Olympiad problems

2000 All-Russian Olympiad, 3

A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary.

2006 Lithuania Team Selection Test, 3

Tags: geometry , geometry unsolved

Inside a convex quadrilateral $ABCD$ there is a point $P$ such that the triangles $PAB, PBC, PCD, PDA$ have equal areas. Prove that the area of $ABCD$ is bisected by one of the diagonals.

2011 Czech and Slovak Olympiad III A, 1

Tags: geometry unsolved , geometry

In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.

2006 India Regional Mathematical Olympiad, 1

Tags: geometry , geometry unsolved

Let $ ABC$ be an acute-angled triangle and let $ D,E,F$ be the feet of perpendiculars from $ A,B,C$ respectively to $ BC,CA,AB .$ Let the perpendiculars from $ F$ to $ CB,CA,AD,BE$ meet them in $ P,Q,M,N$ respectively. Prove that the points $ P,Q,M,N$ are collinear.

1986 IMO Longlists, 21

Tags: geometry unsolved , geometry

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

2006 South africa National Olympiad, 4

Tags: geometry unsolved , geometry

In triangle $ABC$, $AB=AC$ and $B\hat{A}C=100^\circ$. Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$. Prove that $AD+DB=BC$.

2010 Indonesia MO, 8

Tags: geometry , circumcircle , ratio , geometry unsolved

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

1991 Federal Competition For Advanced Students, P2, 1

Tags: geometry , geometry unsolved

Consider a convex solid $ K$ in space and two parallel planes $ \epsilon _1$ and $ \epsilon _2$ on the distance $ 1$ tangent to $ K$. A plane $ \epsilon$ between $ \epsilon _1$ and $ \epsilon _2$ is on the distance $ d_1$ from $ \epsilon _1$. Find all $ d_1$ such that the part of $ K$ between $ \epsilon _1$ and $ \epsilon$ always has a volume not exceeding half the volume of $ K$.

2002 China Team Selection Test, 2

Tags: inequalities , trigonometry , geometry unsolved , geometry

$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1 \]

2010 South East Mathematical Olympiad, 3

Tags: geometry , geometry unsolved

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$

2006 MOP Homework, 5

Tags: search , function , geometry unsolved , geometry

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180$.

2013 Gulf Math Olympiad, 2

Tags: trigonometry , geometry unsolved , geometry

In triangle $ABC$, the bisector of angle $B$ meets the opposite side $AC$ at $B'$. Similarly, the bisector of angle $C$ meets the opposite side $AB$ at $C'$ . Prove that $A=60^{\circ}$ if, and only if, $BC'+CB'=BC$.

2011 Kazakhstan National Olympiad, 1

Tags: ratio , trigonometry , geometry unsolved , geometry

Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

2003 Tournament Of Towns, 6

Tags: geometry , 3D geometry , tetrahedron , parallelogram , inradius , geometry unsolved

Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

2011 Spain Mathematical Olympiad, 1

Tags: ratio , geometry , trapezoid , trigonometry , perpendicular bisector , angle bisector , geometry unsolved

In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

2007 India IMO Training Camp, 1

Tags: Euler , geometry , trigonometry , perpendicular bisector , geometry unsolved

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

2006 MOP Homework, 2

Tags: geometry , circumcircle , search , geometry unsolved

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

2011 JBMO Shortlist, 6

Tags: geometry , rectangle , inequalities , trigonometry , vector , triangle inequality , geometry unsolved

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2012 Indonesia TST, 3

Tags: quadratics , geometry , rectangle , vector , geometry unsolved

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

2004 China Team Selection Test, 1

Tags: geometry unsolved , geometry

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

2005 MOP Homework, 3

Tags: geometry , power of a point , radical axis , geometry unsolved

Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ and $S_2$ again $C$ and $D$ respectively. Circle $S_3$ having $CD$ as its diameter meets $S_1$ and $S_2$ again at $P$ and $Q$ respectively. Prove that lines $CP$, $DQ$, and $AB$ are concurent.

1987 China Team Selection Test, 2

Tags: analytic geometry , geometry , induction , geometry unsolved , Polygons , coordinates , China

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

1994 Hong Kong TST, 1

Tags: trigonometry , geometry unsolved , geometry

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

2006 India IMO Training Camp, 1

Tags: geometry , circumcircle , geometric transformation , reflection , homothety , ratio , geometry unsolved

Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$. [b](a)[/b] Prove that any two of the following statements imply the third. [list] [b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$. [b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$. [b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list] [b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

2000 Brazil Team Selection Test, Problem 1

Tags: geometry unsolved , geometry

Show that if the sides $a, b, c$ of a triangle satisfy the equation \[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\] then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.