Olympiad problems

Croatia MO (HMO) - geometry, 2022.7

Tags: geometry , perpendicular

In the triangle $ABC$ holds $|AB| = |AC|$ and the inscribed circle touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ at the points $D$, $E$ and $F$ respectively . The perpendicular from the point $D$ to the line $EF$ intersects the side $\overline{AB}$ at the point $G$, and the circles circumscribed around the triangles $AEF$ and $ABC$ intersect at the points $A $and $T$. Prove that the lines $T G$ and $T F$ are perpendicular.

2010 Germany Team Selection Test, 1

Tags: geometry , symmetry , triangle , parallelogram

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

1988 AIME Problems, 11

Tags: analytic geometry , graphing lines , slope , geometry , geometric transformation , ratio , complex number

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \[ \sum_{k = 1}^n (z_k - w_k) = 0. \] For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.

2004 India IMO Training Camp, 1

Tags: geometry , circumcircle , trigonometry

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

2005 Today's Calculation Of Integral, 49

Tags: calculus , integration , trigonometry , geometry , function , calculus computations

For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$

2023 Sharygin Geometry Olympiad, 19

Tags: geometry , locus , circle intersections

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.

2002 South africa National Olympiad, 1

Tags: trigonometry , vector , analytic geometry , trig identities , law of cosines , geometry

Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$, prove that $AC \perp BD$.

1989 AMC 12/AHSME, 19

Tags: geometry , trigonometry , circumcircle , perimeter , trig identities , law of sines , area of a triangle

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ \frac{18}{\pi^2} \qquad \textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad \textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad \textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$

2025 Ukraine National Mathematical Olympiad, 11.7

Tags: geometry

Given a triangle $ABC$, an arbitrary point $D$ is chosen on the side $AC$. In triangles $ABD$ and $CBD$, the angle bisectors $BK$ and $BL$ are drawn, respectively. The point $O$ is the circumcenter of $\triangle KBL$. Prove that the second intersection point of the circumcircles of triangles $ABL$ and $CBK$ lies on the line $OD$. [i]Proposed by Anton Trygub[/i]

2016 IFYM, Sozopol, 4

Tags: geometry

Let $ABCD$ be a convex quadrilateral. The circle $\omega_1$ is tangent to $AB$ in $S$ and the continuations after $A$ and $B$ of sides $DA$ and $CB$, circle $\omega_2$ with center $I$ is tangent to $BC$ and the continuations after $B$ and $C$ of sides $AB$ and $DC$, circle $\omega_3$ is tangent to $CD$ in $T$ and the continuations after $C$ and $D$ of sides $BC$ and $AD$, and circle $\omega_4$ with center $J$ is tangent to $DA$ and the continuations after $D$ and $A$ of sides $CD$ and $BA$. Prove that points $S$ and $T$ are on equal distance from the middle point of segment $IJ$.

2013 India Regional Mathematical Olympiad, 3

Tags: perpendicular bisector , geometry

In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.

2007 Iran Team Selection Test, 3

Tags: geometry , incenter , geometric transformation , homothety , trapezoid , analytic geometry , reflection

Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

2018 USA TSTST, 5

Tags: geometry

Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$. [i]Ankan Bhattacharya and Evan Chen[/i]

MIPT student olimpiad spring 2023, 3

Tags: geometry , 3d geometry , sphere

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

2017 USAJMO, 5

Tags: geometry

Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM=CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.

1976 USAMO, 1

Tags: geometry , rectangle

(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with [i]any[/i] such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color. (b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.

VMEO IV 2015, 10.2

Tags: circumcircle , geometry

Given triangle $ABC$ and $P,Q$ are two isogonal conjugate points in $\triangle ABC$. $AP,AQ$ intersects $(QBC)$ and $(PBC)$ at $M,N$, respectively ( $M,N$ be inside triangle $ABC$) 1. Prove that $M,N,P,Q$ locate on a circle - named $(I)$ 2. $MN\cap PQ$ at $J$. Prove that $IJ$ passed through a fixed line when $P,Q$ changed

2003 Paraguay Mathematical Olympiad, 5

Tags: geometry , square , area

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.

2024 Yasinsky Geometry Olympiad, 1

Tags: geometry

Inside triangle $ ABC $, a point $ D $ is chosen such that $ \angle ADB = \angle ADC $. The rays $ BD $ and $ CD $ intersect the circumcircle of triangle $ ABC $ at points $ E $ and $ F $, respectively. On segment $ EF $, points $ K $ and $ L $ are chosen such that \linebreak $ \angle AKD = 180^\circ - \angle ACB $ and $ \angle ALD = 180^\circ - \angle ABC $, with segments $ EL $ and $ FK $ \linebreak not intersecting line $ AD $. Prove that $ AK = AL $. [i]Proposed by Matthew Kurskyi[/i]

2006 Princeton University Math Competition, 2

Tags: geometry

In triangle $ABC$, $R$ is the midpoint of $BC$ and $CS = 3SA$. If $x$ is the area of $CRS$, $y$ is the area of $RBT$, $z$ is the area of $ATS$, and $y^2 = xz$, then what is the value of $\frac{AT}{TB}$? Express your answer in the form $\frac{a+b\sqrt{c}}{d}$ , where $a,b,c,d$ are integers, $d$ is positive and as small as possible, and $c$ is squarefree. [img]https://cdn.artofproblemsolving.com/attachments/f/d/65b443628329610ff41d30b95e5ebd0c914f20.jpg[/img]

Swiss NMO - geometry, 2010.9

Tags: circumcircle , power of a point , radical axis , geometry

Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

2016 CMIMC, 2

Tags: geometry

Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?

2019 Purple Comet Problems, 1

Tags: geometry , rectangle

The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon. [img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]

2005 India National Olympiad, 1

Tags: ratio , geometry

Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$ than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.

2021 Princeton University Math Competition, A4 / B6

Tags: geometry

Let $BCDE$ be a trapezoid with $BE\parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be the intersection of $AM$ and $BE$. If $BX$ can be written as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, find $a + b$