Found problems: 2023
2007 Bulgaria Team Selection Test, 3
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.
1950 Miklós Schweitzer, 2
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
2004 Iran MO (3rd Round), 24
In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA\equal{}AB$ and $ NB\equal{}NC$. Also $ K,L$ lie on line $ BC$ such that $ KA\equal{}KB$ and $ LA\equal{}LC$. It is know that $ KL\equal{}\frac12{BC}$ and $ MN\equal{}AC$. Find angles of triangle $ ABC$.
1983 IMO Longlists, 73
Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
2011 Greece Team Selection Test, 4
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2008 Polish MO Finals, 5
Let $ R$ be a parallelopiped. Let us assume that areas of all intersections of $ R$ with planes containing centers of three edges of $ R$ pairwisely not parallel and having no common points, are equal. Show that $ R$ is a cuboid.
1997 All-Russian Olympiad, 2
Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M.
[i]A. Khrabrov[/i]
1986 Federal Competition For Advanced Students, P2, 1
Show that a square can be inscribed in any regular polygon.
2014 Grand Duchy of Lithuania, 2
An isosceles triangle $ABC$ with $AC = BC$ is given. Let $M$ be the midpoint of the side $AB$ and let $P$ be a point inside the triangle such that $\angle PAB = \angle PBC$. Prove that $\angle APM + \angle BPC = 180 \textdegree $
2009 Cono Sur Olympiad, 3
Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.
2002 China Team Selection Test, 1
Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.
2011 All-Russian Olympiad, 2
Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.
2012 Junior Balkan Team Selection Tests - Moldova, 3
Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.
2008 Sharygin Geometry Olympiad, 12
(A.Myakishev, 9--10) Given a triangle $ ABC$. Point $ A_1$ is chosen on the ray $ BA$ so that segments $ BA_1$ and $ BC$ are equal. Point $ A_2$ is chosen on the ray $ CA$ so that segments $ CA_2$ and $ BC$ are equal. Points $ B_1$, $ B_2$ and $ C_1$, $ C_2$ are chosen similarly. Prove that lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ are parallel.
2001 Brazil National Olympiad, 5
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
2019 Taiwan TST Round 1, 1
Given a triangle $ \triangle{ABC} $ with orthocenter $ H $. On its circumcenter, choose an arbitrary point $ P $ (other than $ A,B,C $) and let $ M $ be the mid-point of $ HP $. Now, we find three points $ D,E,F $ on the line $ BC, CA, AB $, respectively, such that $ AP \parallel HD, BP \parallel HE, CP \parallel HF $. Show that $ D, E, F, M $ are colinear.
2005 Brazil National Olympiad, 3
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$.
1998 Baltic Way, 14
Given triangle $ABC$ with $AB<AC$. The line passing through $B$ and parallel to $AC$ meets the external angle bisector of $\angle BAC$ at $D$. The line passing through $C$ and parallel to $AB$ meets this bisector at $E$. Point $F$ lies on the side $AC$ and satisfies the equality $FC=AB$. Prove that $DF=FE$.
1993 Irish Math Olympiad, 5
$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$.
$ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?
2003 Iran MO (2nd round), 2
$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that:
\[ BT+CT\leq{2R}. \]
2011 Bulgaria National Olympiad, 3
Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.
1998 Turkey Team Selection Test, 2
In a triangle $ABC$, the circle through $C$ touching $AB$ at $A$ and the circle through $B$ touching $AC$ at $A$ have different radii and meet again at $D$. Let $E$ be the point on the ray $AB$ such that $AB = BE$. The circle through $A$, $D$, $E$ intersect the ray $CA$ again at $F$ . Prove that $AF = AC$.
2011 Northern Summer Camp Of Mathematics, 3
Given an acute triangle $ABC$ such that $\angle C< \angle B< \angle A$. Let $I$ be the incenter of $ABC$. Let $M$ be the midpoint of the smaller arc $BC$, $N$ be the midpoint of the segment $BC$ and let $E$ be a point such that $NE=NI$. The line $ME$ intersects circumcircle of $ABC$ at $Q$ (different from $A, B$, and $C$). Prove that
[b](i)[/b] The point $Q$ is on the smaller arc $AC$ of circumcircle of $ABC$.
[b](ii)[/b] $BQ=AQ+CQ$
2014 Kazakhstan National Olympiad, 3
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.
1990 India National Olympiad, 7
Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let
$ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively.
Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles.
For which position of $ P$ will the triangle $ DEF$ become equilateral?