Olympiad problems

2002 Taiwan National Olympiad, 2

Tags: combinatorics , geometry

A lattice point $X$ in the plane is said to be [i]visible[/i] from the origin $O$ if the line segment $OX$ does not contain any other lattice points. Show that for any positive integer $n$, there is square $ABCD$ of area $n^{2}$ such that none of the lattice points inside the square is visible from the origin.

2007 Tournament Of Towns, 2

Tags: perimeter , geometry

$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$, respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$. The perimeter of triangle $KLB$ is equal to $1$. What is the area of triangle $MND$?

2016 Croatia Team Selection Test, Problem 3

Tags: midpoint , geometry , circumcenter , circumcircle

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

2014 Stanford Mathematics Tournament, 4

Tags: geometry

Let $ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $AC = 5$. Let $X$ be a point in the triangle. Compute the minimal possible value of $AX^2 + BX^2 + CX^2$

Swiss NMO - geometry, 2016.1

Tags: circumcircle , geometry , angle , equal angles

Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.

2022 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that: $\bullet$ all four rectangles share a common vertex $P$, $\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$, $\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$. Compute the total area covered by the union of the four rectangles. [img]https://cdn.artofproblemsolving.com/attachments/3/1/e9edd39e60e4a4defdb127b93b19ab0d0f443c.png[/img]

2020 Stanford Mathematics Tournament, 7

Tags: geometry

Let $ABC$ be an acute triangle with $BC = 4$ and $AC = 5$. Let $D$ be the midpoint of $BC$, $E$ be the foot of the altitude from $B$ to $AC$, and $F$ be the intersection of the angle bisector of $\angle BCA$ with segment $AB$. Given that $AD$, $BE$, and $CF$ meet at a single point $P$, compute the area of triangle $ABC$. Express your answer as a common fraction in simplest radical form.

2018 Czech and Slovak Olympiad III A, 5

Tags: geometry

Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.

2014 Online Math Open Problems, 16

Tags: tetrahedron , integration , geometry , calculus , 3d geometry , analytic geometry

Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.) [i]Proposed by Robin Park[/i]

Estonia Open Junior - geometry, 2001.2.2

Tags: consecutive integers , perpendicular , geometry

In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.

1996 Brazil National Olympiad, 4

Tags: ratio , trigonometry , perpendicular bisector , geometry , geometric transformation , circumcircle , vector

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

2024 Brazil Cono Sur TST, 2

Tags: geometry

Let $ABC$ be a triangle with $AB < AC < BC$ and $\Gamma$ its circumcircle. Let $\omega_1$ be the circle with center $B$ and radius $AC$ and $\omega_2$ the circle with center $C$ and radius $AB$. The circles $\omega_1$ and $\omega_2$ intersect at point $E$ such that $A$ and $E$ are on opposite sides of the line $BC$. The circles $\Gamma$ and $\omega_1$ intersect at point $F$ and the circles $\Gamma$ and $\omega_2$ intersect at point $G$ such that the points $F$ and $G$ are on the same side as $E$ in relation to the line $BC$. With $K$ being the point such that $AK$ is a diameter of $\Gamma$, prove that $K$ is circumcenter of triangle $EFG$.

2001 Baltic Way, 9

Tags: circumcircle , rhombus , parallelogram , hyperbola , geometry , cyclic quadrilateral , conic

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

2010 Postal Coaching, 2

Tags: geometry , inequalities

Let $M$ be an interior point of a $\triangle ABC$ such that $\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}$. Let $P, Q, R$ be the circumcentres of the $\triangle AM B, \triangle BM C, \triangle CM A$ respectively. Prove that $[P QR] \ge [ABC]$.

1989 Swedish Mathematical Competition, 3

Tags: number theory , perfect square , 3d geometry , geometry

Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege

2011 ELMO Shortlist, 2

Tags: reflection , trigonometry , angle bisector , geometry , geometric transformation

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$. [i]David Yang.[/i]

2005 Oral Moscow Geometry Olympiad, 5

Tags: geometry , geometric inequality

An arbitrary point $M$ is chosen inside the triangle $ABC$. Prove that $MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)$. (N. Sedrakyan)

2022 China Team Selection Test, 4

Tags: geometry , lattice points , combinatorial geometry , combinatorics

Find all positive integer $k$ such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length $k$ (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.

1986 All Soviet Union Mathematical Olympiad, 431

Tags: combinatorial geometry , geometry , dodecagon , distance , sum

Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.

2011 Dutch IMO TST, 3

Tags: circles , tangent , geometry

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

2024 Malaysian IMO Training Camp, 1

Tags: geometry

Consider $\triangle MAB$ with a right angle at $A$ and circumcircle $\omega$. Take any chord $CD$ perpendicular to $AB$ such that $A, C, B, D, M$ lie on $\omega$ in this order. Let $AC$ and $MD$ intersect at point $E$, and let $O$ be the circumcenter of $\triangle EMC$. Show that if $J$ is the intersection of $BC$ and $OM$, then $JB = JM$. [i](Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)[/i]

1999 Tournament Of Towns, 1

Tags: geometry , speed , algebra

A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son? (Tairova)

2016 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry

Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ ABC$ such that $\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\measuredangle MCB$ .

2022 Sharygin Geometry Olympiad, 9.4

Tags: angle bisector , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$, $P$ be the midpoint of the minor arc $AB$ of its circumcircle, and $Q$ be the midpoint of $AC$. A circumcircle of triangle $APQ$ centered at $O$ meets $AB$ for the second time at point $K$. Prove that lines $PO$ and $KQ$ meet on the bisector of angle $ABC$.

1999 USAMTS Problems, 5

Tags: 3d geometry , geometry , sphere

We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.