Olympiad problems

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorial geometry , combinatorics , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

1982 National High School Mathematics League, 4

Tags: geometry

What's the area defined by equation $|x-1|+|y-1|=1$? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}\pi\qquad\text{(D)}4$

1949-56 Chisinau City MO, 62

Tags: tetrahedron , 3d geometry , volume , fixed , constant , geometry

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.

2016 PUMaC Geometry A, 4

Tags: geometry

Let $\vartriangle ABC$ be a triangle with integer side lengths such that $BC = 2016$. Let $G$ be the centroid of $\vartriangle ABC$ and $I$ be the incenter of $\vartriangle ABC$. If the area of $\vartriangle BGC$ equals the area of $\vartriangle BIC$, find the largest possible length of $AB$.

2023 Baltic Way, 13

Tags: geometry

Let $ABC$ be an acute triangle with $AB<AC$ and incenter $I$. Let $D$ be the projection of $I$ onto $BC$. Let $H$ be the orthocenter of $ABC$ and suppose that $\angle IDH=\angle CBA-\angle ACB$. Prove that $AH=2ID$.

Geometry Mathley 2011-12, 5.4

Tags: concurrency , circles , concurrent circles , geometry

Let $ABC$ be a triangle inscribed in a circle $(O)$. Let $P$ be an arbitrary point in the plane of triangle $ABC$. Points $A',B',C'$ are the reflections of $P$ about the lines $BC,CA,AB$ respectively. $X$ is the intersection, distinct from $A$, of the circle with diameter $AP$ and the circumcircle of triangle $AB'C'$. Points $Y,Z$ are defined in the same way. Prove that five circles $(O), (AB'C')$, $(BC'A'), (CA'B'), (XY Z)$ have a point in common. Nguyễn Văn Linh

2017 All-Russian Olympiad, 2

Tags: circumcircle , tangent , isosceles , geometry

Let $ABC$ be an acute angled isosceles triangle with $AB=AC$ and circumcentre $O$. Lines $BO$ and $CO$ intersect $AC, AB$ respectively at $B', C'$. A straight line $l$ is drawn through $C'$ parallel to $AC$. Prove that the line $l$ is tangent to the circumcircle of $\triangle B'OC$.

1953 Miklós Schweitzer, 4

Tags: real analysis , geometry

[b]4.[/b] Show that every closed curve c of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. [b](G. 8)[/b]

2022 Assara - South Russian Girl's MO, 8

Tags: concurrency , geometry , hexagon

About the convex hexagon $ABCDEF$ it is known that $AB = BC =CD = DE = EF = FA$ and $AD = BE = CF$. Prove that the diagonals $AD$, $BE$, $CF$ intersect at one point.

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9

Tags: trigonometry , geometry

In the triangle $ ABC$ we have $ AB \equal{} 5$ and $ AC \equal{} 6$. The area of the triangle when the $ \angle ACB$ is as large as possible is $ \text{(A)}\ 15 \qquad \text{(B)}\ 5 \sqrt{7} \qquad \text{(C)}\ \frac{7}{2} \sqrt{7} \qquad \text{(D)}\ 3 \sqrt{11} \qquad \text{(E)}\ \frac{5}{2} \sqrt{11}$

2019 Yasinsky Geometry Olympiad, p2

Tags: incircle , centroid , geometry , isosceles

An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$. (Grigory Filippovsky)

2021 CMIMC, 2.1

Tags: geometry

Triangle $ABC$ has a right angle at $A$, $AB=20$, and $AC=21$. Circles $\omega_A$, $\omega_B$, and $\omega_C$ are centered at $A$, $B$, and $C$ respectively and pass through the midpoint $M$ of $\overline{BC}$. $\omega_A$ and $\omega_B$ intersect at $X\neq M$, and $\omega_A$ and $\omega_C$ intersect at $Y\neq M$. Find $XY$. [i]Proposed by Connor Gordon[/i]

2005 Sharygin Geometry Olympiad, 12

Tags: construction , sidelengths , geometry

Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.

2006 China Second Round Olympiad, 4

Tags: analytic geometry , 3d geometry , prism , graphing lines , slope , geometry

Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is ${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $

2004 China Team Selection Test, 1

Tags: geometry

Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$

1972 IMO Shortlist, 4

Tags: combinatorial geometry , geometry , partition , point set , combinatorics

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

2009 Belarus Team Selection Test, 3

Tags: trigonometry , angle bisector , altitude , geometry

Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$ I. Voronovich

Kyiv City MO Juniors Round2 2010+ geometry, 2016.9.2

Tags: angle bisector , circumcircle , geometry

The bisector of the angle $BAC$of the acute triangle $ABC$ ( $AC \ne AB$) intersects its circumscribed circle for the second time at the point $W$. Let $O$ be the center of the circumscribed circle $\Delta ABC$. The line $AW$ intersects for the second time the circumcribed circles of triangles $OWB$ and $OWC$ at the points $N$ and $M$, respectively. Prove that $BN + MC = AW$. (Mitrofanov V., Hilko D.)

1997 Italy TST, 2

Tags: geometry

Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.

1979 IMO Longlists, 75

Tags: geometry

Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$. $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

2024 Rioplatense Mathematical Olympiad, 1

Tags: geometry , rioplatense

Let $ \triangle ABC $ be a triangle such that $ BC > AC > AB $. A point $ X $ is marked on side $ BC $ such that $ AX = XC $. Let $ Y $ be a point on segment $ AX $ such that $ CY = AB $. Prove that $ \angle CYX = \angle ABC $.

2011 Postal Coaching, 3

Tags: geometry

Let $ABC$ be a scalene triangle. Let $l_A$ be the tangent to the nine-point circle at the foot of the perpendicular from $A$ to $BC$, and let $l_A'$ be the tangent to the nine-point circle from the mid-point of $BC$. The lines $l_A$ and $l_A'$ intersect at $A'$ . Deﬁne $B'$ and $C'$ similarly. Show that the lines $AA' , BB'$ and $CC'$ are concurrent.

2024 Austrian MO Regional Competition, 2

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$. Prove that $H$ is the circumcenter of the triangle $APQ$. [i](Karl Czakler)[/i]

2024 Middle European Mathematical Olympiad, 6

Tags: incenter , geometry

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

Denmark (Mohr) - geometry, 2010.1

Tags: circles , area , geometry , right triangle

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]