Olympiad problems

2021 Mexico National Olympiad, 4

Tags: geometry , orthocenter , euler line , circumcircle , tangent circle

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. [list] [*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. [*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. [/list] [i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

1961 AMC 12/AHSME, 36

Tags: pythagorean theorem , geometry

In triangle $ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$. ${{ \textbf{(A)}\ 4\qquad\textbf{(B)}\ \sqrt{17} \qquad\textbf{(C)}\ 4.25\qquad\textbf{(D)}\ 2\sqrt{5} }\qquad\textbf{(E)}\ 4.5} $

1962 All Russian Mathematical Olympiad, 020

Tags: pentagon , minimum , maximum , geometry

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

2022 South East Mathematical Olympiad, 6

Tags: geometry

$H$ is the orthocenter of $\triangle ABC$,the circle with center $H$ passes through $A$,and it intersects with $AC,AB$ at two other points $D,E$.The orthocenter of $\triangle ADE$ is $H'$,line $AH'$ intersects with $DE$ at point $F$.Point $P$ is inside the quadrilateral $BCDE$,so that $\triangle PDE\sim\triangle PBC$.Let point $K$ be the intersection of line $HH'$ and line $PF$.Prove that $A,H,P,K$ lie on one circle. [img]https://i.ibb.co/mcyhxRM/graph.jpg[/img]

2016 Fall CHMMC, 9

Tags: geometry

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2013 NIMO Problems, 2

Tags: geometry , perimeter , probability , analytic geometry , graphing lines , slope , geometric transformation

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

2021 Ukraine National Mathematical Olympiad, 4

Tags: geometry , circumcenter

Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$. (Fedir Yudin)

2009 Sharygin Geometry Olympiad, 2

Tags: perimeter , inequalities , geometry

Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?

2000 Harvard-MIT Mathematics Tournament, 20

Tags: geometry , perimeter , hmmt , trigonometry

What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

2022 Kyiv City MO Round 2, Problem 3

Tags: ratio , geometry

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. [i](Proposed by Mykhailo Bondarenko)[/i]

2015 District Olympiad, 4

Tags: pure geometry , 3d geometry , geometry

Consider the rectangular parallelepiped $ ABCDA'B'C'D' $ and the point $ O $ to be the intersection of $ AB' $ and $ A'B. $ On the edge $ BC, $ pick a point $ N $ such that the plane formed by the triangle $ B'AN $ has to be parallel to the line $ AC', $ and perpendicular to $ DO'. $ Prove, then, that this parallelepiped is a cube.

Novosibirsk Oral Geo Oly VIII, 2021.2

Tags: geometry , angle

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

Mathematical Minds 2024, P8

Tags: olympiad geometry , excircle , incircle , collinearity , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$, incircle $\omega$, and $A$-excircle $\omega_A$. Let $X$ and $Y$ be the tangency points of $\omega_A$ with $AB$ and $AC$. Lines $XY$ and $BC$ intersect in $T$. The tangent from $T$ to $\omega$ different from $BC$ intersects $\omega$ at $K$. The radical axis of $\omega_A$ and $\Omega$ intersects $BC$ in $S$. The tangent from $S$ to $\omega_A$ different from $BC$ intersects $\omega_A$ at $L$. Prove that $A$, $K$ and $L$ are collinear. [i]Proposed by Ana Boiangiu[/i]

2007 Tournament Of Towns, 7

Tags: ratio , trigonometry , geometric transformation , reflection , geometry

$T$ is a point on the plane of triangle $ABC$ such that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Prove that the lines symmetric to $AT, BT$ and $CT$ with respect to $BC, CA$ and $AB$, respectively, are concurrent.

2022 Brazil EGMO TST, 2

Tags: geometry , angle

Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?

1984 IMO, 2

Tags: algebra , geometry , convex polygon , perimeter , geometric inequality

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

2018 India PRMO, 24

Tags: integer , geometry , angle , combinatorics

If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

2007 Sharygin Geometry Olympiad, 9

Tags: geometry , angle chasing , convex quadrilateral , angle chase

Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,

2022 LMT Spring, 1

Tags: algebra , geometry

Derek and Jacob have a cake in the shape a rectangle with dimensions $14$ inches by $9$ inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let $D : J$ be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where $D$ and $J$ are relatively prime positive integers. Find $D + J$.

2012 Baltic Way, 11

Tags: geometry

Let $ABC$ be a triangle with $\angle A = 60^\circ$. The point $T$ lies inside the triangle in such a way that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Let $M$ be the midpoint of $BC$. Prove that $TA + TB + TC = 2AM$.

2006 Junior Tuymaada Olympiad, 1

Tags: geometry , right triangle , isosceles , equal segments

On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.

1992 Tournament Of Towns, (347) 5

Tags: geometry , fixed , fixed point

An angle with vertex $O$ and a point $A$ inside it are placed on a plane. Points $M$ and $N$ are chosen on different sides of the angle so that the angles $CAM$ and $CAN$ are equal. Prove that the straight line $MN$ always passes through a fixed point (or is always parallel to a fixed line). (S Tokarev)

Novosibirsk Oral Geo Oly VII, 2023.5

Tags: geometric inequality , geometry

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

2012 USA TSTST, 2

Tags: geometry

Let $ABCD$ be a quadrilateral with $AC = BD$. Diagonals $AC$ and $BD$ meet at $P$. Let $\omega_1$ and $O_1$ denote the circumcircle and the circumcenter of triangle $ABP$. Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$. Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$), respectively. Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$) and $\widehat {TP}$ (not including $C$). Prove that $MN \parallel O_1O_2$.

2019 CHMMC (Fall), 1

Tags: geometry

Let $ABC$ be an equilateral triangle of side length $6$. Points $D, E$ and $F$ are on sides $AB$, $BC$, and $AC$ respectively such that $AD = BE = CF = 2$. Let circle $O$ be the circumcircle of $DEF$, that is, the circle that passes through points $D, E$, and $F$. What is the area of the region inside triangle $ABC$ but outside circle $O$?