Olympiad problems

2012 China Team Selection Test, 1

In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.

Kyiv City MO Seniors Round2 2010+ geometry, 2022.11.4

Tags: geometry

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

2019 Purple Comet Problems, 2

Tags: geometry

The large square in the diagram below with sides of length $8$ is divided into $16$ congruent squares. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/6/e/cf828197aa2585f5eab2320a43b80616072135.png[/img]

2007 Princeton University Math Competition, 1

Tags: calculus , logarithm , probability , integration , trapezoid , geometry

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

1962 All-Soviet Union Olympiad, 6

Tags: geometry

Given the lengths $AB$ and $BC$ and the fact that the medians to those two sides are perpendicular, construct the triangle $ABC$.

Estonia Open Junior - geometry, 2016.1.5

Tags: isosceles , right triangle , geometry

A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.

V Soros Olympiad 1998 - 99 (Russia), 9.6

Tags: geometry , ratio

On side $AB$ of triangle $ABC$, points $M$ and $K$ are taken ($M$ on segment $AK$). It is known that $AM: MK: MB = a: b: c$. Straight lines $CM$ and $CK$ intersect for the second time the circumscribed circle of the triangle $ABC$ at points $E$ and $F$, respectively. In what ratio does the circumscribed circle of the triangle $BMF$ divide the segment $BE$?

1957 Moscow Mathematical Olympiad, 370

Tags: equal circles , tangent circle , geometry

* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.

JBMO Geometry Collection, 2001

Tags: geometry , trigonometry

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

2023 Ukraine National Mathematical Olympiad, 10.3

Tags: tangency , geometry

Let $I$ be the incenter of the triangle $ABC$, and $P$ be any point on the arc $BAC$ of its circumcircle. Points $K$ and $L$ are chosen on the tangent to the circumcircle $\omega$ of triangle $API$ at point $I$, so that $BK = KI$ and $CL = LI$. Show that the circumcircle of triangle $PKL$ is tangent to $\omega$. [i]Proposed by Mykhailo Shtandenko[/i]

1951 Moscow Mathematical Olympiad, 198

Tags: protractor , geometric construction , point construction , geometry

* On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$

1997 Mexico National Olympiad, 5

Tags: ratio , geometry , area

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.

2024 Assara - South Russian Girl's MO, 2

Tags: geometry

Prove that in any described $8$-gon there is a side that does not exceed the diameter of the inscribed circle in length. [i]P.A.Kozhevnikov[/i]

2000 Romania National Olympiad, 3

Tags: 3d geometry , pyramid , geometry

Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation $$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$

2009 Sharygin Geometry Olympiad, 6

Tags: angle , geometry

Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$. (A.Akopjan)

2010 Oral Moscow Geometry Olympiad, 3

Tags: circles , collinear , geometry

Two circles $w_1$ and $w_2$ intersect at points $A$ and $B$. Tangents $\ell_1$ and $\ell_2$ respectively are drawn to them through point $A$. The perpendiculars dropped from point $B$ to $\ell_2$ and $\ell_1$ intersects the circles $w_1$ and $w_2$, respectively, at points $K$ and $N$. Prove that points $K, A$ and $N$ lie on one straight line.

2022 Flanders Math Olympiad, 1

Tags: geometry , equal angles

The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/7ec65246ff5ecfebc25ca13f3709d1791ceb6c.png[/img] =

1972 IMO Longlists, 45

Tags: geometry

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.

2014 Contests, 2

Tags: parabola , calculus , integration , geometry , conic , calculus computations

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2010 Sharygin Geometry Olympiad, 6

Tags: midpoint , angle chasing , geometry , angle , square

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2002 China Team Selection Test, 2

Tags: circumcircle , incenter , reflection , geometry , parallelogram , inequalities , geometric transformation

Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: equal angles , tangent , geometry , circles

Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

2011 Korea Junior Math Olympiad, 5

Tags: midpoint , circumcircle , geometry , ratio

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.

2008 South East Mathematical Olympiad, 3

Tags: circumcircle , geometry , perpendicular bisector

In $\triangle ABC$, side $BC>AB$. Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$. Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$. $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$. Prove that $O,H,E,M$ lie on a circle.

2010 Princeton University Math Competition, 6

Tags: geometry , circumcircle

All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?