Olympiad problems

2016 Iranian Geometry Olympiad, 1

In trapezoid $ABCD$ with $AB || CD$, $\omega_1$ and $\omega_2$ are two circles with diameters $AD$ and $BC$, respectively. Let $X$ and $Y$ be two arbitrary points on $\omega_1$ and $\omega_2$, respectively. Show that the length of segment $XY$ is not more than half the perimeter of $ABCD$. [i]Proposed by Mahdi Etesami Fard[/i]

2018 Finnish National High School Mathematics Comp, 2

Tags: angle bisector , sidelengths , geometry

The sides of triangle $ABC$ are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.

1997 Baltic Way, 15

Tags: incenter , circumcircle , geometry

In the acute triangle $ABC$, the bisectors of $A,B$ and $C$ intersect the circumcircle again at $A_1,B_1$ and $C_1$, respectively. Let $M$ be the point of intersection of $AB$ and $B_1C_1$, and let $N$ be the point of intersection of $BC$ and $A_1B_1$. Prove that $MN$ passes through the incentre of $\triangle ABC$.

2001 USAMO, 6

Tags: reflection , geometry

Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

2025 Ukraine National Mathematical Olympiad, 9.7

Tags: geometry

In a quadrilateral $ABCD$, it is known that $\angle ABC = \angle ADC = 90^{\circ}$. On the ray $AB$ beyond point $B$, a point $K$ is chosen such that $\angle AKD = \angle ADB$. Point $L$ is the projection of point $K$ onto the line $AD$, and point $N$ is the projection of point $D$ onto the line $CL$. Find the degree measure of $\angle ANK$. [i]Proposed by Mykhailo Shtandenko[/i]

1988 IberoAmerican, 1

Tags: trigonometry , arithmetic sequence , geometry

The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

1986 Spain Mathematical Olympiad, 6

Tags: product , trigonometry , geometry

Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$

2005 Paraguay Mathematical Olympiad, 5

Tags: chord , geometry , equal segments

Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.

1994 All-Russian Olympiad, 7

Tags: altitude , geometry , 3d geometry , tetrahedron , sphere

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)

2016 South East Mathematical Olympiad, 2

Tags: geometry

Suppose $PAB$ and $PCD$ are two secants of circle $O$. Lines $AD \cap BC=Q$. Point $T$ lie on segment $BQ$ and point $K$ is intersection of segment $PT$ with circle $O$, $S=QK\cap PA$ Given that $ST \parallel PQ$, prove that $B,S,K,T$ lie on a circle.

1986 IMO Longlists, 76

Tags: number theory , diophantine equation , geometry , pythagorean theorem , algebra

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

2008 China Girls Math Olympiad, 3

Tags: ratio , inequalities , geometry

Determine the least real number $ a$ greater than $ 1$ such that for any point $ P$ in the interior of the square $ ABCD$, the area ratio between two of the triangles $ PAB$, $ PBC$, $ PCD$, $ PDA$ lies in the interval $ \left[\frac {1}{a},a\right]$.

2023 Regional Olympiad of Mexico West, 5

Tags: geometry , rhombus , equilateral , parallelogram

We have a rhombus $ABCD$ with $\angle BAD=60^\circ$. We take points $F,H,G$ on the sides $AD,DC$ and the diagonal $AC$, respectively, such that $DFGH$ is a parallelogram. Prove that $BFH$ is equilateral.

2005 Germany Team Selection Test, 3

Tags: inequalities , geometry

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

1994 Dutch Mathematical Olympiad, 3

Tags: modular arithmetic , geometry , 3d geometry , number theory

$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

1998 Iran MO (2nd round), 2

Tags: circumcircle , perpendicular bisector , cyclic quadrilateral , geometry

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: \[ AQ+CQ=BP. \]

2005 JHMT, 9

Tags: geometry

A square with side length $1$ is inscribed in a hemisphere such that one side of the square is on the hemisphere’s diameter. What is the semicircle’s perimeter?

1995 Hungary-Israel Binational, 2

Tags: geometry

Let $ P_1$, $ P_2$, $ P_3$, $ P_4$ be five distinct points on a circle. The distance of $ P$ from the line $ P_iP_k$ is denoted by $ d_{ik}$. Prove that $ d_{12}d_{34} \equal{} d_{13}d_{24}$.

2007 AMC 10, 11

Tags: circumcircle , trigonometry , trig identities , law of sines , geometry

A circle passes through the three vertices of an isosceles triangle that has two sides of length $ 3$ and a base of length $ 2$. What is the area of this circle? $ \textbf{(A)}\ 2\pi\qquad \textbf{(B)}\ \frac {5}{2}\pi\qquad \textbf{(C)}\ \frac {81}{32}\pi\qquad \textbf{(D)}\ 3\pi\qquad \textbf{(E)}\ \frac {7}{2}\pi$

2019 Stanford Mathematics Tournament, 9

Tags: geometry

Let $ABCD$ be a quadrilateral with $\angle ABC = \angle CDA = 45^o$ , $AB = 7$, and $BD = 25$. If $AC$ is perpendicular to $CD$, compute the length of $BC$.

2013 Tournament of Towns, 5

Tags: polygon , perpendicular , combinatorial geometry , geometry

A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.

2005 Junior Balkan Team Selection Tests - Romania, 11

Tags: circumcircle , geometric transformation , reflection , homothety , ratio , geometry

Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

1992 AMC 12/AHSME, 19

Tags: geometry , 3d geometry , tetrahedron , ratio

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these? $ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $

2021 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , equal segments

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

1981 National High School Mathematics League, 5

Tags: geometry , 3d geometry

Given a cube $ABCD-A'B'C'D'$, in the $12$ lines:$AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'$, how many sets of lines are skew lines? $\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48$