This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 25757

2004 Polish MO Finals, 5
Tags: combinatorial geometry , geometry
Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.

1975 Chisinau City MO, 110
Tags: octagon , symmetry , geometry
Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.

2016 Tournament Of Towns, 2
Tags: geometry , locus , circles
On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.

2004 Alexandru Myller, 2
Tags: geometry , 3d geometry , tetrahedron
The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular. Show that the edges from $ A $ have equal lengths. [i]Dinu Șerbănescu[/i]

2024 Malaysia IMONST 2, 5
Tags: geometry
A duck drew a square $ABCD$, then he reflected $C$ across $B$ to obtain a point $E$. He also drew the center of the square to be $F$. Then, he drew a point $G$ on ray $EF$ beyond $F$ such that $\angle AGC = 135^{\circ}$. Help the Duck prove that $\angle CGD = 135^{\circ}$ as well.

2024 Brazil EGMO TST, 4
Tags: geometry , bicentric quadrilateral , tangent , projective geometry
Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.

2015 Albania JBMO TST, 2
Tags: geometry
The triangle $ABC$ has $\angle BCA=90^{\circ}.$ Bisector of angle $\angle CAB$ intersects the side $BC$ in point $P$ and bisector of angle $\angle ABC$ intersects the side $AC$ in point $Q.$ If $M$ and $N$ are projections of $P$ and $Q$ on side $AB$, find the measure of the angle $\angle MCN.$

2018 Azerbaijan JBMO TST, 2
Tags: geometry
Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

Cono Sur Shortlist - geometry, 2020.G2
Tags: geometry , incircle , circumcircle
Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

2023 New Zealand MO, 6
Tags: geometry
Let triangle $ABC$ be right-angled at $A$. Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$. Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}$.

2024 Iberoamerican, 2
Tags: geometry , circumcircle
Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

2025 Bulgarian Spring Mathematical Competition, 9.2
Tags: geometry , angle chasing , similarity , circumcircle
Let $ABC$ be an acute scalene triangle inscribed in a circle \( \Gamma \). The angle bisector of \( \angle BAC \) intersects \( BC \) at \( L \) and \( \Gamma \) at \( S \). The point \( M \) is the midpoint of \( AL \). Let \( AD \) be the altitude in \( \triangle ABC \), and the circumcircle of \( \triangle DSL \) intersects \( \Gamma \) again at \( P \). Let \( N \) be the midpoint of \( BC \), and let \( K \) be the reflection of \( D \) with respect to \( N \). Prove that the triangles \( \triangle MPS \) and \( \triangle ADK \) are similar.

2012 India Regional Mathematical Olympiad, 5
Tags: ratio , geometry , trigonometry , area of a triangle
Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

1997 Turkey Junior National Olympiad, 2
Tags: symmetry , pythagorean theorem , power of a point , geometry
Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

2020 Belarusian National Olympiad, 11.7
Tags: geometry
Line $AL$ is an angle bisector in the triangle $ABC$ ($L \in BC$), and $\omega$ is its circumcircle. Chords $X_1X_2$ and $Y_1Y_2$ pass through $L$ such that points $X_1,Y_1$ and $A$ lie in the same half-plane with respect to $BC$. Lines $X_1Y_2$ and $Y_1X_2$ intersect side $BC$ in points $Z_1$ and $Z_2$ respectively. Prove that $\angle BAZ_1=\angle CAZ_2$.

2018 Malaysia National Olympiad, A1
Tags: geometry , area
Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.

1993 Dutch Mathematical Olympiad, 5
Tags: geometry
Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.

2022 LMT Spring, 2
Tags: geometry , combinatorics
Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

2010 IMO Shortlist, 4
Tags: geometry , inversion , p2
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

1976 Vietnam National Olympiad, 3
Tags: geometry , geometric inequality , tetrahedron , area of a triangle , distance
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

2025 Harvard-MIT Mathematics Tournament, 2
Tags: geometry
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

2021 Azerbaijan Junior NMO, 5
Tags: geometry
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$. $\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$

2022 Sharygin Geometry Olympiad, 10.7
Tags: geometry , combinatorics , combinatorial geometry
Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?

2005 Portugal MO, 5
Tags: geometry , tangential , cyclic quadrilateral
Considers a quadrilateral $[ABCD]$ that has an inscribed circle and a circumscribed circle. The sides $[AD]$ and $[BC]$ are tangent to the circle inscribed at points $E$ and $F$, respectively. Prove that $AE \cdot F C = BF \cdot ED$. [img]https://1.bp.blogspot.com/-6o1fFTdZ69E/X4XMo98ndAI/AAAAAAAAMno/7FXiJnWzJgcfSn-qSRoEAFyE8VgxmeBjwCLcBGAsYHQ/s0/2005%2BPortugal%2Bp5.png[/img]

Denmark (Mohr) - geometry, 1993.2
Tags: geometry , rectangle , paper folding
A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]