Olympiad problems

2015 Tuymaada Olympiad, 7

Tags: geometry , incenter

$CL$ is bisector of $\angle C$ of $ABC$ and intersect circumcircle at $K$. $I$ - incenter of $ABC$. $IL=LK$. Prove, that $CI=IK$ [i]D. Shiryaev [/i]

2012 Switzerland - Final Round, 10

Tags: geometry , concyclic , projection

Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.

1997 Tournament Of Towns, (563) 4

Tags: geometry , combinatorial geometry , combinatorics , regular

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

2013 Germany Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

2024 JBMO TST - Turkey, 1

Tags: geometry

In the acute-angled triangle $ABC$, $P$ is the midpoint of segment $BC$ and the point $K$ is the foot of the altitude from $A$. $D$ is a point on segment $AP$ such that $\angle BDC=90$. Let $(ADK) \cap BC=E$ and $(ABC) \cap AE=F$. Prove that $\angle AFD=90$.

LMT Team Rounds 2021+, B19

Tags: geometry

Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk. [i]Proposed by Kevin Zhao[/i]

2009 Today's Calculation Of Integral, 419

Tags: calculus , integration , parabola , geometry , calculus computations , conic

In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant. (1) Find the equation of $ l$. (2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.

2021 Honduras National Mathematical Olympiad, Problem 4

Tags: geometry

Consider parallelogram $ABCD$ and let $E$ be the midpoint of $BC$. In segment $DE$ a point $F$ is chosen such that $AF$ is perpendicular to $DE$. Prove that $\angle CDE=\angle EFB$.

2013 Harvard-MIT Mathematics Tournament, 3

Tags: circumcircle , rectangle , geometry

Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.

2021 Canadian Mathematical Olympiad Qualification, 3

Tags: geometry , pentagon

$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$. The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$. Prove that $AX = AY$.

2018 JBMO TST-Turkey, 3

Tags: geometry , concyclic , arc midpoint , circles

Let $H$ be the orthocenter of an acute angled triangle $ABC$. Circumcircle of the triangle $ABC$ and the circle of diameter $[AH]$ intersect at point $E$, different from $A$. Let $M$ be the midpoint of the small arc $BC$ of the circumcircle of the triangle $ABC$ and let $N$ the midpoint of the large arc $BC$ of the circumcircle of the triangle $BHC$ Prove that points $E, H, M, N$ are concyclic.

1991 Baltic Way, 17

Tags: geometric transformation , reflection , analytic geometry , geometry

Let the coordinate planes have the reflection property. A ray falls onto one of them. How does the final direction of the ray after reflecting from all three coordinate planes depend on its initial direction?

1963 Dutch Mathematical Olympiad, 1

Tags: geometry , perpendicular bisector , tangent

In a plane are given a straight line $\ell$ and a point $P$ not located on $\ell$. Is there a circle in this plane such that there exist more than three different points $S$ on $\ell$ with the property that the perpendicular bisector of $PS$ is tangent to the circle ? Explain the answer.

2018 CMIMC Individual Finals, 2

Tags: geometry , trapezoid , angle bisector

Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.

2002 Romania National Olympiad, 4

Tags: 3d geometry , prism , geometry

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that: $a)$ $n=3$; $b)$ the prism is regular.

2014 Balkan MO Shortlist, G1

Tags: geometry

Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$, respectively such that $\angle BAM = \angle MNC$. Suppose that the lines $MN$ and $AB$ intersects at $P$. Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$

2017 Sharygin Geometry Olympiad, 1

Tags: geometry , circumcircle

If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint

1980 Bundeswettbewerb Mathematik, 2

Tags: geometry , distance , triangle , bisector

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.

2024 Sharygin Geometry Olympiad, 10.1

Tags: geo , geometry

The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. The bisector of angle $ABD$ meets $AC$ at point $E$, and the bisector of angle $ACD$ meets $BD$ at point $F$. Prove that the lines $AF$ and $DE$ meet on the median of triangle $APD$.

2004 National High School Mathematics League, 1

Tags: geometry

In acute triangle $ABC$, point $H$ is the intersection point of heights $CE$ on side $AB$ and $BD$ on side $AC$. A circle with diameter $DE$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. $FG$ and $AH$ intersect at $K$. If $BC=25,BD=20, BE=7$, find the length of $AK$.

2025 Kyiv City MO Round 2, Problem 4

Tags: geometry

Let $ H $ be the orthocenter, and $ O $ be the circumcenter of $ \triangle ABC $. The line $ AH $ intersects the circumcircle of $ \triangle ABC $ at point $ N $ for the second time. The circumcircle of $ \triangle BOC $, with center at point $ Q $, intersects the line $ OH $ at point $ X $ for the second time. Prove that the points $ O, Q, N, X $ lie on the same circle. [i]Proposed by Matthew Kurskyi[/i]

1969 IMO Shortlist, 10

Tags: trigonometric equation , trigonometry , geometry , median

$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

1986 Poland - Second Round, 6

Tags: geometry , geometric inequalities , area

In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

2015 Iran Geometry Olympiad, 2

Tags: geometry

let $ ABC $ an equilateral triangle with circum circle $ w $ let $ P $ a point on arc $ BC $ ( point $ A $ is on the other side ) pass a tangent line $ d $ through point $ P $ such that $ P \cap AB = F $ and $ AC \cap d = L $ let $ O $ the center of the circle $ w $ prove that $ \angle LOF > 90^{0} $

2007 Junior Balkan MO, 2

Tags: trigonometry , incenter , function , circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.