Olympiad problems

2011 Math Prize For Girls Problems, 9

Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?

2021 Stanford Mathematics Tournament, 5

Tags: geometry

Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be on the lines $AB$ and $AD$, respectively, so that $B$ lies between $A$ and $E$, and $D$ lies between $A$ and $F$. Suppose that $\angle BCE = 20^o$ and $\angle DCF = 25^o$. Find the area of triangle $\vartriangle EAF$.

2012 Today's Calculation Of Integral, 854

Tags: analytic geometry , ellipse , conic , geometry , integration , limit , calculus

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

2017 Saudi Arabia Pre-TST + Training Tests, 5

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN, BC$ and line through $H$ and perpendicular to $AN$ intersect each other and form triangle $X Y Z$. Prove that the circumcircle of triangle $X Y Z$ is tangent to $(O)$.

2020 USOMO, 1

Tags: area of a triangle , minimize , olympiad geometry , geometry

Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized. [i]Proposed by Zuming Feng[/i]

2021 Romania Team Selection Test, 3

Tags: quadrilateral , geometry

Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.

1965 German National Olympiad, 6

Tags: inequalities , geometric inequalities , geometry , trigonometry

Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

1981 Vietnam National Olympiad, 3

Tags: geometry , ratio

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

2023 South East Mathematical Olympiad, 5

Tags: geometry

Let $AB$ be a chord of the semicircle $O$ (not the diameter). $M$ is the midpoint of $AB$, and $D$ is a point lies on line $OM$ ($D$ is outside semicircle $O$). Line $l$ passes through $D$ and is parallel to $AB$. $P, Q$ are two points lie on $l$ and $PO$ meets semicircle $O$ at $C$. If $\angle PCD=\angle DMC$, and $M$ is the orthocentre of $\triangle OPQ$. Prove that the intersection of $AQ$ and $PB$ lies on semicircle $O$.

1996 Romania National Olympiad, 4

Tags: right triangle , geometry

In the right triangle $ABC$ ($m ( \angle A) = 90^o$) $D$ is the foot of the altitude from $A$. The bisectors of the angles $ABD$ and $ADB$ intersect in $I_1$ and the bisectors of the angles $ACD$ and $ADC$ in $I_2$. Find the angles of the triangle if the sum of distances from $I_1$ and $I_2$ to $AD$ is equal to $\frac14$ of the length of $BC$.

1998 Romania Team Selection Test, 2

Tags: rectangle , geometry , pigeonhole principle

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

2007 Estonia Team Selection Test, 2

Tags: altitude , geometric inequalities , circumradius , incircle , circumcircle , inradius , geometry

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

1994 Mexico National Olympiad, 5

Tags: geometry , altitude , convex quadrilateral

$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$. Show that at least one of these points must lie on the sides of $ABCD$.

2013 Bulgaria National Olympiad, 3

Tags: geometry , combinatorics

The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$. [i]Proposed by Nikolay Beluhov[/i] EDIT: It was the problem 3 (not 2), corrected the source title.

1991 Bundeswettbewerb Mathematik, 4

Tags: geometry , combinatorial geometry , rectangle , combinatorics

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

1994 National High School Mathematics League, 11

Tags: 3d geometry , geometry

Intersections between a plane and 12 edges of a cube are all $\alpha$, then $\sin\alpha=$________.

2019 India Regional Mathematical Olympiad, 2

Tags: p2 , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

2021 Brazil EGMO TST, 5

Tags: number theory , geometry , 3d geometry , combinatorics

Let $S$ be a set, such that for every positive integer $n$, we have $|S\cap T|=1$, where $T=\{n,2n,3n\}$. Prove that if $2\in S$, then $13824\notin S$.

1977 Chisinau City MO, 137

Tags: geometry , angle , equal angles

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

III Soros Olympiad 1996 - 97 (Russia), 9.5

Tags: geometric inequality , geometry

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

Ukrainian From Tasks to Tasks - geometry, 2014.4

Tags: right triangle , geometry , square

In the triangle $ABC$ it is known that $AC = 21$ cm, $BC = 28$ cm and $\angle C = 90^o$. On the hypotenuse $AB$, we construct a square $ABMN$ with center $O$ such that the segment $CO$ intersects the hypotenuse $AB$ at the point $K$. Find the lengths of the segments $AK$ and $KB$.

2021 Canadian Junior Mathematical Olympiad, 1

Tags: angle bisector , concentric , geometry

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.

2023 New Zealand MO, 3

Tags: geometry , collinear

Let $ABCD$ be a square (vertices labelled in clockwise order). Let $Z$ be any point on diagonal $AC$ between $A$ and $C$ such that $AZ > ZC$. Points $X$ and $Y$ exist such that $AXY Z $ is a square (vertices labelled in clockwise order) and point $B$ lies inside $AXY Z$. Let $M$ be the point of intersection of lines $BX$ and $DZ$ (extended if necessary). Prove that $C$, $M$ and $Y$ are colinear

1994 Spain Mathematical Olympiad, 5

Tags: grid , rectangle , geometry , combinatorics

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

2021-IMOC, G2

Tags: geometry , circumcircle

Let the midline of $\triangle ABC$ parallel to $BC$ intersect the circumcircle $\Gamma$ of $\triangle ABC$ at $P$, $Q$, and the tangent of $\Gamma$ at $A$ intersects $BC$ at $T$. Show that $\measuredangle BTQ = \measuredangle PTA$.