Found problems: 25757
2003 District Olympiad, 1
Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.
1996 Tournament Of Towns, (489) 2
An exterior common tangent to two non-intersecting circles with centers and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point on the line $A_1A_2$ such that $CD$ is perpendicular to $B_1B_2$. Prove that $A_1D = DA_2$.
2018 Vietnam Team Selection Test, 1
Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly.
a. Prove that if $PO=PO'$ then $O\in(A'B'C')$;
b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.
2009 Nordic, 1
A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given triangle. Prove that $f\ge\frac{1}{3}$ and determine those points $P$ for which $f =\frac{1}{3}$ .
2010 Contests, 1
Let $\gamma,\Gamma$ be two concentric circles with radii $r,R$ with $r<R$. Let $ABCD$ be a cyclic quadrilateral inscribed in $\gamma$. If $\overrightarrow{AB}$ denotes the Ray starting from $A$ and extending indefinitely in $B's$ direction then Let $\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} , \overrightarrow{DA}$ meet $\Gamma$ at the points $C_1,D_1,A_1,B_1$ respectively. Prove that
\[\frac{[A_1B_1C_1D_1]}{[ABCD]} \ge \frac{R^2}{r^2}\]
where $[.]$ denotes area.
2001 Turkey MO (2nd round), 2
Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.
2014 NIMO Problems, 5
In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$?
[i]Proposed by Evan Chen[/i]
2014 Contests, 2
Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered [u]clockwise[/u]. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$.
Prove that $\Delta A_3MN$ is an equilateral triangle.
2019 Belarus Team Selection Test, 5.2
Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.
Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.
[i](M. Karpuk)[/i]
Estonia Open Senior - geometry, 2014.1.4
In a plane there is a triangle $ABC$. Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$. Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$. The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$. Prove that the triangle $ABC$ is equilateral.
1963 Polish MO Finals, 2
In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.
2008 Bosnia And Herzegovina - Regional Olympiad, 1
Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that:
(i) Lines $ BC$ and $ AA'$ are orthogonal.
(ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid
2009 Ukraine National Mathematical Olympiad, 4
In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$
2007 South africa National Olympiad, 4
Let $ ABC$ be a triangle and $ PQRS$ a square with $ P$ on $ AB$, $ Q$ on $ AC$, and $ R$ and $ S$ on $ BC$. Let $ H$ on $ BC$ such that $ AH$ is the altitude of the triangle from $ A$ to base $ BC$. Prove that:
(a) $ \frac{1}{AH} \plus{}\frac{1}{BC}\equal{}\frac{1}{PQ}$
(b) the area of $ ABC$ is twice the area of $ PQRS$ iff $ AH\equal{}BC$
2008 JBMO Shortlist, 1
Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.
2010 Cuba MO, 3
Let $ABC$ be a right triangle at $B$. Let $D$ be a point such that $BD\perp AC$ and $DC = AC$. Find the ratio $\frac{AD}{AB}$.
2025 AIME, 14
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$
2002 Flanders Junior Olympiad, 4
Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?
[img]https://cdn.artofproblemsolving.com/attachments/a/8/807bb5b760caaa600f0bac95358963a902b1e7.png[/img]
2017 CCA Math Bonanza, T7
Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$.
[i]2017 CCA Math Bonanza Team Round #7[/i]
2021 Saint Petersburg Mathematical Olympiad, 6
Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent.
[i]A. Kuznetsov[/i]
2021 ITAMO, 5
Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$.
Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.
2009 Harvard-MIT Mathematics Tournament, 6
Let $ABC$ be a triangle in the coordinate plane with vertices on lattice points and with $AB = 1$. Suppose the perimeter of $ABC$ is less than $17$. Find the largest possible value of $1/r$, where $r$ is the inradius of $ABC$.
2024 New Zealand MO, 3
Let $A,B,C,D,E$ be five different points on the circumference of a circle in that (cyclic) order. Let $F$ be the intersection of chords $BD$ and $CE$. Show that if $AB=AE=AF$ then lines $AF$ and $CD$ are perpendicular.
2000 Romania National Olympiad, 4
In the rectangular parallelepiped $ABCDA'B'C'D'$, the points $E$ and $F$ are the centers of the faces $ABCD$ and $ADD' A'$, respectively, and the planes $(BCF)$ and $(B'C'E)$ are perpendicular. Let $A'M \perp B'A$, $M \in B'A$ and $BN \perp B'C$, $N \in B'C$. Denote $n = \frac{C'D}{BN}$.
a) Show that $n \ge \sqrt2$. .
b) Express and in terms of $n$, the ratio between the volume of the tetrahedron $BB'M N$ and the volume of the parallelepiped $ABCDA'B'C'D'$.
2014 JBMO Shortlist, 6
Let $ABCD$ be a quadrilateral whose diagonals are not perpendicular and whose sides $AB$ and $CD$ are not parallel.Let $O$ be the intersection of its diagonals.Denote with $H_1$ and $H_2$ the orthocenters of triangles $AOB$ and $COD,$ respectively.If $M$ and $N$ are the midpoints of the segment lines $AB$ and $CD,$ respectively.Prove that the lines $H_1H_2$ and $MN$ are parallel if and only if $AC=BD.$