Olympiad problems

Kvant 2021, M2635

In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$. [i]Porposed by O. Titov[/i]

2023 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , cyclic quadrilateral

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$. Segments $AC$ and $BD$ intersect at $E$. Circle $\gamma$ passes through $E$ and is tangent to $\Gamma$ at $A$. Suppose the circumcircle of triangle $BCE$ is tangent to $\gamma$ at $E$ and is tangent to line $CD$ at $C$. Suppose that $\Gamma$ has radius $3$ and $\gamma$ has radius $2$. Compute $BD$.

1953 Moscow Mathematical Olympiad, 255

Tags: pyramid , 3d geometry , cube , geometry

Divide a cube into three equal pyramids.

2024 ELMO Shortlist, G2

Tags: geometry

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

2015 Iran MO (3rd round), 2

Tags: geometry , circumcircle

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.

2015 South East Mathematical Olympiad, 6

Tags: geometry

In $\triangle ABC$, we have three edges with lengths $BC=a, \, CA=b \, AB=c$, and $c<b<a<2c$. $P$ and $Q$ are two points of the edges of $\triangle ABC$, and the straight line $PQ$ divides $\triangle ABC$ into two parts with the same area. Find the minimum value of the length of the line segment $PQ$.

2021 Saudi Arabia Training Tests, 6

Tags: perpendicular , tangent , geometry

Let $A$ be a point lies outside circle $(O)$ and tangent lines $AB$, $AC$ of $(O)$. Consider points $D, E, M$ on $(O)$ such that $MD = ME$. The line $DE$ cuts $MB$, $MC$ at $R, S$. Take $X \in OB$, $Y \in OC$ such that $RX, SY \perp DE$. Prove that $XY \perp AM$.

2009 Ukraine National Mathematical Olympiad, 4

Tags: geometry , angle bisector

In the triangle $ABC$ given that $\angle ABC = 120^\circ .$ The bisector of $\angle B$ meet $AC$ at $M$ and external bisector of $\angle BCA$ meet $AB$ at $P.$ Segments $MP$ and $BC$ intersects at $K$. Prove that $\angle AKM = \angle KPC .$

2017 Adygea Teachers' Geometry Olympiad, 1

Tags: trapezoid , geometry , area

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

2006 India National Olympiad, 4

Tags: geometry , combinatorics , rectangle

Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].

2009 South East Mathematical Olympiad, 4

Tags: geometry , combinatorics

Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points . Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .

1985 IMO, 5

Tags: angle , geometry , circumcircle , triangle

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

1977 IMO Longlists, 52

Tags: geometry , calculus

Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$

2021 OMpD, 5

Tags: angle , circumcircle , geometry

Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.

2006 Cuba MO, 9

Tags: equal angles , perpendicular , geometry

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

2008 Baltic Way, 16

Tags: geometry , rhombus , parallelogram

Let $ABCD$ be a parallelogram. The circle with diameter $AC$ intersects the line $BD$ at points $P$ and $Q$. The perpendicular to the line $AC$ passing through the point $C$ intersects the lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P,Q,X$ and $Y$ lie on the same circle.

2018 CMIMC Geometry, 6

Tags: geometry

Let $\omega_1$ and $\omega_2$ be intersecting circles in the plane with radii $12$ and $15$, respectively. Suppose $\Gamma$ is a circle such that $\omega_1$ and $\omega_2$ are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Similarly, $\ell$ is a line that is tangent to $\omega_1$ and $\omega_2$ at $Y_1$ and $Y_2$, respectively. If $X_1X_2=18$ and $Y_1Y_2=9$, what is the radius of $\Gamma$?

2004 Harvard-MIT Mathematics Tournament, 7

Tags: algebra , geometry

Farmer John is grazing his cows at the origin. There is a river that runs east to west $50$ feet north of the origin. The barn is $100$ feet to the south and $80$ feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.

1970 Czech and Slovak Olympiad III A, 5

Tags: rotation , geometry , line , geometric transformation , locus , homothety

Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$

2012 Macedonia National Olympiad, 4

Tags: power of a point , geometry , projective geometry

A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.

Novosibirsk Oral Geo Oly VII, 2020.4

Tags: bisects segment , geometry

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

2006 Lithuania Team Selection Test, 4

Tags: combinatorics , geometry

Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.

2021 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , concyclic

The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.

2022 JHMT HS, 9

Tags: geometry

In convex quadrilateral $KALE$, angles $\angle KAL$, $\angle AKL$, and $\angle ELK$ measure $110^\circ$, $50^\circ$, and $10^\circ$, respectively. Given that $KA = LE$ and that $\overline{KL}$ and $\overline{AE}$ intersect at point $X$, compute the value of $\tfrac{KX^2}{AL\cdot EX}$.

1974 IMO Shortlist, 11

Tags: dissection , geometry , chessboard , rectangle , combinatorics

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]