Found problems: 25757
1992 India Regional Mathematical Olympiad, 5
$ABCD$ is a quadrilateral and $P,Q$ are the midpoints of $CD, AB, AP, DQ$ meet at $X$ and $BP, CQ$ meet at $Y$. Prove that $A[ADX]+A[BCY] = A[PXOY]$.
2009 Harvard-MIT Mathematics Tournament, 3
Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?
1949-56 Chisinau City MO, 26
Formulate a criterion for the conguence of triangles by two medians and an altitude.
2007 Thailand Mathematical Olympiad, 3
A triangle $\vartriangle ABC$ has $\angle B = 90^o$ . A circle is tangent to $AB$ at $B$ and also tangent to $AC$. Another circle is tangent to the first circle as well as the two sides $AB$ and $AC$. Suppose that $AB =\sqrt3$ and $BC = 3$. What is the radius of the second circle?
1983 Spain Mathematical Olympiad, 2
Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.
2022 Cyprus TST, 3
Let $ABC$ be an obtuse-angled triangle with $ \angle ABC>90^{\circ}$, and let $(c)$ be its circumcircle. The internal angle bisector of $\angle BAC$ meets again the circle $(c)$ at the point $E$, and the line $BC$ at the point $D$. The circle of diameter $DE$ meets the circle $(c)$ at the point $H$.
If the line $HE$ meets the line $BC$ at the point $K$, prove that:
(a) the points $K, H, D$ and $A$ are concyclic
(b) the line $AH$ passes through the point of intersection of the tangents to the circle $(c)$ at the points $B$ and $C$.
2012 NIMO Problems, 5
In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.
[i]Proposed by Lewis Chen[/i]
1980 AMC 12/AHSME, 26
Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals
$\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6$
2013 National Olympiad First Round, 17
Let $ABC$ be an equilateral triangle with side length $10$ and $P$ be a point inside the triangle such that $|PA|^2+ |PB|^2 + |PC|^2 = 128$. What is the area of a triangle with side lengths $|PA|,|PB|,|PC|$?
$
\textbf{(A)}\ 6\sqrt 3
\qquad\textbf{(B)}\ 7 \sqrt 3
\qquad\textbf{(C)}\ 8 \sqrt 3
\qquad\textbf{(D)}\ 9 \sqrt 3
\qquad\textbf{(E)}\ 10 \sqrt 3
$
2012 ELMO Shortlist, 6
In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear.
[i]Ray Li.[/i]
MBMT Team Rounds, 2020.42
$\vartriangle ABC$ has side lengths $AB = 4$ and $AC = 9$. Angle bisector $AD$ bisects angle $A$ and intersects $BC$ at $D$. Let $k$ be the ratio $\frac{BD}{AB}$ . Given that the length $AD$ is an integer, find the sum of all possible $k^2$
.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
1978 Vietnam National Olympiad, 3
The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4} AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.
2022 May Olympiad, 5
Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that:
$\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed,
$\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed,
$\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed.
Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.
2013 Sharygin Geometry Olympiad, 6
Dear Mathlinkers,
1. A, B the end points of an arch circle
2. (O) a circle tangent to AB intersecting the arch in question
3. T the point of contact of (O) and AB
4. C, D the points of intersection of (O) with the arch in the order A, D, C, B
5. E, F the points of intersection of AC and DT, BD and CT.
Prove : EF is parallel to AB.
Sincerely
Jean-Louis
Durer Math Competition CD Finals - geometry, 2015.D1
From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.
2004 Putnam, B4
Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.
2018 Singapore Senior Math Olympiad, 2
In a convex quadrilateral $ABCD, \angle A < 90^o, \angle B < 90^o$ and $AB > CD$. Points $P$ and $Q$ are on the segments $BC$ and $AD$ respectively. Suppose the triangles $APD$ and $BQC$ are similar. Prove that $AB$ is parallel to $CD$.
2012 Iran Team Selection Test, 1
Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.
[i]Proposed by Hesam Rajabzade[/i]
1998 French Mathematical Olympiad, Problem 1
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of
$$BC^6+BD^6-AC^6-AD^6.$$
2015 China Girls Math Olympiad, 6
Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$. $F$ is a point on $\Gamma_1$, the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$. $MG$ is a line tangent to $\Gamma_2$ at $G$. Prove that $MF=MG$.
1986 AMC 12/AHSME, 18
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$
2014 May Olympiad, 4
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$
2019 Lusophon Mathematical Olympiad, 3
Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.
2010 Sharygin Geometry Olympiad, 2
Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle.
(Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)