Olympiad problems

1996 AIME Problems, 14

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

2009 Dutch IMO TST, 5

Tags: equal segments , rational , polygon , analytic geometry , geometry

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

2010 IMAC Arhimede, 4

Tags: segment , tangential quadrilateral , inradius , geometry

Let $M$ and $N$ be two points on different sides of the square $ABCD$. Suppose that segment $MN$ divides the square into two tangential polygons. If $R$ and $r$ are radii of the circles inscribed in these polygons ($R> r$), calculate the length of the segment $MN$ in terms of $R$ and $r$. (Moldova)

2012 China Second Round Olympiad, 5

Tags: pyramid , 3d geometry , geometry , sphere

Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.

1996 Korea National Olympiad, 8

Tags: angle bisector , circumcircle , geometry

Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.

2024 USAJMO, 1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral. [i]Proposed by Evan O'Dorney[/i]

2011 Turkey Junior National Olympiad, 2

Tags: analytic geometry , slope , power of a point , radical axis , circumcircle , graphing lines , geometry

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

1951 Miklós Schweitzer, 15

Tags: geometric transformation , advanced fields , rotation , geometry

Let the line $ z\equal{}x, \, y\equal{}0$ rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.

2010 ELMO Shortlist, 2

Tags: trapezoid , parallelogram , geometry

Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$. [i]Brian Hamrick.[/i]

2023 Belarusian National Olympiad, 10.5

Tags: geometry , area

On hyperbola $y=\frac{1}{x}$ points $A_1,\ldots,A_{10}$ are chosen such that $(A_i)_x=2^{i-1}a$, where $a$ is some positive constant. Find the area of $A_1A_2 \ldots A_{10}$

1997 Akdeniz University MO, 5

Tags: perimeter , geometry , incenter

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

1999 Canada National Olympiad, 2

Tags: geometry

Let $ABC$ be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$. Prove that the arc of the circle that is inside the triangle always has the same length.

2014 Sharygin Geometry Olympiad, 3

Tags: kosnita , circumcircle , geometry

Let $ABC$ be an isosceles triangle with base $AB$. Line $\ell$ touches its circumcircle at point $B$. Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$. Prove that $D$, $E$, and $F$ are collinear.

1969 IMO Shortlist, 33

Tags: covering , geometry , circles

$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)

2022 USA TSTST, 2

Tags: geometry

Let $ABC$ be a triangle. Let $\theta$ be a fixed angle for which \[\theta<\frac12\min(\angle A,\angle B,\angle C).\] Points $S_A$ and $T_A$ lie on segment $BC$ such that $\angle BAS_A=\angle T_AAC=\theta$. Let $P_A$ and $Q_A$ be the feet from $B$ and $C$ to $\overline{AS_A}$ and $\overline{AT_A}$ respectively. Then $\ell_A$ is defined as the perpendicular bisector of $\overline{P_AQ_A}$. Define $\ell_B$ and $\ell_C$ analogously by repeating this construction two more times (using the same value of $\theta$). Prove that $\ell_A$, $\ell_B$, and $\ell_C$ are concurrent or all parallel.

1989 IMO Longlists, 8

Tags: rotation , geometric transformation , geometry

Let $ Ax,By$ be two perpendicular semi-straight lines, being not complanar, (non-coplanar rays) such that $ AB$ is the their common perpendicular, and let $ M$ and $ N$ be the two variable points on $ Ax$ and $ Bx,$ respectively, such that $ AM \plus{} BN \equal{} MN.$ [b](a)[/b] Prove that there exist infinitely many lines being co-planar with each of the straight lines $ MN.$ [b](b)[/b] Prove that there exist infinitely many rotations around a fixed axis $ \delta$ mapping the line $ Ax$ onto a line coplanar with each of the lines $ MN.$

2004 Germany Team Selection Test, 2

Tags: angle bisector , geometry , menelaus

In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

1990 IMO Shortlist, 11

Tags: trigonometry , geometry , similar triangles , tangent

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

1999 Tournament Of Towns, 6

Tags: rectangle , geometry

Inside a rectangular piece of paper $n$ rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper? (A Shapovalov)

2007 Sharygin Geometry Olympiad, 2

Tags: diagonal , isosceles , isosceles triangle , quadrilateral , geometry

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

2000 Moldova National Olympiad, Problem 8

Tags: geometric inequality , geometry

A circle with radius $r$ touches the sides $AB,BC,CD,DA$ of a convex quadrilateral $ABCD$ at $E,F,G,H$, respectively. The inradii of the triangles $EBF,FCG,GDH,HAE$ are equal to $r_1,r_2,r_3,r_4$. Prove that $$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$

Kyiv City MO 1984-93 - geometry, 1991.9.3

Tags: geometry , area

The point $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of the triangle $BKM$.

2004 Tournament Of Towns, 1

Tags: geometry , collinear

Segments $AB, BC$ and $CD$ of the broken line $ABCD$ are equal and are tangent to a circle with centre at the point $O$. Prove that the point of contact of this circle with $BC$, the point $O$ and the intersection point of $AC$ and $BD$ are collinear.

2023 VN Math Olympiad For High School Students, Problem 3

Tags: geometry

Given a triangle $ABC$ isosceles at $A.$ A point $P$ lying inside the triangle such that $\angle PBC=\angle PCA$ and let $M$ be the midpoint of $BC.$ Prove that: $\angle APB+ \angle MPC =180^{\circ}.$

2001 Irish Math Olympiad, 3

Tags: geometry

In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.