Found problems: 25757
2017 Taiwan TST Round 3, 2
Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$
2003 AIME Problems, 4
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2012 Saint Petersburg Mathematical Olympiad, 2
Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$
2016 Tournament Of Towns, 3
The quadrilateral $ABCD$ is inscribed in circle $\Omega$ with center $O$, not lying on either of the diagonals. Suppose that the circumcircle of triangle $AOC$ passes through the midpoint of the diagonal $BD$. Prove that the circumcircle of triangle $BOD$ passes through the midpoint of diagonal $AC$.
[i](A. Zaslavsky)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
2007 Polish MO Finals, 1
1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$
LMT Team Rounds 2021+, 15
In triangle $ABC$ with $AB = 26$, $BC = 28$, and $C A = 30$, let $M$ be the midpoint of $AB$ and let $N$ be the midpoint of $C A$. The circumcircle of triangle $BCM$ intersects $AC$ at $X\ne C$, and the circumcircle of triangle $BCN $intersects $AB$ at $Y\ne B$. Lines $MX$ and $NY$ intersect $BC$ at $P$ and $Q$, respectively. The area of quadrilateral $PQY X$ can be expressed as $\frac{p}{q}$ for positive integers $p$ and $q$ such that gcd$(p,q) = 1$. Find $q$.
2023 BMT, 8
A circle intersects equilateral triangle $\vartriangle XY Z$ at $A,$ $B$, $C$, $D$, $E$, and $F$ such that points $X$, $A$, $B$, $Y$ , $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. If $AC^2 +CE^2 +EA^2 = 1900$ and $BD^2 + DF^2 + FB^2 = 2092$, compute the positive difference between the areas of triangles $\vartriangle ACE$ and $\vartriangle BDF$.
2013 India IMO Training Camp, 3
For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.
1997 IMO Shortlist, 5
Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.
2004 Germany Team Selection Test, 2
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
STEMS 2021-22 Math Cat A-B, A5 B5
Let $\triangle ABC$ be an acute angled triangle. Let $G$ be the centroid and let $D$ be the foot of the altitude from $A$ onto $BC$. Let ray $GD$ intersect $(ABC)$ at $X$ and let $AG$ intersect nine point circle at $Y$ not on $BC$. Let $Z$ be the intersection of the $\text{A-tangent}$ to $(ABC)$ and $\text{A-midline}$. Prove that perpendicular from $Z$ to the Euler line, $AX$ and $DY$ concur.
The line joining the midpoints of $AB$ and $AC$ is called the $\text{A-midline}$.
$(ABC)$ denotes the circumcircle of $\triangle ABC$
2000 Hong kong National Olympiad, 1
Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.
1955 Moscow Mathematical Olympiad, 317
A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?
2023 Paraguay Mathematical Olympiad, 2
Aidée draws ten squares of different sizes. The diagonal of the first square measures $1$ cm, the diagonal of the second measures $2$ cm, the diagonal of the third measures $3$ cm, and so on until the diagonal of the tenth square measures $10$ cm. How much are the areas of the ten squares?
2020 Brazil Cono Sur TST, 2
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
1979 Chisinau City MO, 182
Prove that a section of a cube by a plane cannot be a regular pentagon.
2012 India PRMO, 7
In $\vartriangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?
1999 AIME Problems, 14
Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2014 Canadian Mathematical Olympiad Qualification, 7
A bug is standing at each of the vertices of a regular hexagon $ABCDEF$. At the same time each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in originally to the vertex it picked. All bugs travel at the same speed and are of negligible size. Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move to the vertices so that no two bugs are ever in the same spot at the same time?
2005 Baltic Way, 15
Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.
2019 ASDAN Math Tournament, 4
Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?
2016 JBMO TST - Turkey, 4
In a trapezoid $ABCD$ with $AB<CD$ and $AB \parallel CD$, the diagonals intersect each other at $E$. Let $F$ be the midpoint of the arc $BC$ (not containing the point $E$) of the circumcircle of the triangle $EBC$. The lines $EF$ and $BC$ intersect at $G$. The circumcircle of the triangle $BFD$ intersects the ray $[DA$ at $H$ such that $A \in [HD]$. The circumcircle of the triangle $AHB$ intersects the lines $AC$ and $BD$ at $M$ and $N$, respectively. $BM$ intersects $GH$ at $P$, $GN$ intersects $AC$ at $Q$. Prove that the points $P, Q, D$ are collinear.
2021 Taiwan TST Round 2, 3
Let $ABC$ be a scalene triangle, and points $O$ and $H$ be its circumcenter and orthocenter, respectively. Point $P$ lies inside triangle $AHO$ and satisfies $\angle AHP = \angle POA$. Let $M$ be the midpoint of segment $\overline{OP}$. Suppose that $BM$ and $CM$ intersect with the circumcircle of triangle $ABC$ again at $X$ and $Y$, respectively.
Prove that line $XY$ passes through the circumcenter of triangle $APO$.
[i]Proposed by Li4[/i]
2005 Vietnam Team Selection Test, 3
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
2020 Taiwan TST Round 2, 1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)