Olympiad problems

2011 Tournament of Towns, 2

On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.

2016 China Team Selection Test, 6

Tags: geometry , harmonic division , cyclic quadrilateral

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.

2012 Spain Mathematical Olympiad, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

2014 Online Math Open Problems, 17

Tags: circumcircle , ratio , analytic geometry , homothety , geometry , geometric transformation

Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ [i]Proposed by Ray Li[/i]

2010 Contests, 1

Tags: geometry

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.41

Tags: geometric inequality , right angle , perimeter , geometry

Point $C$ lies inside the right angle $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2 OC$.

2020 Harvest Math Invitational Team Round Problems, HMI Team #5

Tags: geometry , euclidean geometry

5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by winnertakeover and Monkey_king1[/i]

2015 APMO, 1

Tags: circumcircle , geometry

Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively. Prove that $AB = V W$ [i]Proposed by Warut Suksompong, Thailand[/i]

2007 Gheorghe Vranceanu, 2

Tags: algebra , complex number , imaginary numbers , complex geometry , analytic geometry , geometry

Let be a natural number $ n\ge 2 $ and an imaginary number $ z $ having the property that $ |z-1|=|z+1|\cdot\sqrt[n]{2} . $ Denote with $ A,B,C $ the points in the Euclidean plane whose representation in the complex plane are the affixes of $ z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} , $ respectively. Prove that $ AB $ is perpendicular to $ AC. $

2021/2022 Tournament of Towns, P2

Tags: analytic geometry , function , ruler , compass , geometry

On a blank paper there were drawn two perpendicular axes $x$ and $y$ with the same scale. The graph of a function $y=f(x)$ was drawn in this coordinate system. Then the $y$ axis and all the scale marks on the $x$ axis were erased. Provide a way how to draw again the $y$ axis using pencil, ruler and compass: (a) $f(x)= 3^x$; (b) $f(x)= \log_a x$, where $a>1$ is an unknown number.

2015 Canada National Olympiad, 2

Tags: geometry , inequalities , geometric inequality

Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]

2008 Sharygin Geometry Olympiad, 2

Tags: geometry

(F.Nilov) Given quadrilateral $ ABCD$. Find the locus of points such that their projections to the lines $ AB$, $ BC$, $ CD$, $ DA$ form a quadrilateral with perpendicular diagonals.

1955 Moscow Mathematical Olympiad, 289

Tags: geometry , locus , equilateral , equal segments

Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.

1985 All Soviet Union Mathematical Olympiad, 395

Tags: area , hexagon , geometry

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

STEMS 2024 Math Cat B, P4

Tags: config geo , geometry

Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying $AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from A satisfying $\angle AQH = 90$ and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$. • Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$. • Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$ such that $(APN)$ is tangent to $AB$. [i]Proposed by Krutarth Shah[/i]

2019 Stanford Mathematics Tournament, 2

Tags: geometry

Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 6$. Point $E$ is outside of the rectangle such that $CE = DE$. Point $D$ is reflected over line $AE$ so that its image, $D'$ , lies on the interior of the rectangle. Point $D'$ is then reflected over diagonal $AC$, and its image lies on side $AB$. What is the length of $DE$?

2021 Moldova EGMO TST, 9

Tags: geometry

Let $ABCD$ be a square and $E$ a on point diagonal $(AC)$, different from its midpoint. $H$ and $K$ are the orthoceneters of triangles $ABE$ and $ADE$. Prove that $AH$ and $CK$ are parallel.

1977 IMO Shortlist, 14

Tags: combinatorics , 3d geometry , pyramid , geometry

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

2016 Middle European Mathematical Olympiad, 5

Tags: circumcircle , law of cosines , geometry

Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.

TNO 2024 Junior, 2

Tags: geometry

Prove that the area enclosed by three semicircles, tangent at their ends, is equal to the area of the circle whose diameter is $CD$, perpendicular to the diameter $AB$.

2011 Today's Calculation Of Integral, 698

Tags: rectangle , limit , integration , perimeter , analytic geometry , calculus , geometry

For a positive integer $n$, let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$-$y$ plane. Denote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\ y$ coordinates of 4 vertices are integers, and the vertices are included in $C_n$. Prove that $\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi$.

2008 AMC 12/AHSME, 20

Tags: analytic geometry , ratio , inradius , incenter , angle bisector , geometry , trigonometry

Triangle $ ABC$ has $ AC\equal{}3$, $ BC\equal{}4$, and $ AB\equal{}5$. Point $ D$ is on $ \overline{AB}$, and $ \overline{CD}$ bisects the right angle. The inscribed circles of $ \triangle ADC$ and $ \triangle BCD$ have radii $ r_a$ and $ r_b$, respectively. What is $ r_a/r_b$? $ \textbf{(A)}\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)$

2011 Mediterranean Mathematics Olympiad, 4

Tags: trigonometry , incenter , angle bisector , geometry

Let $D$ be the foot of the internal bisector of the angle $\angle A$ of the triangle $ABC$. The straight line which joins the incenters of the triangles $ABD$ and $ACD$ cut $AB$ and $AC$ at $M$ and $N$, respectively. Show that $BN$ and $CM$ meet on the bisector $AD$.

2017 Yasinsky Geometry Olympiad, 4

Tags: geometry , construction , distance

Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

2014 Math Prize for Girls Olympiad, 1

Tags: rhombus , geometry

Say that a convex quadrilateral is [i]tasty[/i] if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.