Olympiad problems

2019 Junior Balkan Team Selection Tests - Romania, 3

A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.

2001 Croatia National Olympiad, Problem 1

Tags: parallelogram , geometry

Let $O$ and $P$ be fixed points on a plane, and let $ABCD$ be any parallelogram with center $O$. Let $M$ and $N$ be the midpoints of $AP$ and $BP$ respectively. Lines $MC$ and $ND$ meet at $Q$. Prove that the point $Q$ lies on the lines $OP$, and show that it is independent of the choice of the parallelogram $ABCD$.

2016 AMC 12/AHSME, 8

Tags: geometry

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece? $\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$

2006 National Olympiad First Round, 33

Tags: geometry , angle bisector

Let $ABCD$ be a convex quadrileteral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. What is $m(\widehat{CAD})$? $ \textbf{(A)}\ 25^\circ \qquad\textbf{(B)}\ 30^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 38^\circ \qquad\textbf{(E)}\ 40^\circ $

2013 Korea National Olympiad, 6

Tags: geometry , circumcircle

Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.

2013 VTRMC, Problem 2

Tags: triangle , geometry

Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$?

2007 Iran MO (3rd Round), 5

Tags: ratio , geometry , trapezoid , rectangle , search , parallelogram , circumcircle

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

2008 Bosnia Herzegovina Team Selection Test, 2

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

Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$. If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.

2017 ELMO Problems, 2

Tags: orthocenter , geometry , circumcircle

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

1987 National High School Mathematics League, 8

Tags: tetrahedron , geometry , 3d geometry

We have two triangles that lengths of its sides are $3,4,5$, one triangle that lengths of its sides are $4,5,\sqrt{41}$, one triangle that lengths of its sides are $\frac{5}{6}\sqrt2,4,5$. The number of tetrahedrons with such four surfaces is________.

2006 Sharygin Geometry Olympiad, 10.1

Tags: geometry , broken line

Five lines go through one point. Prove that there exists a closed five-segment polygonal line, the vertices and the middle of the segments of which lie on these lines, and each line has exactly one vertex.

2008 Oral Moscow Geometry Olympiad, 6

Tags: circumcenter , geometry , similar triangles , collinear

Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$. (A. Zaslavsky)

2006 Romania National Olympiad, 3

Tags: circumcircle , geometry

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$. (a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions. (b) Prove that $PD=r$. [i]Virgil Nicula[/i]

2020 Durer Math Competition Finals, 11

Tags: geometry , angle

The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?

2015 Brazil National Olympiad, 6

Tags: circumcircle , geometry

Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$.

1998 National High School Mathematics League, 1

Tags: incenter , circumcircle , geometry

Circumcenter and incentre of $\triangle ABC$ are $O,I$. $AD$ is the height on side $BC$. If $I$ is on line $OC$, prove that the radius of circumcircle and escribed circle (in \angle BAC) are equal.

2016 German National Olympiad, 3

Tags: orthocenter , parallel , excenter , geometry , circumcircle

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

1989 Tournament Of Towns, (234) 2

Tags: geometric construction , midpoint , geometry

Three points $K, L$ and $M$ are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.

2023 Oral Moscow Geometry Olympiad, 5

Tags: geometry

Altitudes $BB_1$ and $CC_1$ of acute triangle $ABC$ intersect at $H$, and $\angle A = 60^{o}$, $AB < AC$. The median $AM$ intersects the circumcircle of $ABC$ at point $K$; $L$ is the midpoint of the arc $BC$ of the circumcircle that does not contain point $A$; lines $B_1C_1$ and $BC$ intersect at point $E$. Prove that $\angle EHL = \angle ABK$.

Estonia Open Junior - geometry, 2005.1.3

Tags: concurrency , angle bisector , geometry

In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.

2011 Peru IMO TST, 4

Tags: geometry

Let $ABC$ be an acute triangle, and $AA_1$, $BB_1$, and $CC_1$ its altitudes. Let $A_2$ be a point on segment $AA_1$ such that $\angle{BA_2C} = 90^{\circ}$. The points $B_2$ and $C_2$ are defined similarly. Let $A_3$ be the intersection point of segments $B_2C$ and $BC_2$. The points $B_3$ and $C_3$ are defined similarly. Prove that the segments $A_2A_3$, $B_2B_3$, and $C_2C_3$ are concurrent.

2005 IMO Shortlist, 4

Tags: geometry , spiral similarity

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

2023 Greece Junior Math Olympiad, 3

Tags: combinatorial geometry , lattice points , geometry , combinatorics

Find the number of rectangles who have the following properties: a) Have for vertices, points $(x,y)$ of plane $Oxy$ with $x,y$ non negative integers and $ x \le 8$ , $y\le 8$ b) Have sides parallel to axes c) Have area $E$, with $30<E\le 40$

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , line , locus , circles

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

2006 Taiwan National Olympiad, 2

Tags: inequalities , geometry

In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.