Found problems: 25757
Champions Tournament Seniors - geometry, 2003.1
Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.
2017 Sharygin Geometry Olympiad, P23
Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
2015 BMT Spring, 7
$X_1, X_2, . . . , X_{2015}$ are $2015$ points in the plane such that for all $1 \le i, j \le 2015$, the line segment $X_iX_{i+1} = X_jX_{j+1}$ and angle $\angle X_iX_{i+1}X_{i+2} = \angle X_jX_{j+1}X_{j+2}$ (with cyclic indices such that $X_{2016} = X_1$ and $X_{2017} = X_2$). Given fixed $X_1$ and $X_2$, determine the number of possible locations for $X_3$.
2020 Israel National Olympiad, 5
Two triangles $ACE, BDF$ are given which intersect at six points: $G, H, I, J, K, L$ as in the picture. It is known that in each of the quadrilaterals
\[ABIK ,BCJL ,CDKG ,DELH ,EFGI\]
it is possible to inscribe a circle. Is it possible for the quadrilateral $FAHJ$ is also circumscribed around a circle?
2008 AIME Problems, 9
A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.
1993 All-Russian Olympiad Regional Round, 11.6
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2001 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.
2022 JHMT HS, 8
In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.
1982 IMO Shortlist, 2
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$
2008 ISI B.Stat Entrance Exam, 4
Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$
2014 AMC 10, 14
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
$ \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 $
Ukraine Correspondence MO - geometry, 2013.9
Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.
Geometry Mathley 2011-12, 14.2
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.
Đỗ Thanh Sơn
1998 Baltic Way, 13
In convex pentagon $ABCDE$, the sides $AE,BC$ are parallel and $\angle ADE=\angle BDC$. The diagonals $AC$ and $BE$ intersect at $P$. Prove that $\angle EAD=\angle BDP$ and $\angle CBD=\angle ADP$.
2011 Sharygin Geometry Olympiad, 4
Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
VMEO III 2006, 12.1
Given a triangle $ABC$ and a point $K$ . The lines $AK$,$BK$,$CK$ hit the opposite side of the triangle at $D,E,F$ respectively. On the exterior of $ABC$, we construct three pairs of similar triangles: $BDM$,$DCN$ on $BD$,$DC$, $CEP$,$EAQ$ on $CE$,$EA$, and $AFR$,$FBS$ on $AF$, $FB$. The lines $MN$,$PQ$,$RS$ intersect each other form a triangle $XYZ$. Prove that $AX$,$BY$,$CZ$ are concurrent.
2007 Romania National Olympiad, 2
Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if \[\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}\]
2020 Malaysia IMONST 1, 18
In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest
side has length $12$. What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side?
2015 Saint Petersburg Mathematical Olympiad, 4
$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram
2018 Moscow Mathematical Olympiad, 5
On the sides of the convex hexagon $ABCDEF$ into the outer side were built equilateral triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$. The triangle $B_1D_1F_1$ is equilateral too. Prove that, the triangle $A_1C_1E_1$ is also equilateral.
1999 USAMTS Problems, 4
There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.
2016 PUMaC Geometry B, 3
Let $ABCD$ be a square with side length $8$. Let $M$ be the midpoint of $BC$ and let $\omega$ be the circle passing through $M, A$, and $D$. Let $O$ be the center of $\omega, X$ be the intersection point (besides A) of $\omega$ with $AB$, and $Y$ be the intersection point of $OX$ and $AM$. If the length of $OY$ can be written in simplest form as $\frac{m}{n}$ , compute $m + n$.
2017 Math Prize for Girls Problems, 10
Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.
1957 Polish MO Finals, 2
Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle there is a relationship $$ a^2 \cos^2 A = b^2 \cos^2 B + c^2 \cos^2 C + 2bc \cos B \cos C \cos 2A.$$