Olympiad problems

2011 Belarus Team Selection Test, 1

Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon. a) Prove thas $S \le \frac15 A$. b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ? I.Voronovich

OMMC POTM, 2024 11

Tags: geometry , equal segments

Rectangle $ABCD$ with $AB>BC$ has point $P$ inside of it and $Q$ outside of it, such that $PQCD$ is a parallelogram with $PD=AD$. Let $M$ be the midpoint of $CD$. Give that $\angle AMP=\angle BMQ$, prove that $AB=2BC$.

Ukraine Correspondence MO - geometry, 2004.8

Tags: geometry , ratio , trapezoid , area

The extensions of the sides $AB$ and $CD$ of the trapezoid $ABCD$ intersect at point $E$. Denote by $H$ and $G$ the midpoints of $BD$ and $AC$. Find the ratio of the area $AEGH$ to the area $ABCD$.

1985 IMO, 1

Tags: trigonometry , geometry , circumcircle , cyclic quadrilateral

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

2017 Cono Sur Olympiad, 4

Tags: geometry , circumcircle

Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that: [list] [*]$\angle XAB = \angle YCB = 90^\circ$[/*] [*]$\angle ABC = \angle BXA = \angle BYC$[/*] [*]$X$ and $C$ are in different half-planes with respect to $AB$[/*] [*]$Y$ and $A$ are in different half-planes with respect to $BC$[/*] [/list] Prove that $O$ is the midpoint of $XY$.

2019 Moldova Team Selection Test, 2

Tags: trigonometry , easy , geometry

Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$. (A natural number is a positive integer)

2017 Peru Iberoamerican Team Selection Test, P5

Tags: geometry , trapezoid

Let $ABCD$ be a trapezoid of bases $AD$ and $BC$ , with $AD> BC$, whose diagonals are cut at point $E$. Let $P$ and $Q$ be the feet of the perpendicular drawn from $E$ on the sides $AD$ and $BC$, respectively, with $P$ and $Q$ in segments $AD$ and $BC,$ respectively. Let $I$ be the center of the triangle $AED$ and let $K$ be the point of intersection of the lines $AI$ and $CD$. If $AP + AE = BQ + BE$, show that $AI = IK$.

2022 Turkey Team Selection Test, 8

Tags: orthocenter , incircle , geometry

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

2016 USAJMO, 5

Tags: geometry

Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, respectively. Given that $$AH^2=2\cdot AO^2,$$ prove that the points $O,P,$ and $Q$ are collinear.

2003 India IMO Training Camp, 1

Tags: circumcircle , geometry

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

2020 Indonesia MO, 1

Tags: circles , bisector , geometry

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here. Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.

2009 JBMO Shortlist, 3

Tags: geometry

Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.

2025 Malaysian IMO Team Selection Test, 8

Tags: geometry

Let $ABC$ be an equilateral triangle, and $P$ is a point on its incircle. Let $\omega_a$ be the circle tangent to $AB$ passing through $P$ and $A$. Similarly, let $\omega_b$ be the circle tangent to $BC$ passing through $P$ and $B$, and $\omega_c$ be the circle tangent to $CA$ passing through $P$ and $C$. Prove that the circles $\omega_a$, $\omega_b$, $\omega_c$ has a common tangent line. [i]Proposed by Ivan Chan Kai Chin[/i]

2004 Italy TST, 1

Tags: geometry

Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$. $(a)$ Prove that $t$ is parallel to $AC$. $(b)$ Prove that the lines $r,s,t$ are concurrent.

1970 Canada National Olympiad, 8

Tags: parameterization , ellipse , geometry , conic

Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.

2015 Postal Coaching, Problem 5

Tags: geometry , incenter

Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.

2009 Saint Petersburg Mathematical Olympiad, 7

Tags: geometry , circumcircle

Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$

2007 Indonesia TST, 2

Tags: geometry

Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.

Indonesia MO Shortlist - geometry, g2.3

Tags: ratio , geometry

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that: \[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]

1979 Swedish Mathematical Competition, 6

Tags: geometric inequalities , geometry , centroid

Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

2001 Moldova National Olympiad, Problem 2

Tags: geometry

A regular $n$-gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.

2019 IberoAmerican, 4

Tags: trapezoid , projective geometry , miquel point , geometry

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.

2010 German National Olympiad, 1

Tags: tangent , geometry , circles , chord , length

Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$

1985 Tournament Of Towns, (096) 5

Tags: rectangle , combinatorics , combinatorial geometry , projection , parallelepiped , 3d geometry , geometry

A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

2008 AIME Problems, 13

Tags: geometry , geometric transformation , reflection , calculus , trigonometry , integration , analytic geometry

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.