Olympiad problems

2014 IMAC Arhimede, 2

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

1993 Vietnam Team Selection Test, 1

Tags: geometry , rectangle , combinatorics

We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.

2017 India PRMO, 20

Tags: geometry

Attached below:

2016 PAMO, 5

Tags: geometry , midpoint , trapezoid

Let $ABCD$ be a trapezium such that the sides $AB$ and $CD$ are parallel and the side $AB$ is longer than the side $CD$. Let $M$ and $N$ be on the segments $AB$ and $BC$ respectively, such that each of the segments $CM$ and $AN$ divides the trapezium in two parts of equal area. Prove that the segment $MN$ intersects the segment $BD$ at its midpoint.

2020 JHMT, 5

Tags: geometry

Let $A$ and $B$ be fixed points in the Euclidean plane with $AB = 6$. Let $R$ be the region of points in the plane such that, for each $P \in R$, there exists a point $C$ such that $AC = 3$ and $P$ does not lie outside $\vartriangle ABC$. Compute the greatest integer less than or equal to the area of $R$.

Novosibirsk Oral Geo Oly VII, 2020.1

Tags: geometry , dodecagon , area

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. [img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]

2021 Sharygin Geometry Olympiad, 10

Tags: geometry , isotomic

Prove that two isotomic lines of a triangle cannot meet inside its medial triangle. [i](Two lines are isotomic lines of triangle $ABC$ if their common points with $BC, CA, AB$ are symmetric with respect to the midpoints of the corresponding sides.)[/i]

1970 IMO Longlists, 31

Tags: inequalities , geometry

Prove that for any triangle with sides $a, b, c$ and area $P$ the following inequality holds: \[P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.\] Find all triangles for which equality holds.

2009 Tournament Of Towns, 4

Tags: reflection , trigonometry , geometry , geometric transformation

A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$ [i](4 points)[/i]

Novosibirsk Oral Geo Oly IX, 2017.1

Tags: geometry , grid , min

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

2019 Durer Math Competition Finals, 5

Tags: concurrency , arc midpoint , geometry

Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 8

Tags: geometry

Place three discs with radius $ r$ in a square with sides of length 1 so that the discs do not intersect: as on the figure. What is the greatest possible value of $ r$? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number8.jpg[/img] A. $ \frac {1}{3}$ B. $ \frac {1}{4}$ C. $ \frac {\sqrt {2}}{6}$ D. $ 2 \sqrt {2} \minus{} \sqrt {6}$ E. $ \frac {\sqrt {2}}{1 \plus{} 2 \sqrt {2} \plus{} \sqrt {3}}$

2025 Euler Olympiad, Round 1, 9

Tags: geometry

Three circles with radii $1$, $2$, and $3$ are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles. [i]Proposed by Tamar Turashvili, Georgia [/i]

2008 Sharygin Geometry Olympiad, 2

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

(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.

2016 Bundeswettbewerb Mathematik, 2

Tags: area of a triangle , game theory , area , geometry

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

1954 AMC 12/AHSME, 26

Tags: geometry , similar triangles

The straight line $ \overline{AB}$ is divided at $ C$ so that $ AC\equal{}3CB$. Circles are described on $ \overline{AC}$ and $ \overline{CB}$ as diameters and a common tangent meets $ AB$ produced at $ D$. Then $ BD$ equals: $ \textbf{(A)}\ \text{diameter of the smaller circle} \\ \textbf{(B)}\ \text{radius of the smaller circle} \\ \textbf{(C)}\ \text{radius of the larger circle} \\ \textbf{(D)}\ CB\sqrt{3}\\ \textbf{(E)}\ \text{the difference of the two radii}$

2020 JHMT, 10

Tags: geometry

Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/d/6/5e3faa5755628cceb2b5c39c95f6126669a3c6.png[/img]

2016 Sharygin Geometry Olympiad, 3

Tags: locus , circumcircle , geometry , inradius

Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

2022 Yasinsky Geometry Olympiad, 2

Tags: angle bisector , geometry , bisector , angle

In the triangle $ABC$, angle $C$ is four times smaller than each of the other two angle The altitude $AK$ and the angle bisector $AL$ are drawn from the vertex of the angle $A$. It is known that the length of $AL$ is equal to $\ell$. Find the length of the segment $LK$. (Gryhoriy Filippovskyi)

Indonesia MO Shortlist - geometry, g5

Tags: geometry , collinear

Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

1967 AMC 12/AHSME, 20

Tags: geometry , geometric series , ratio

A circle is inscribed in a square of side $m$, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches: $\textbf{(A)}\ \frac{\pi m^2}{2}\qquad \textbf{(B)}\ \frac{3\pi m^2}{8}\qquad \textbf{(C)}\ \frac{\pi m^2}{3}\qquad \textbf{(D)}\ \frac{\pi m^2}{4}\qquad \textbf{(E)}\ \frac{\pi m^2}{8}$

2007 Kazakhstan National Olympiad, 2

Tags: geometry , circumcircle

Let $ABC$ be an isosceles triangle with $AC = BC$ and $I$ is the center of the inscribed circle. The point $P$ lies on the circle circumscribed about the triangle $AIB$ and lies inside the triangle $ABC$. Straight lines passing through point $P$ parallel to $CA$ and $CB$ intersect $AB$ at points $D$ and $E$, respectively. The line through $P$ which is parallel to $AB$ intersects $CA$ and $CB$ at points $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ meet at the circumcircle of triangle $ABC$.

2014 PUMaC Geometry B, 1

Tags: angle bisector , geometry

Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.

2014 Math Prize For Girls Problems, 8

Tags: trig identities , area of a triangle , geometry , heron's formula , law of cosines , trigonometry

A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

2002 AMC 10, 20

Tags: rotation , symmetry , linear algebra , geometric transformation , geometry

Let $ a$, $ b$, and $ c$ be real numbers such that $ a \minus{} 7b \plus{} 8c \equal{} 4$ and $ 8a \plus{} 4b \minus{} c \equal{} 7$. Then $ a^2 \minus{} b^2 \plus{} c^2$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$