Olympiad problems

Ukrainian TYM Qualifying - geometry, 2010.15

On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.

1985 Vietnam National Olympiad, 3

Tags: geometry

A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.

1997 German National Olympiad, 5

Tags: geometry , combinatorics , area , combinatorial geometry

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.

2004 AMC 8, 14

Tags: geometry

What is the area enclosed by the geoboard quadrilateral below? [asy] int i,j; for(i=0; i<11; i=i+1) { for(j=0; j<11; j=j+1) { dot((i,j)); } } draw((0,5)--(4,0)--(10,10)--(3,4)--cycle, linewidth(0.7)); [/asy] $\textbf{(A)} 15\qquad \textbf{(B)} 18\tfrac12\qquad \textbf{(C)} 22\tfrac12\qquad \textbf{(D)} 27\qquad \textbf{(E)} 41\qquad$

2021 Israel TST, 1

Tags: geometry , regular polygon

Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.

1988 All Soviet Union Mathematical Olympiad, 464

Tags: equal area , geometry , convex , area

$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

Brazil L2 Finals (OBM) - geometry, 2003.5

Tags: trigonometry , inequalities , triangle inequality , geometry

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

1991 All Soviet Union Mathematical Olympiad, 547

Tags: geometry , circumradius , circumcircle , equal circles

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

2018 Bangladesh Mathematical Olympiad, 2

Tags: geometry , circles , triangle

BdMO National 2018 Higher Secondary P2 $AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$

2017 CMIMC Individual Finals, 1

Tags: trapezoid , geometry

Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$.

1997 Brazil National Olympiad, 6

Tags: geometry

$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1. Show that $f$ preserves distances.

2016 SDMO (Middle School), 2

Tags: geometry , ratio

Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?

1981 USAMO, 3

Tags: trigonometry , inequalities , geometry

If $A,B,C$ are the angles of a triangle, prove that \[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\] and determine when equality holds.

2015 Sharygin Geometry Olympiad, P19

Tags: geometry , concurrency

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

Denmark (Mohr) - geometry, 1997.2

Tags: geometry , square , area

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

2009 Thailand Mathematical Olympiad, 8

Tags: geometry , geometric inequalities , inequalities

Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that $$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$

2011 Vietnam National Olympiad, 3

Tags: trigonometry , projective geometry , geometry

Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then, [b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$ [b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$

2021 ELMO Problems, 1

Tags: triangle , geometry , circles , cyclic

In $\triangle ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD = EC$. Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$, and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$. Prove that $D$, $E$, $X$, and $Y$ are concyclic.

Estonia Open Junior - geometry, 2013.1.4

Tags: geometry , tangent circle

Inside a circle $c$ with the center $O$ there are two circles $c_1$ and $c_2$ which go through $O$ and are tangent to the circle $c$ at points $A$ and $B$ crespectively. Prove that the circles $c_1$ and $c_2$ have a common point which lies in the segment $AB$.

2022 Israel TST, 2

Tags: number theory , geometry , lattice points

Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point. [b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$. [b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.

2022 Brazil Team Selection Test, 2

Tags: geometry , circumcircle , orthocenter , simson line

Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.

2025 All-Russian Olympiad Regional Round, 10.5

Tags: inequalities , geometry

The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

2010 Stanford Mathematics Tournament, 1

Tags: geometry , geometric transformation , reflection

Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.

2024 Switzerland - Final Round, 8

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2011 Gheorghe Vranceanu, 1

Tags: circumcircle , geometry