Found problems: 25757
2017 Istmo Centroamericano MO, 1
Let $ABC$ be a triangle with $\angle ABC = 90^o$ and $AB> BC$. Let $D$ be a point on side $AB$ such that $BD = BC$. Let $E$ be the foot of the perpendicular from $D$ on $AC$, and $F$ the reflection of $B$ wrt $CD$. Show that $EC$ is the bisector of angle $\angle BEF$.
1987 AMC 12/AHSME, 27
A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.)
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $
Durer Math Competition CD 1st Round - geometry, 2014.C4
$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?
Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3.1
Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.
PEN R Problems, 6
Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.
1997 All-Russian Olympiad, 3
The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then
the number of possible coverings is even.
[i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]
2015 Iran MO (3rd round), 1
Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.
1975 Chisinau City MO, 94
A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.
2000 Belarus Team Selection Test, 1.4
A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$.
Prove that $\ell \le \sin \frac{2\pi}{5}$
.
2021 BMT, 8
Let $\overline{AB}$ be a line segment with length $10$. Let $P$ be a point on this segment with $AP = 2$. Let $\omega_1$ and $\omega_2$ be the circles with diameters $\overline{AP}$ and $\overline{P B}$, respectively. Let $XY$ be a line externally tangent to $\omega_1$ and $\omega_2$ at distinct points $X$ and $Y$ , respectively. Compute the area of $\vartriangle XP Y$ .
1999 All-Russian Olympiad, 3
A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.
1995 India National Olympiad, 1
In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.
1924 Eotvos Mathematical Competition, 3
Let $A$, $B$, and $C$ be three given points in the plane; construct three cirdes, $k_1$, $k_2$, and $k_3$, such that $k_2$ and $k_3$ have a common tangent at $A$, $k_3$ and $k_1$ at $B$, and $k_1$ and $k_2$ at $C$.
2014 Saint Petersburg Mathematical Olympiad, 2
All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$
2002 AIME Problems, 15
Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB=12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF=AG=8,$ and $GF=6;$ and face $CDE$ has $CE=DE=14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
1969 IMO Shortlist, 21
$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$
2010 Sharygin Geometry Olympiad, 4
Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.
2005 Bulgaria Team Selection Test, 1
Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.
2009 South africa National Olympiad, 2
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?
KoMaL A Problems 2017/2018, A. 702
Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z $). Of all the elegant triangles, which one has the smallest perimeter?
2013 Singapore Senior Math Olympiad, 1
In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC
2024 PErA, P2
Let $ABCD$ be a fixed convex quadrilateral. Say a point $K$ is [i]pastanaga[/i] if there's a rectangle $PQRS$ centered at $K$ such that $A\in PQ, B\in QR, C\in RS, D\in SP$. Prove there exists a circle $\omega$ depending only on $ABCD$ that contains all pastanaga points.
2015 Korea - Final Round, 2
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively.
Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively.
Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$.
Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.
1955 AMC 12/AHSME, 14
The length of rectangle R is $ 10$ percent more than the side of square S. The width of the rectangle is $ 10$ percent less than the side of the square. The ratio of the areas, R:S, is:
$ \textbf{(A)}\ 99: 100 \qquad
\textbf{(B)}\ 101: 100 \qquad
\textbf{(C)}\ 1: 1 \qquad
\textbf{(D)}\ 199: 200 \qquad
\textbf{(E)}\ 201: 200$
Revenge EL(S)MO 2024, 2
In $\triangle ABC$ let $D$ and $E$ be points on $AB$ and $AC$ respectively. The circumcircle of $\triangle CDE$ meets $AB$ again at $F$, and the circumcircle of $\triangle ACD$ meets $BC$ again at $G$. Show that if the circumcircles of $DFG$ and $ADE$ meet at $H$, then the three lines $AG$, $BE$, and $DH$ concur.
Proposed by [i]Oron Wang[/i] inspired by [i]Tiger Zhang[/i]