Olympiad problems

1990 AMC 12/AHSME, 25

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? $ \textbf{(A)}\ 1-\frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ \frac{2\sqrt{3}-3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(D)}\ \frac{1}{4} \qquad\textbf{(E)}\ \frac{\sqrt{3}(2-\sqrt{2})}{4} $

1995 All-Russian Olympiad Regional Round, 11.2

Tags: geometry , 3d geometry , cube , parallelepiped

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

2007 National Olympiad First Round, 29

Tags: analytic geometry , graphing lines , rotation , slope , geometry

Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, of a square $ABCD$. If $|BM|=21$, $|DN|=4$, and $|NC|=24$, what is $m(\widehat{MAN})$? $ \textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 30^\circ \qquad\textbf{(C)}\ 37^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ 60^\circ $

1989 IMO Longlists, 62

Tags: area , geometry , convex polygon , algebra

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

2009 Belarus Team Selection Test, 1

Tags: geometric inequality , midpoint , area , geometry

Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$. I. Voronovich

2010 Bulgaria National Olympiad, 1

Tags: rectangle , combinatorics , geometry

A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

2006 Singapore Senior Math Olympiad, 3

Tags: geometry , tangent circle , angle bisector

Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Prove that the line TP bisects $\angle ATB$.

2011 Mediterranean Mathematics Olympiad, 3

Tags: trigonometry , tetrahedron , 3d geometry , sphere , integration , geometry

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

1970 IMO Longlists, 16

Tags: 3d geometry , geometry

Show that the equation $\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$ has no real roots.

2010 Harvard-MIT Mathematics Tournament, 3

Tags: geometry

For $0\leq y\leq 2$, let $D_y$ be the half-disk of diameter 2 with one vertex at $(0,y)$, the other vertex on the positive $x$-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of $D_y$ for all $0\leq y\leq 2$.

2016 Sharygin Geometry Olympiad, 5

Tags: paper , equilateral triangle , geometry

Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle. by A.Khachaturyan

2021 IMO Shortlist, G4

Tags: angle chasing , concurrency , geometry , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2007 Estonia Math Open Junior Contests, 4

Tags: perimeter , ratio , integration , calculus , number theory , geometry

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

2013 239 Open Mathematical Olympiad, 3

Tags: geometry

Inside a regular triangle $ABC$, points $X$ and $Y$ are chosen such that $\angle{AXC} = 120^{\circ}$, $2\angle{XAC} + \angle{YBC} = 90^{\circ}$and $XY = YB = \frac{AC}{\sqrt{3}}$. Prove that point $Y$ lies on the incircle of triangle $ABC$.

2006 Baltic Way, 14

Tags: geometry

There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.

Ukrainian TYM Qualifying - geometry, 2018.17

Tags: construction , incenter , geometry

Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.

2005 Alexandru Myller, 2

Tags: geometry , circumcircle

Let $ ABC $ be a triangle with $ \angle BAC <90^{\circ } . $ In the exterior of $ ABC, $ choose the points $ D,E $ such that $ DA=DB,EA=EC $ and $ \angle ADB =\angle AEC =2\angle BAC . $ Show that the symmetric of $ A $ with respect to the midpoint of the segment $ DE $ is the circumcircle of $ ABC. $

2020 Austrian Junior Regional Competition, 3

Tags: trapezoid , geometry

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$. (Karl Czakler)

2011 NIMO Problems, 5

Tags: circumcircle , perimeter , perpendicular bisector , geometry

In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle. [i]Proposed by Isabella Grabski [/i]

2009 Middle European Mathematical Olympiad, 2

Tags: geometric transformation , combinatorics , geometry

Suppose that we have $ n \ge 3$ distinct colours. Let $ f(n)$ be the greatest integer with the property that every side and every diagonal of a convex polygon with $ f(n)$ vertices can be coloured with one of $ n$ colours in the following way: (i) At least two colours are used, (ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours. Show that $ f(n) \le (n\minus{}1)^2$ with equality for infintely many values of $ n$.

1989 China Team Selection Test, 1

Tags: geometry

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

2014 China Girls Math Olympiad, 1

Tags: circumcircle , geometry , perpendicular bisector

In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url] $\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$, and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$. If $DE \parallel O_1A$, prove that $DC \perp CO_2$.

2017 Taiwan TST Round 3, 2

Tags: orthocenter , geometry , incenter

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

2006 MOP Homework, 4

Tags: pythagorean theorem , tetrahedron , 3d geometry , geometry

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

1990 IMO Longlists, 97

Tags: geometry

In convex hexagon $ABCDEF$, we know that $\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.$ Prove that $AB = CD = EF.$