Olympiad problems

Indonesia Regional MO OSP SMA - geometry, 2007.4

In acute triangles $ABC$, $AD, BE ,CF$ are altitudes, with $D, E, F$ on the sides $BC, CA, AB$, respectively. Prove that $$DE + DF \le BC$$

2017 Moldova Team Selection Test, 7

Tags: inequalities , geometric inequality , incenter , circumcircle , geometry

Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality: $$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.

2024 Moldova Team Selection Test, 2

Tags: geometry

In the acute-angled triangle $ABC$, let $AD$, $D \in BC$ be the $A$-angle bisector. The perpenducular to $BC$ through $D$ and the perpendicular to $AD$ through $A$ meet at $I$. The circle with center $I$ and radius $ID$, intersects $AB$ and $AC$ at $F$ and $E$ respectively. On the arc $FE$, which does not contain $A$, of the circumcircle of $AFE$, consider a point $X$, such that $\frac{XF}{XE}=\frac{AF}{AE}$. Prove that the circumcircles of triangles $AFE$ and $BXC$ are tangent.

1969 Kurschak Competition, 2

Tags: trigonometry , equilateral , geometry

A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.

2010 Turkey Junior National Olympiad, 1

Tags: power of a point , circumcircle , ratio , rectangle , geometry

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

2022 Mexican Girls' Contest, 7

Tags: parallelogram , geometry , circumcircle

Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that $$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$

2022 Durer Math Competition (First Round), 2

Tags: geometry , circumcircle

In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP \cdot OQ$.

1998 AMC 12/AHSME, 18

Tags: 3d geometry , geometry , sphere

A right circular cone of volume $ A$, a right circular cylinder of volume $ M$, and a sphere of volume $ C$ all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then $ \textbf{(A)}\ A \minus{} M \plus{} C \equal{} 0 \qquad \textbf{(B)}\ A \plus{} M \equal{} C \qquad \textbf{(C)}\ 2A \equal{} M \plus{} C$ $ \textbf{(D)}\ A^2 \minus{} M^2 \plus{} C^2 \equal{} 0 \qquad \textbf{(E)}\ 2A \plus{} 2M \equal{} 3C$

2023 European Mathematical Cup, 2

Tags: right triangle , geometry , circumcircle

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius. [i]Kyprianos-Iason Prodromidis[/i]

1968 Dutch Mathematical Olympiad, 4

Tags: reflection , construction , geometry

Given is a triangle $ABC$. A line $\ell$ passes through reflection wrt $BC$ changes into the line $\ell'$, $\ell'$ changes into $\ell''$ through reflection wrt $AC$ and $\ell''$ through reflection wrt $AB$ changes into $\ell'''$. Construct the line $\ell$ given that $\ell'''$ coincides with $\ell$.

2007 Sharygin Geometry Olympiad, 17

Tags: circumradius , geometry , circumcircle

What triangles can be cut into three triangles having equal radii of circumcircles?

2021 All-Russian Olympiad, 6

Tags: 3d geometry , perpendicular bisector , tetrahedron , geometry

In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.

2012 Sharygin Geometry Olympiad, 4

Tags: angle chasing , geometry , isosceles

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$. (A.Akopyan, D.Shvetsov)

2023 Sharygin Geometry Olympiad, 8.1

Tags: geometry

Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$.

2015 AMC 8, 6

Tags: geometry

In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$? $\textbf{(A) }100\qquad\textbf{(B) }420\qquad\textbf{(C) }500\qquad\textbf{(D) }609\qquad \textbf{(E) }701$

2007 Pre-Preparation Course Examination, 2

Tags: 3d geometry , sphere , geometry , combinatorial geometry

a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere. b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.

2010 All-Russian Olympiad Regional Round, 9.6

Tags: circumcircle , geometry

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.

2013 Tuymaada Olympiad, 8

Tags: trigonometry , inequalities , analytic geometry , perimeter , geometry

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

Kvant 2020, M2600

Tags: circles , concurrency , geometry , cyclic quadrilateral

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

2022 Germany Team Selection Test, 2

Tags: geometry

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

2011 USAMO, 3

Tags: reflection , geometric transformation , 3d geometry , geometry

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.

Estonia Open Junior - geometry, 2019.2.1

Tags: equilateral , angle , pentagon , geometry

A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

2017-IMOC, G6

Tags: geometric inequality , geometry

A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that $$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$$

1994 Brazil National Olympiad, 6

Tags: perimeter , inradius , geometry , circumcircle

A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.

2012 Morocco TST, 4

Tags: marvio , radical axis , symmetry , homothety , circumcircle , geometry

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]