Olympiad problems

2006 AMC 8, 18

Tags: geometry , 3d geometry

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? $ \textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{5}{9} \qquad \textbf{(E)}\ \dfrac{19}{27}$

1993 Tournament Of Towns, (390) 2

Tags: geometry , right triangle , angle

Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees. (Folklore)

2023 Thailand TSTST, 1

Tags: geometry

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.

2023 Bangladesh Mathematical Olympiad, P4

Tags: geometry , orthocenter

Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.

XMO (China) 2-15 - geometry, 11.1

Tags: geometry

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

2020 CHKMO, 3

Tags: geometry , incircle , circumcircle

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

2003 Tournament Of Towns, 6

Tags: geometry

An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?

2015 Romania Team Selection Test, 2

Tags: circles , triangle inequality , inradius , inequalities , geometry

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2023 Kyiv City MO Round 1, Problem 3

Tags: circumcircle , geometry

Let $I$ be the incenter of triangle $ABC$ with $AB < AC$. Point $X$ is chosen on the external bisector of $\angle ABC$ such that $IC = IX$. Let the tangent to the circumscribed circle of $\triangle BXC$ at point $X$ intersect the line $AB$ at point $Y$. Prove that $AC = AY$. [i]Proposed by Oleksiy Masalitin[/i]

2002 Junior Balkan MO, 2

Tags: geometry , trapezoid

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

2016 Romanian Masters in Mathematic, 1

Tags: geometry

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).

2024 Romanian Master of Mathematics, 3

Tags: geometry , combinatorics , linear algebra , combinatorial geometry , set

Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

1990 Tournament Of Towns, (276) 4

Tags: combinatorics , combinatorial geometry , geometry

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

2009 Germany Team Selection Test, 1

Tags: geometry

Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with \[ \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}.\] How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD}$ is minimal?

2005 APMO, 3

Tags: algorithm , geometry

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

2014 Portugal MO, 2

Tags: geometry

Let $[ABCD]$ be a square, $M$ a point on the segment $[AD]$, and $N$ a point on the segment $[DC]$ such that $B\hat{M}A = N\hat{M}D = 60^{\circ}$. Calculate $M\hat{B}N$.

2015 JBMO Shortlist, 5

Tags: geometry , similar triangles , incircle

Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$ onto ${BC}$ meets ${EF}$ at ${M}$, and similarly the perpendicular line erected at ${B}$ onto ${BC}$ meets ${EF}$ at ${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$. Ruben Dario & Leo Giugiuc (Romania)

2014 Iran Team Selection Test, 6

Tags: geometry , incenter , circumcircle , trigonometry , projective geometry , geometric transformation , homothety

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

2022 Junior Balkan Team Selection Tests - Romania, P1

Tags: geometry

Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\][list=a] [*]Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$ [*]Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$ [/list]

2016 Sharygin Geometry Olympiad, 4

Tags: combinatorics , geometry

One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

2005 Argentina National Olympiad, 4

Tags: number theory , perfect cube , perfect square , geometry , 3d geometry

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

2008 Oral Moscow Geometry Olympiad, 5

Tags: geometry , polyhedron , 3d geometry

There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)? (S. Markelov).

1977 Chisinau City MO, 137

Tags: equal angles , geometry , angle

Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertex, divide this angle into four equal parts.

2016 South African National Olympiad, 3

Tags: geometry

The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$ and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.

2008 Bulgarian Autumn Math Competition, Problem 10.2

Tags: geometry , midpoint , area of a triangle

Let $\triangle ABC$ have $M$ as the midpoint of $BC$ and let $P$ and $Q$ be the feet of the altitudes from $M$ to $AB$ and $AC$ respectively. Find $\angle BAC$ if $[MPQ]=\frac{1}{4}[ABC]$ and $P$ and $Q$ lie on the segments $AB$ and $AC$.