Olympiad problems

1997 Dutch Mathematical Olympiad, 5

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$. a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

2014 USA Team Selection Test, 2

Tags: analytic geometry , cyclic quadrilateral , orthocenter , area , geometry

Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

2019 Olympic Revenge, 3

Tags: geometry , incenter , euler line , construction

Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.

1986 IMO Shortlist, 14

Tags: geometry , incenter , circumcircle , collinearity

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2006 MOP Homework, 3

Tags: vector , geometry

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

2006 AMC 8, 7

Tags: geometry

Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius. $ \textbf{(A)}\ X, Y, Z \qquad \textbf{(B)}\ Z, X, Y \qquad \textbf{(C)}\ Y, X, Z \qquad \textbf{(D)}\ Z, Y, X \qquad \textbf{(E)}\ X, Z, Y$

2017 239 Open Mathematical Olympiad, 6

Tags: geometry , angle bisector , circumcircle

Given a circumscribed quadrilateral $ABCD$ in which $$\sqrt{2}(BC-BA)=AC.$$ Let $X$ be the midpoint of $AC$ and $Y$ a point on the angle bisector of $B$ such that $XD$ is the angle bisector of $BXY$. Prove that $BD$ is tangent to the circumcircle of $DXY$.

2019 Balkan MO Shortlist, G2

Tags: geometry

Let be a triangle $\triangle ABC$ with $m(\angle ABC) = 75^{\circ}$ and $m(\angle ACB) = 45^{\circ}$. The angle bisector of $\angle CAB$ intersects $CB$ at point $D$. We consider the point $E \in (AB)$, such that $DE = DC$. Let $P$ be the intersection of lines $AD$ and $CE$. Prove that $P$ is the midpoint of segment $AD$.

1999 All-Russian Olympiad Regional Round, 10.6

Tags: geometry , incircle , equal segments

Triangle $ABC$ has an inscribed circle tangent to sides $AB$, $AC$ and $BC$ at points $C_1$, $B_1$ and $A_1 $ respectively. Let $K$ be a point on the circle diametrically opposite to point $C_1$, $D$ be the intersection point of lines $B_1C_1$ and $A_1K$. Prove that $CD = CB_1$.

2007 Germany Team Selection Test, 1

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2001 Moldova National Olympiad, Problem 8

Tags: geometry

Let $P$ be the midpoint of the arc $AC$ of a circle, and $B$ be a point on the arc $AP$. Let $M$ and $N$ be the projections of $P$ onto the segments $AC$ and $BC$ respectively. Prove that if $D$ is the intersection of the bisector of $\angle ABC$ and the segment $AC$, then every diagonal of the quadrilateral $BDMN$ bisects the area of the triangle $ABC$.

2023 Taiwan TST Round 1, G

Tags: geometry

Let $ABC$ be a triangle. Let $ABC_1, BCA_1, CAB_1$ be three equilateral triangles that do not overlap with $ABC$. Let $P$ be the intersection of the circumcircles of triangle $ABC_1$ and $CAB_1$. Let $Q$ be the point on the circumcircle of triangle $CAB_1$ so that $PQ$ is parallel to $BA_1$. Let $R$ be the point on the circumcircle of triangle $ABC_1$ so that $PR$ is parallel to $CA_1$. Show that the line connecting the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to $BC$. [i]Proposed by usjl[/i]

2019 Centroamerican and Caribbean Math Olympiad, 4

Tags: geometry , circumcircle

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$. The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$, respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$, respectively. Show that the triangle $APQ$ is isosceles.

2012 Today's Calculation Of Integral, 814

Tags: calculus , integration , geometry , calculus computations

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

2007 India IMO Training Camp, 1

Tags: trapezoid , circumcircle , ratio , homothety , geometry

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

2006 Sharygin Geometry Olympiad, 8

Tags: geometry , square , segment , tangential quadrilateral , inradius

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

Estonia Open Junior - geometry, 1999.1.2

Tags: geometry , locus , isosceles triangle , isosceles

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

1992 Irish Math Olympiad, 4

Tags: geometry

A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.

BIMO 2022, 1

Tags: geometry

A pentagon $ABCDE$ is such that $ABCD$ is cyclic, $BE\parallel CD$, and $DB=DE$. Let us fix the points $B,C,D,E$ and vary $A$ on the circumcircle of $BCD$. Let $P=AC\cap BE$, and $Q=BC\cap DE$. Prove that the second intersection of circles $(ABE)$ and $(PQE)$ lie on a fixed circle.

2020 Hong Kong TST, 5

Tags: geometry , incircle

In $\Delta ABC$, let $D$ be a point on side $BC$. Suppose the incircle $\omega_1$ of $\Delta ABD$ touches sides $AB$ and $AD$ at $E,F$ respectively, and the incircle $\omega_2$ of $\Delta ACD$ touches sides $AD$ and $AC$ at $F,G$ respectively. Suppose the segment $EG$ intersects $\omega_1$ and $\omega_2$ again at $P$ and $Q$ respectively. Show that line $AD$, tangent of $\omega_1$ at $P$ and tangent of $\omega_2$ at $Q$ are concurrent.

2009 China Team Selection Test, 1

Tags: geometry

In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

2014 NZMOC Camp Selection Problems, 2

Tags: geometry , concurrency , perpendicular

Let $ABC$ be a triangle in which the length of side $AB$ is $4$ units, and that of $BC$ is $2$ units. Let $D$ be the point on $AB$ at distance $3$ units from $A$. Prove that the line perpendicular to $AB$ through $D$, the angle bisector of $\angle ABC$, and the perpendicular bisector of $BC$ all meet at a single point.

2022 Princeton University Math Competition, B2

Tags: geometry , 3d geometry

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

1967 IMO Shortlist, 6

Tags: geometry , circumcircle , reflection , triangle , concurrency

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

1998 Korea Junior Math Olympiad, 5

Tags: complex number , geometry

Regular $2n$-gon is inscribed in the unit circle. Find the sum of the squares of all sides and diagonal lengths in the $2n$-gon.