Olympiad problems

2024 Argentina Cono Sur TST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle. The point $B'$ of the line $CA$ is such that $A$, $C$ and $B'$ are in that order on the line and $B'C=AB$; the point $C'$ of the line $AB$ is such that $A$, $B$ and $C'$ are in that order on the line and $C'B=AC$. Prove that the circumcenter of triangle $AB'C'$ belongs to the circumcircle of triangle $ABC$.

1998 National Olympiad First Round, 33

Tags: symmetry , geometry

Chord $ \left[AD\right]$ is perpendicular to the diameter $ \left[BC\right]$ of a circle. Let $ E$ and $ F$ be the midpoints of the arcs $ AC$ and $ CD$, respectively. If $ AD\bigcap BE\equal{}\left\{G\right\}$, $ AF\bigcap BC\equal{}\left\{H\right\}$ and $ m(AC)\equal{}\alpha$, find the measure of angle $ BHC$ in terms of $ \alpha$. $\textbf{(A)}\ 90{}^\circ \minus{}\frac{\alpha }{2} \qquad\textbf{(B)}\ 60{}^\circ \minus{}\frac{\alpha }{3} \qquad\textbf{(C)}\ \alpha \minus{}30{}^\circ \\ \qquad\textbf{(D)}\ 15{}^\circ \plus{}\frac{\alpha }{2} \qquad\textbf{(E)}\ \frac{180{}^\circ \minus{}2\alpha }{3}$

2006 All-Russian Olympiad, 2

Tags: geometry , 3d geometry , modular arithmetic , relatively prime , number theory

If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.

1948 Moscow Mathematical Olympiad, 150

Tags: symmetry , geometry

Can a figure have a greater than $1$ and finite number of centers of symmetry?

2006 Kyiv Mathematical Festival, 4

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2008 Tournament Of Towns, 3

Tags: perimeter , geometry

Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$

2007 Mathematics for Its Sake, 1

Tags: trapezoid , geometry

Consider a trapezium $ ABCD $ in which $ AB\parallel CD. $ Show that $$ (AC^2+AB^2-BC^2)(BD^2-BC^2+CD^2) =(AC^2-AD^2+CD^2)(BD^2+AB^2-AD^2) . $$

2010 Vietnam Team Selection Test, 2

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

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

2014 Romania Team Selection Test, 1

Tags: geometric transformation , homothety , geometry

Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.

LMT Team Rounds 2021+, 14

Tags: geometry

In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.

2012 Indonesia TST, 3

Tags: geometry

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

2016 Sharygin Geometry Olympiad, 1

Tags: geometry , angle chasing , parallelogram , circumcircle

The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.

Ukrainian TYM Qualifying - geometry, VI.14

Tags: geometry , bicentric quadrilateral , geometric inequality

A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds. Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.

2007 Kyiv Mathematical Festival, 2

Tags: geometric transformation , reflection , trapezoid , geometry

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

Kvant 2020, M600

Tags: geometry

Two cyclists ride on two intersecting circles. Each of them rides on his own circle at a constant speed. Having left at the same time from one of the points of intersection of the circles and having made one lap each, the cyclists meet again at this point. Prove that there exists a fixed point in the plane, the distances from which to cyclists are the same all the time, regardless of the directions they travel in. [i]Proposed by N. Vasiliev and I. Sharygin[/i]

1995 National High School Mathematics League, 3

Tags: geometry , rhombus

Inscribed Circle of rhombus $ABCD$ touches $AB,BC,CD,DA$ at $E,F,G,H$. $l_1,l_2$ are two lines that are tangent to the circle. $l_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q$. Prove that $MQ/\! /NP$.

2001 JBMO ShortLists, 10

Tags: ratio , incenter , geometry

A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.

2011 Morocco National Olympiad, 4

Tags: geometry

Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.

2003 Italy TST, 2

Tags: circumcircle , geometry

Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.

2016 Chile National Olympiad, 2

Tags: geometry , area

For a triangle $\vartriangle ABC$, determine whether or not there exists a point $P$ on the interior of $\vartriangle ABC$ in such a way that every straight line through $P$ divides the triangle $\vartriangle ABC$ in two polygons of equal area.

1952 AMC 12/AHSME, 16

Tags: geometry , rectangle

If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by: $ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$

2016 BMT Spring, 5

Tags: geometry

Convex pentagon $ABCDE$ has the property that $\angle ADB = 20^o$, $\angle BEC = 16^o$, $\angle CAD = 3^o$,and $\angle DBE = 12^o$. What is the measure of $\angle ECA$?

2008 Iran MO (3rd Round), 3

Tags: projective geometry , geometry

Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.

PEN N Problems, 4

Tags: 3d geometry , algebra , polynomial , arithmetic sequence , more sequences , geometry

Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.

1958 AMC 12/AHSME, 40

Tags: invariant , algebra , polynomial , linear algebra , matrix , vector , geometry

Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals: $ \textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \minus{}17$