Olympiad problems

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 $\frac{r_1}{r_2}$ .

2018 Chile National Olympiad, 6

Tags: collinear , orthocenter , geometry

Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.

2018 Polish Junior MO First Round, 7

Tags: 3d geometry , volume , plane , square , geometry

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2018 Sharygin Geometry Olympiad, 7

Tags: geometry

A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$?

2006 Iran Team Selection Test, 6

Tags: geometry , combinatorics , perimeter , absolute value , induction

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

2006 All-Russian Olympiad, 6

Tags: circumcircle , geometry

Let $P$, $Q$, $R$ be points on the sides $AB$, $BC$, $CA$ of a triangle $ABC$ such that $AP=CQ$ and the quadrilateral $RPBQ$ is cyclic. The tangents to the circumcircle of triangle $ABC$ at the points $C$ and $A$ intersect the lines $RQ$ and $RP$ at the points $X$ and $Y$, respectively. Prove that $RX=RY$.

2009 Cono Sur Olympiad, 6

Tags: rectangle , combinatorial geometry , geometry

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

1988 Irish Math Olympiad, 2

Tags: trigonometry , circumcircle , geometry

A; B; C; D are the vertices of a square, and P is a point on the arc CD of its circumcircle. Prove that $ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $ Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.

2018-2019 SDML (High School), 15

Tags: geometry

Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to $ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$

2010 Irish Math Olympiad, 3

Tags: geometry

In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.

2024 Olympic Revenge, 3

Tags: geometry

Let $A_1A_2 \dots A_n$ a cyclic $n$-agon with center $O$ and $P$, $Q$ being two isogonal conjugates of it (i.e, $\angle PA_{i+1}A_i = \angle QA_{i+1}A_{i+2}$ for all $i$). Let $P_i$ be the circumcenter of $\triangle PA_iA_{i+1}$ and $Q_i$ the circumcenter of $\triangle QA_iA_{i+1}$ for all $i$. Prove that: $a) ~P_1P_2 \dots P_n$ and $Q_1Q_2 \dots Q_n$ are cyclic, with centers $O_P$ and $O_Q$, respectively. $b)~O, O_P$ and $O_Q$ are collinears. $c)~O_PO_Q \mid \mid PQ.$ Remark: indices are taken modulo $n$.

1997 Tournament Of Towns, (549) 3

Tags: geometry , square , equal segments

In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$. (Folklore)

1994 Tournament Of Towns, (399) 1

Tags: geometry , geometric construction , quadrilateral construction , construction

Construct a convex quadrilateral given the lengths of all its sides and the length of the segment between the midpoints of its diagonals. (Folklore)

2008 239 Open Mathematical Olympiad, 6

Tags: geometry

$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.

2013 Brazil Team Selection Test, 1

Tags: geometry , integer

Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.

2012 USAJMO, 2

Tags: fibonacci , geometry

Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.

2007 Estonia National Olympiad, 5

Tags: geometry

Some circles of radius 2 are drawn on the plane. Prove that the numerical value of the total area covered by these circles is at least as big as the total length of arcs bounding the area.

Brazil L2 Finals (OBM) - geometry, 2009.2

Tags: geometry , circumcircle , perpendicular bisector

Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.

2010 All-Russian Olympiad, 4

Tags: geometry , combinatorics , rectangle

In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more. Find maximum $d$ for which we can guaranteed paint at least $d$ points.

Swiss NMO - geometry, 2011.2

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

Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively. [i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]

2010 Sharygin Geometry Olympiad, 17

Tags: geometry

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

2021 Brazil Team Selection Test, 2

Tags: right triangle , isosceles triangle , geometry

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

2019 Oral Moscow Geometry Olympiad, 6

Tags: geometry , 3d geometry , solid geometry , angle

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.

2010 Stars Of Mathematics, 2

Tags: geometry , reflection , perpendicular bisector , geometric transformation

Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$. 1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$; 2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$. (Dan Schwarz)

2021 Final Mathematical Cup, 2

Tags: tangent , geometry

The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .