Found problems: 25757
1976 Polish MO Finals, 3
Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.
2011 Kazakhstan National Olympiad, 5
Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.
2006 MOP Homework, 2
Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.
2017 Oral Moscow Geometry Olympiad, 5
The inscribed circle of the non-isosceles triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint of the $AC$ lie on one straight line.
2023 Germany Team Selection Test, 1
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.
1984 IMO Longlists, 65
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
1983 National High School Mathematics League, 4
In a tetrahedron, lengths of six edges are $2,3,3,4,5,5$. Find its largest volume.
II Soros Olympiad 1995 - 96 (Russia), 9.9
Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.
2019 ELMO Shortlist, G5
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
1954 AMC 12/AHSME, 27
A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:
$ \textbf{(A)}\ \frac{1}{1} \qquad
\textbf{(B)}\ \frac{1}{2} \qquad
\textbf{(C)}\ \frac{2}{3} \qquad
\textbf{(D)}\ \frac{2}{1} \qquad
\textbf{(E)}\ \sqrt{\frac{5}{4}}$
2018 Sharygin Geometry Olympiad, 5
Let $\omega$ be the incircle of a triangle $ABC$. The line passing though the incenter $I$ and parallel to $BC$ meets $\omega$ at $A_b$ and $A_c$ ($A_b$ lies in the same semi plane with respect to $AI$ as $B$). The lines $BA_b$ and $CA_c$ meet at $A_1$. The points $B_1$ and $C_1$ are defined similarly. prove that $AA_1,BB_1,CC_1$ concur.
2014 AIME Problems, 11
In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.
2009 Puerto Rico Team Selection Test, 1
By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
2018 Korea - Final Round, 6
Twenty ants live on the faces of an icosahedron, one ant on each side, where the icosahedron have each side with length 1. Each ant moves in a counterclockwise direction on each face, along the side/edges. The speed of each ant must be no less than 1 always. Also, if two ants meet, they should meet at the vertex of the icosahedron. If five ants meet at the same time at a vertex, we call that a [i]collision[/i]. Can the ants move forever, in a way that no [i]collision[/i] occurs?
2002 Romania National Olympiad, 3
Let $ABCD$ be a trapezium and $AB$ and $CD$ be it's parallel edges. Find, with proof, the set of interior points $P$ of the trapezium which have the property that $P$ belongs to at least two lines each intersecting the segments $AB$ and $CD$ and each dividing the trapezium in two other trapezoids with equal areas.
2014 Balkan MO Shortlist, G3
Let $\triangle ABC$ be an isosceles.$(AB=AC)$.Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$,$E\in DC$ and $2\angle DAE = \angle BAC$.Prove that we can construct a triangle $XYZ$ such that $XY=BD$,$YZ=DE$ and $ZX=EC$.Find $\angle BAC + \angle YXZ$.
1996 Tournament Of Towns, (493) 6
In an equilateral triangle $ABC$, let $D$ be a point on the side $AB$ such that $AD = AB /n$. Prove that the sum of $n - 1$ angles $\angle DP_lA$, $\angle DP_2A$, $...$, $\angle DP_nA$ where $P_1$, $P_2$, $...$ ,$P_{n-1}$ are the points dividing the side $BC$ into $n$ equal parts, is equal to $30$ degrees if
(a) $n = 3$
(b) $n$ is an arbitrary integer, $n > 2$.
(V Proizvolov)
1952 AMC 12/AHSME, 33
A circle and a square have the same perimeter. Then:
$ \textbf{(A)}\ \text{their areas are equal} \qquad\textbf{(B)}\ \text{the area of the circle is the greater}$
$ \textbf{(C)}\ \text{the area of the square is the greater}$
$ \textbf{(D)}\ \text{the area of the circle is } \pi \text{ times the area of the square} \\
\qquad\textbf{(E)}\ \text{none of these}$
2021 Science ON grade VI, 1
Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$.
[i](Vlad Robu)[/i]
2010 Contests, 3
Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.
2023 China Girls Math Olympiad, 8
Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define
$$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$.
Prove that $\lambda \geq 44$ and provide an example in which the equality holds.
2013 Romanian Masters In Mathematics, 1
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
2021 Bundeswettbewerb Mathematik, 3
We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$.
Show that the line $DE$ bisects the angle $\angle BFC$.
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
2020 CCA Math Bonanza, T3
Five unit squares are arranged in a plus shape as shown below:
[asy]
size(3cm);
real s=0.1;
draw(s*(0,1)--s*(0,2));
draw(s*(1,0)--s*(1,3));
draw(s*(2,0)--s*(2,3));
draw(s*(3,1)--s*(3,2));
draw(s*(1,0)--s*(2,0));
draw(s*(0,1)--s*(3,1));
draw(s*(0,2)--s*(3,2));
draw(s*(1,3)--s*(2,3));
[/asy]
What is the area of the smallest circle containing the interior and boundary of the plus shape?
[i]2020 CCA Math Bonanza Team Round #3[/i]