Found problems: 1546
1986 China Team Selection Test, 2
Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that:
i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}.
ii) The same as above replacing "area" for "perimeter".
1988 IMO Longlists, 43
Find all plane triangles whose sides have integer length and whose incircles have unit radius.
1988 China National Olympiad, 5
Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.
2005 China Team Selection Test, 2
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
1992 APMO, 2
In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$.
Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
1987 IMO Longlists, 47
Through a point $P$ within a triangle $ABC$ the lines $l, m$, and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$, $m$ intersects $AC$ in $R$, and $n$ intersects $AB$ in $S$, then the points $Q, R$, and $S$ are collinear.
2004 Postal Coaching, 16
Find all regular $n$-gons with the following properties:
(a) a diagonal is equal to the sum of two other diagonals
(b) a diagonal is equal to the sum of a side and another diagonal
2005 MOP Homework, 6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.
2018 Bundeswettbewerb Mathematik, 3
Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.
2012 Indonesia TST, 3
Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.
1998 Iran MO (3rd Round), 2
Let $ABCD$ be a convex pentagon such that
\[\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.\]
Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that
\[\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.\]
1991 IMTS, 5
Show that it is impossible to dissect an arbitary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.
2011 Kazakhstan National Olympiad, 5
On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?
2003 Bulgaria Team Selection Test, 1
Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.
2001 Korea - Final Round, 2
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.
2006 Czech-Polish-Slovak Match, 1
Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.
2009 Germany Team Selection Test, 2
Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$
2013 Kazakhstan National Olympiad, 3
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
1986 IMO Longlists, 49
Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows:
$C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
2014 Iran Team Selection Test, 1
suppose that $O$ is the circumcenter of acute triangle $ABC$.
we have circle with center $O$ that is tangent too $BC$ that named $w$
suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$)
$T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$.
$S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$.
prove that $ST$ is tangent $ABC$.
2013 Romania Team Selection Test, 2
Let $K$ be a convex quadrangle and let $l$ be a line through the point of intersection of the diagonals of $K$. Show that the length of the segment of intersection $l\cap K$ does not exceed the length of (at least) one of the diagonals of $K$.
2011 Czech-Polish-Slovak Match, 3
Points $A$, $B$, $C$, $D$ lie on a circle (in that order) where $AB$ and $CD$ are not parallel. The length of arc $AB$ (which contains the points $D$ and $C$) is twice the length of arc $CD$ (which does not contain the points $A$ and $B$). Let $E$ be a point where $AC=AE$ and $BD=BE$. Prove that if the perpendicular line from point $E$ to the line $AB$ passes through the center of the arc $CD$ (which does not contain the points $A$ and $B$), then $\angle ACB = 108^\circ$.
1996 South africa National Olympiad, 3
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.
2007 Olympic Revenge, 3
The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.