Olympiad problems

2017 Princeton University Math Competition, B2

Tags: geometry

A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$.

1905 Eotvos Mathematical Competition, 2

Tags: geometry , area

Divide the unit square into $9$ equal squares by means of two pairs of lines parallel to the sides (see figure). Now remove the central square. Treat the remaining $8$ squares the same way, and repeat the process $n$ times. (a) How many squares of side length $1/3^n$ remain? (b) What is the sum of the areas of the removed squares as $n$ becomes infinite? [center][img]https://cdn.artofproblemsolving.com/attachments/7/d/3e6e68559919583c24d4457f946bc4cef3922f.png[/img][/center]

2002 Vietnam National Olympiad, 2

Tags: geometric transformation , reflection , geometry

An isosceles triangle $ ABC$ with $ AB \equal{} AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$. Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$, respectively. Find the locus of the orthocenter of triangle $ AMN$.

2009 Moldova Team Selection Test, 3

Tags: circumcircle , search , cyclic quadrilateral , geometry

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

2003 Denmark MO - Mohr Contest, 1

Tags: geometry , right triangle

In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.

VII Soros Olympiad 2000 - 01, 10.6

Tags: geometry , circumcircle , incircle

A circle is inscribed in triangle $ABC$. $M$ and $N$ are the points of its tangency with the sides $BC$ and $CA$, respectively. The segment $AM$ intersects $BN$ at point $P$ and the inscribed circle at point $Q$. It is known that $MP = a$, $PQ = b$. Find $AQ$.

2018 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , 3d geometry , triangle , distance , geometry

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

2022 Korea National Olympiad, 5

Tags: geometry , incenter

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

1977 Czech and Slovak Olympiad III A, 1

Tags: geometry , 3d geometry , combinatorial geometry , cube

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

2000 Bulgaria National Olympiad, 2

Tags: geometric transformation , reflection , geometry

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.

1993 Bulgaria National Olympiad, 2

Tags: inequalities , geometry , circumcircle , trigonometry

Let $M$ be an interior point of the triangle $ABC$ such that $AMC = 90^\circ$, $AMB = 150^\circ$, and $BMC = 120^\circ$. The circumcenters of the triangles $AMC$, $AMB$, and $BMC$ are $P$, $Q$, and $R$ respectively. Prove that the area of $\Delta PQR$ is greater than or equal to the area of $\Delta ABC$.

2018 Purple Comet Problems, 18

Tags: geometry , probability

Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2019 Stanford Mathematics Tournament, 6

Tags: geometry

Let the altitude of $\vartriangle ABC$ from $A$ intersect the circumcircle of $\vartriangle ABC$ at $D$. Let $E$ be a point on line $AD$ such that $E \ne A$ and $AD = DE$. If $AB = 13$, $BC = 14$, and $AC = 15$, what is the area of quadrilateral $BDCE$?

2018 BMT Spring, 9

Tags: geometry

What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area?

1999 Korea Junior Math Olympiad, 7

Tags: point , geometry , combinatorics

$A_0B, A_0C$ rays that satisfy $\angle BA_0C=14^{\circ}$. You are to place points $A_1, A_2, ...$ by the following rules. [b]Rules[/b] (1) On the first move, place $A_1$ on any point on $A_0B$(except $A_0$). (2) On the $n>1$th move, place $A_n$ on $A_0B$ iff $A_{n-1}$ is on $A_0C$, and place $A_n$ on $A_0C$ iff $A_{n-1}$ is one $A_0B$. $A_n$ must be place on the point that satisfies $A_{n-2}A_n{n-1}=A_{n-1}A_n$. All the points must be placed in different locations. What is the maximum number of points that can be placed?

1968 Spain Mathematical Olympiad, 5

Tags: geometry , rectangle , locus , inscribed

Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.

1998 IMO Shortlist, 7

Tags: trigonometry , geometry , trapezoid , parallelogram

Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]

2014 Korea National Olympiad, 3

Tags: geometry , incenter

$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.

1979 VTRMC, 1

Tags: geometry

Show that the right circular cylinder of volume $V$ which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)

2023 Taiwan TST Round 2, 6

Tags: combinatorics , geometry

There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns. [i] Proposed by usjl[/i]

2012 Polish MO Finals, 2

Tags: 3d geometry , cube , geometry , combinatorics

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.

1994 Tuymaada Olympiad, 4

Tags: convex , polyhedron , sphere , geometry

Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

2011 Indonesia TST, 2

Tags: geometry , probability , acute , combinatorics

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

2016 JBMO Shortlist, 2

Tags: geometry

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

2023 Euler Olympiad, Round 1, 1

Tags: geometry , rectangle , euler

Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. Determine the ratio $\frac{AK}{KD}$. [i]Proposed by Giorgi Arabidze, Georgia[/i]