Olympiad problems

2008 Mexico National Olympiad, 2

Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.

1988 ITAMO, 3

Tags: radius , geometry , distance , regular , pentagon

A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.

2013 Harvard-MIT Mathematics Tournament, 4

Tags: geometry

Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, and radii $r_1$ and $r_2$, respectively. Suppose that $O_2$ is on $\omega_1$. Let $A$ be one of the intersections of $\omega_1$ and $\omega_2$, and $B$ be one of the two intersections of line $O_1O_2$ with $\omega_2$. If $AB = O_1A$, find all possible values of $\frac{r_1}{r_2}$ .

2007 Moldova Team Selection Test, 3

Tags: ratio , trigonometry , circumcircle , trapezoid , radical axis , projective geometry , geometry

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

1999 Romania National Olympiad, 3

Tags: geometry , geometric inequality

Let $ABCD$ be a convex quadrilateral with $\angle BAC = \angle CAD$, $\angle ABC =\angle ACD$, $(AD \cap (BC =\{E\}$, $(AB \cap (DC = \{F\}$. Prove that: a) $AB\cdot DE = BC \cdot CE$ b) $AC^2 < \frac12 (AD \cdot AF + AB \cdot AE).$

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , incenter , arc midpoint , bisects segment , circumcircle

Let $ABC$ a triangle, $I$ the incenter, $D$ the contact point of the incircle with the side $BC$ and $E$ the foot of the bisector of the angle $A$. If $M$ is the midpoint of the arc $BC$ which contains the point $A$ of the circumcircle of the triangle $ABC$ and $\{F\} = DI \cap AM$, prove that $MI$ passes through the midpoint of $[EF]$.

2010 Morocco TST, 4

Tags: circumcircle , pentagon , geometry

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

2011 NZMOC Camp Selection Problems, 2

Tags: geometry , diameter

Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

2023 VIASM Summer Challenge, Problem 4

Tags: geometry

Let $ABC$ be a non-isosceles acute triangle with $(I)$ be it's incircle. $D, E, F$ are the touchpoints of $(I)$ and $BC, CA, AB,$ respectively. $P$ is the perpendicular projection of $D$ on $EF.$ $DP$ intersects $(I)$ at the second point $K,L$ is the perpendicular projection of $A$ on $IK.$ $(LEC), (LFB) $ intersects $(I)$ the second time at $M, N,$ respectively. Prove that $M, N, P$ are collinear.

2009 Italy TST, 2

Tags: geometry , circumcircle , trapezoid , geometric transformation , reflection , power of a point , radical axis

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

1991 Vietnam National Olympiad, 2

Tags: geometry , circumcircle , inequalities

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that: $\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

Denmark (Mohr) - geometry, 2001.3

Tags: geometry , min , square

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

2004 Federal Math Competition of S&M, 2

Tags: geometry

Let $r$ be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed $3r$

1976 IMO Longlists, 22

Tags: 3d geometry , sphere , geometry

A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.

2013 NIMO Summer Contest, 13

Tags: geometry , trapezoid , trigonometry

In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$. [i]Proposed by Lewis Chen[/i]

1952 AMC 12/AHSME, 29

Tags: pythagorean theorem , geometry

In a circle of radius $ 5$ units, $ CD$ and $ AB$ are perpendicular diameters. A chord $ CH$ cutting $ AB$ at $ K$ is $ 8$ units long. The diameter $ AB$ is divided into two segments whose dimensions are: $ \textbf{(A)}\ 1.25, 8.75 \qquad\textbf{(B)}\ 2.75,7.25 \qquad\textbf{(C)}\ 2,8 \qquad\textbf{(D)}\ 4,6$ $ \textbf{(E)}\ \text{none of these}$

Swiss NMO - geometry, 2022.1

Tags: ratio , geometry

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

2015 JBMO Shortlist, 3

Tags: geometry

Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

2008 Tuymaada Olympiad, 4

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

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

1974 Canada National Olympiad, 2

Tags: geometry , rectangle , trigonometry , parallelogram

Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then \[\angle DBC+\angle DPC = \angle DQC.\]

2015 Cuba MO, 8

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle and $D$ be the foot of the altiutude from $A$ on $BC$, $E$ and $F$ are the midpoints of $BD$ and $DC$ respectively. $O$ and $Q$ are the circumcenters of the triangles $\vartriangle BF$ and $\vartriangle ACE$ respectively. $P$ is the intersection point of $OE$ and $QF$, show that $PB = PC$.

2021 Final Mathematical Cup, 2

Tags: geometry , circumcenter , equal angles

Let $ABC$ be an acute triangle, where $AB$ is the smallest side and let $D$ be the midpoint of $AB$. Let $P$ be a point in the interior of the triangle $ABC$ such that $\angle CAP = \angle CBP = \angle ACB$. From the point $P$, we draw perpendicular lines on $BC$ and $AC$ where the intersection point with $BC$ is $M$, and with $AC$ is $N$ . Through the point $M$ we draw a line parallel to $AC$, and through $N$ parallel to $BC$. These lines intercept at the point $K$. Prove that $D$ is the center of the circumscribed circle for the triangle $MNK$.

2008 Postal Coaching, 1

Tags: geometry , concyclic , parallelogram , cyclic quadrilateral

Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.

2010 Sharygin Geometry Olympiad, 1

Tags: circumcircle , angle bisector , tangent , geometry

For a nonisosceles triangle $ABC$, consider the altitude from vertex $A$ and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex $A$.

2009 Sharygin Geometry Olympiad, 8

Tags: symmetry , geometry

Some polygon can be divided into two equal parts by three different ways. Is it certainly valid that this polygon has an axis or a center of symmetry?