Olympiad problems

2000 Greece National Olympiad, 1

Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and suﬃcient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.

1985 IMO Longlists, 57

Tags: geometry , 3D geometry , sphere , tetrahedron , geometry unsolved

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$

1988 Vietnam National Olympiad, 3

Tags: geometry , geometric transformation , reflection , geometry unsolved

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

2005 Italy TST, 2

Tags: geometry , geometric transformation , reflection , power of a point , geometry unsolved

The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.

1976 IMO Longlists, 22

Tags: geometry , 3D geometry , sphere , geometry unsolved

A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.

2010 Germany Team Selection Test, 1

Tags: geometry , rhombus , geometry unsolved

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

2007 Germany Team Selection Test, 3

Tags: trigonometry , inequalities , geometry , search , geometry unsolved

Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove: \[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F \] When does equality occur?

2000 Polish MO Finals, 1

Tags: geometry , 3D geometry , pyramid , inequalities , geometry unsolved

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

2004 Croatia Team Selection Test, 3

Tags: geometry , circumcircle , geometry unsolved

A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.

1995 Polish MO Finals, 3

Tags: geometry unsolved , geometry

$PA, PB, PC$ are three rays in space. Show that there is just one pair of points $B', C$' with $B'$ on the ray $PB$ and $C'$ on the ray $PC$ such that $PC' + B'C' = PA + AB'$ and $PB' + B'C' = PA + AC'$.

2007 Bundeswettbewerb Mathematik, 3

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

In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$

2010 Contests, 2

Tags: geometry , circumcircle , trigonometry , parallelogram , geometry unsolved

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

2006 Bundeswettbewerb Mathematik, 3

Tags: geometry , circumcircle , geometry unsolved

A point $P$ is given inside an acute-angled triangle $ABC$. Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

2013 Kurschak Competition, 2

Tags: geometry , geometry unsolved

Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.) (a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$. (b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.

2006 MOP Homework, 5

Tags: geometry , incenter , circumcircle , trigonometry , geometric transformation , reflection , geometry unsolved

Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.

1992 Taiwan National Olympiad, 5

Tags: geometry , circumcircle , incenter , geometry unsolved

A line through the incenter $I$ of triangle $ABC$, perpendicular to $AI$, intersects $AB$ at $P$ and $AC$ at $Q$. Prove that the circle tangent to $AB$ at $P$ and to $AC$ at $Q$ is also tangent to the circumcircle of triangle $ABC$.

2003 Tournament Of Towns, 5

Tags: inequalities , geometry unsolved , geometry

A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

2009 ISI B.Stat Entrance Exam, 3

Tags: geometry , geometry unsolved

Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR, APQ$ and $PQCR$. Find the minimum possible value of $M$.

2010 Kazakhstan National Olympiad, 3

Tags: geometry , trigonometry , geometry unsolved

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

2005 MOP Homework, 7

Tags: inequalities , trigonometry , geometry unsolved , geometry

Let $ABC$ be a triangle. Prove that \[\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).\]

2001 Brazil National Olympiad, 3

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

$ABC$ is a triangle $E, F$ are points in $AB$, such that $AE = EF = FB$ $D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$ $AD$ is perpendicular to $CF$. The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$) Determine the ratio $\frac{DB}{DC}$. %Edited!%

2014 Postal Coaching, 3

Tags: geometry , geometric transformation , homothety , perpendicular bisector , radical axis , power of a point , geometry unsolved

The circles $\mathcal{K}_1,\mathcal{K}_2$ and $\mathcal{K}_3$ are pairwise externally tangent to each other; the point of tangency betwwen $\mathcal{K}_1$ and $\mathcal{K}_2$ is $T$. One of the external common tangents of $\mathcal{K}_1$ and $\mathcal{K}_2$ meets $\mathcal{K}_3$ at points $P$ and $Q$. Prove that the internal common tangent of $\mathcal{K}_1$ and $\mathcal{K}_2$ bisects the arc $PQ$ of $\mathcal{K}_3$ which is closer to $T$.

2009 Poland - Second Round, 1

Tags: geometry , cyclic quadrilateral , geometry unsolved

$ABCD$ is a cyclic quadrilateral inscribed in the circle $\Gamma$ with $AB$ as diameter. Let $E$ be the intersection of the diagonals $AC$ and $BD$. The tangents to $\Gamma$ at the points $C,D$ meet at $P$. Prove that $PC=PE$.

2002 APMO, 3

Tags: geometry unsolved , geometry

Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that the triangle $TRS$ is equilateral.

2011 Mongolia Team Selection Test, 3

Tags: geometry , circumcircle , IMO Shortlist , power of a point , radical axis , geometry unsolved

We are given an acute triangle $ABC$. Let $(w,I)$ be the inscribed circle of $ABC$, $(\Omega,O)$ be the circumscribed circle of $ABC$, and $A_0$ be the midpoint of altitude $AH$. $w$ touches $BC$ at point $D$. $A_0 D$ and $w$ intersect at point $P$, and the perpendicular from $I$ to $A_0 D$ intersects $BC$ at the point $M$. $MR$ and $MS$ lines touch $\Omega$ at $R$ and $S$ respectively [note: I am not entirely sure of what is meant by this, but I am pretty sure it means draw the tangents to $\Omega$ from $M$]. Prove that the points $R,P,D,S$ are concyclic. (proposed by E. Enkzaya, inspired by Vietnamese olympiad problem)