Olympiad problems

Brazil L2 Finals (OBM) - geometry, 2015.6

Let $ABC$ a scalene triangle and $AD, BE, CF$ your angle bisectors, with $D$ in the segment $BC, E$ in the segment $AC$ and $F$ in the segment $AB$. If $\angle AFE = \angle ADC$. Determine $\angle BCA$.

2014 Baltic Way, 5

Tags: algebra , cyclic quadrilateral , trigonometry , geometry

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$

2023 Czech and Slovak Olympiad III A., 3

Tags: geometry

In acute triangle $ABC$ let $H$ be its orthocenter and $I$ be its incenter. Let $D$ be the projection of point $I$ onto the line $BC$ and $E$ be the reflection of point $A$ in point $I$. Further, let $F$ be the projection of point $H$ onto the line $ED$. Prove that points $B, H, F$ and $C$ lie on circle.

2023 Azerbaijan National Mathematical Olympiad, 2

Tags: geometry

Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.

Croatia MO (HMO) - geometry, 2016.3

Tags: geometry , altitude , concyclic , cyclic quadrilateral

Given a cyclic quadrilateral $ABCD$ such that the tangents at points $B$ and $D$ to its circumcircle $k$ intersect at the line $AC$. The points $E$ and $F$ lie on the circle $k$ so that the lines $AC, DE$ and $BF$ parallel. Let $M$ be the intersection of the lines $BE$ and $DF$. If $P, Q$ and $R$ are the feet of the altitides of the triangle $ABC$, prove that the points $P, Q, R$ and $M$ lie on the same circle

2021 Dutch IMO TST, 2

Tags: right triangle , geometry , right angle , centroid

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

2014 Tuymaada Olympiad, 6

Tags: geometry , trigonometry

Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other. [i](L. Emelyanov)[/i]

2020 HMNT (HMMO), 6

Tags: geometry

Regular hexagon $P_1P_2P_3P_4P_5P_6$ has side length $2$. For $1 \le i \le 6$, let $C_i$ be a unit circle centered at $P_i$ and $\ell_i$ be one of the internal common tangents of $C_i$ and $C_{i+2}$, where $C_7 = C_1$ and $C_8 = C_2$. Assume that the lines $\{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5,\ell_6\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Mathley 2014-15, 2

Tags: algebra , geometry , integer , triangle , sequence , area of a triangle

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

2019 Czech and Slovak Olympiad III A, 2

Tags: rectangle , geometry , constructive geometry

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

2022 Austrian MO National Competition, 5

Tags: geometry , collinear , isosceles

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

1985 IMO Shortlist, 2

Tags: angle , 3d geometry , polyhedron , euler s theorem , geometry

A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$

Novosibirsk Oral Geo Oly VIII, 2020.2

Tags: perimeter , chessboard , combinatorics , geometry

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

1986 Traian Lălescu, 1.3

Tags: geometry

Let be three distinct squares $ ABCD, BCEF, EFGH. $ Show that $ \angle EDF +\angle HDG =45^{\circ } . $

2016 Canada National Olympiad, 5

Tags: circumcircle , geometry

Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.

2017 Nordic, 4

Tags: geometry , combinatorics

Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.

1988 Bulgaria National Olympiad, Problem 3

Tags: inequalities , geometric inequality , geometry , 3d geometry , tetrahedron

Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?

2025 Romania EGMO TST, P3

Tags: geometry , ratio

$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\dfrac{AM}{AO}$. [i]Proposed by Fedir Yudin[/i]

2020 HMNT (HMMO), 2

Tags: geometry

In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is $160$ meters tall, the Stata Center is $90$ meters tall, and the two buildings are $120$ meters apart. Furthermore, both zipline at $10$ meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100a$.

2019 Iran Team Selection Test, 3

Tags: geometry , parallelogram

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

2011 Armenian Republican Olympiads, Problem 2

Tags: rhombus , geometry , hexagon

Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)

1982 AMC 12/AHSME, 4

Tags: perimeter , geometry

The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is $\text{(A)} \pi \qquad \text{(B)} \frac{2}{\pi} \qquad \text{(C)} 1 \qquad \text{(D)} \frac{1}{2} \qquad \text{(E)} \frac{4}{\pi} + 2$

2023 Kyiv City MO Round 1, Problem 3

Tags: cyclic quadrilateral , analytic geometry , grid , geometry

Consider all pairs of distinct points on the Cartesian plane $(A, B)$ with integer coordinates. Among these pairs of points, find all for which there exist two distinct points $(X, Y)$ with integer coordinates, such that the quadrilateral $AXBY$ is convex and inscribed. [i]Proposed by Anton Trygub[/i]

1984 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , trapezoid , trigonometry

Let $\alpha, \beta, \gamma, \delta$ be the interior angles of a convex quadrilateral, If $$ \cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 , $$ then this quadrilateral is cyclic or a trapezium. Prove it.

Indonesia MO Shortlist - geometry, g1

Tags: geometry , rectangle , angle , geometric inequality

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.