D'A'$ of the quadrilateral $A'B'CD$' and it is known that $AB \parallel CD$ and $B'C' \parallel D'A'$. Prove that both quadrilaterals are parallelograms.
(V Proizvolov, Moscow)
2014 Harvard-MIT Mathematics Tournament, 4
In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$
2013 ELMO Shortlist, 2
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2012 CentroAmerican, 2
Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
II Soros Olympiad 1995 - 96 (Russia), 9.3
Two straight lines are drawn on a plane, intersecting at an angle of $40^o$. A circle with center at point $O$ touches these lines. Let's consider a line, different from the given ones, tangent to the same circle and intersecting the given lines at points $B$ and $C$. What can the angle $\angle BOC$ be equal to?
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 $CD$.
2010 Kazakhstan National Olympiad, 6
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$
Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$
2024 Sharygin Geometry Olympiad, 9.7
Let $P$ and $Q$ be arbitrary points on the side $BC$ of triangle ABC such that $BP = CQ$. The common points of segments $AP$ and $AQ$ with the incircle form a quadrilateral $XYZT$. Find the locus of common points of diagonals of such quadrilaterals.
1997 Croatia National Olympiad, Problem 4
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.
1985 Traian Lălescu, 1.2
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
1985 Dutch Mathematical Olympiad, 4
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
1955 AMC 12/AHSME, 9
A circle is inscribed in a triangle with sides $ 8$, $ 15$, and $ 17$. The radius of the circle is:
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 7$
2022 Novosibirsk Oral Olympiad in Geometry, 4
Fold the next seven corners into a rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]
2020 Baltic Way, 13
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.
1996 AMC 8, 22
The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(3,2)--(4,3)--cycle);
label("$A$",(0,0),SW);
label("$B$",(3,2),SE);
label("$C$",(4,3),NE);
[/asy]
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$
1994 Tournament Of Towns, (426) 3
Two-mutually perpendicular lines $\ell$ and $m$ intersect each other at a point of the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1$,$2$,$3$) is taken on each arc so that the tangent line to the circumference at the point $M_i$ intersects $\ell$ and $m$ in two points at the same distance from $M_i$ (that is $M_i$ is the midpoint of the segment between them). Prove that the triangle $M_1M_2M_3$ is equilateral.
(Przhevalsky)
2010 Switzerland - Final Round, 2
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
2023 Bulgarian Autumn Math Competition, 11.2
The points $A_1, B_1, C_1$ are chosen on the sides $BC, CA, AB$ of a triangle $ABC$ so that $BA_1=BC_1$ and $CA_1=CB_1$. The lines $C_1A_1$ and $A_1B_1$ meet the line through $A$, parallel to $BC$, at $P, Q$. Let the circumcircles of the triangles $APC_1$ and $AQB_1$ meet at $R$. Given that $R$ lies on $AA_1$, show that $R$ lies on the incircle of $ABC$.
2011 Hanoi Open Mathematics Competitions, 11
Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$. Calculate the area of the quadrilateral.
2008 Germany Team Selection Test, 3
A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary.
[i]Author: Kei Irie, Japan[/i]