Olympiad problems

2009 JBMO Shortlist, 1

Tags: geometry

Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ intersection point of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.

2000 Bulgaria National Olympiad, 2

Tags: circumcircle , geometry

Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.

2006 QEDMO 2nd, 14

Tags: geometric transformation , vector , parallelogram , geometry

On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$. Prove that $B_uC_u\parallel BC$. [i]Comment.[/i] This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more: $B_uC_u=4\cdot BC$. Darij

1998 Mediterranean Mathematics Olympiad, 2

Tags: polynomial , geometry , 3d geometry , algebra

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

2011 USAMTS Problems, 4

Tags: geometry , perpendicular bisector

A $\emph{luns}$ with vertices $X$ and $Y$ is a region bounded by two circular arcs meeting at the endpoints $X$ and $Y$. Let $A$, $B$, and $V$ be points such that $\angle AVB=75^\circ$, $AV=\sqrt{2}$ and $BV=\sqrt{3}$. Let $\mathcal{L}$ be the largest area luns with vertices $A$ and $B$ that does not intersect the lines $VA$ or $VB$ in any points other than $A$ and $B$. Define $k$ as the area of $\mathcal{L}$. Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]

2018 Sharygin Geometry Olympiad, 23

Tags: geometry , combinatorial geometry

The plane is divided into convex heptagons with diameters less than 1. Prove that an arbitrary disc with radius 200 intersects most than a billion of them.

2017 Dutch IMO TST, 4

Tags: geometry , combinatorics

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

2012 Online Math Open Problems, 27

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

Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$. [i]Ray Li.[/i]

2015 ISI Entrance Examination, 7

Tags: circles , triangle , geometry

Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$

2018 IFYM, Sozopol, 5

Tags: geometry

Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that $AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$ and find for which points $X$ the equality is met.

2022 ELMO Revenge, 1

Tags: geometry

Let $ABC$ and $DBC$ be triangles with incircles touching at a point $P$ on $BC.$ Points $A,D$ lie on the same side of $BC$ and $DB < AB < DC < AC.$ The bisector of $\angle BDC$ meets line $AP$ at $X,$ and the altitude from $A$ meets $DP$ at $Y.$ Point $Z$ lies on line $XY$ so $ZP \perp BC.$ Show the reflection of $A$ over $BC$ is on line $ZD.$ [i]Proposed by squareman (Evan Chang), USA[/i]

1977 Vietnam National Olympiad, 3

Tags: combinatorial geometry , circles , geometry

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

2020 Romanian Masters In Mathematics, 1

Tags: geometry

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

1986 National High School Mathematics League, 2

Tags: geometry , circumcircle

In acute triangle $ABC$, $D\in BC,E\in CA,F\in AB$. Prove that the necessary and sufficient condition of $AD,BE,CF$ are heights of $\triangle ABC$ is that $S=\frac{R}{2}(EF+FD+DE)$. Note: $S$ is the area of $\triangle ABC$, $R$ is the circumradius of $\triangle ABC$.

2019 IOM, 5

Tags: geometry , 3d geometry , pyramid , sphere

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic. [i] Tibor Bakos and Géza Kós [/i]

JBMO Geometry Collection, 2003

Tags: geometry , circumcircle , angle bisector

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2014 Contests, 2

Tags: geometry , angle bisector , concurrency , parallelogram

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

Swiss NMO - geometry, 2012.3

Tags: equal angles , geometry , circles , tangent

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

2007 All-Russian Olympiad, 2

Tags: geometry , rectangle , combinatorics

The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves. [i]N. Agakhanov[/i]

2009 Oral Moscow Geometry Olympiad, 1

Tags: diagonal , geometry , sidelengths

Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

1998 Tournament Of Towns, 1

Tags: geometry , similar triangles , tiling , combinatorics , combinatorial geometry , rectangle

Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying? (A Fedotov)

2015 Stars Of Mathematics, 2

Tags: abstract algebra , geometry

Let $\gamma,\gamma_0,\gamma_1,\gamma_2$ be four circles in plane,such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$,and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$,$i=0,1,2$(the indexes are reduced modulo $3$).The tangent in $B_i$,common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$,intersects circle $\gamma$ in point $C_i$,situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$.Prove that the three lines $A_iC_i$ are concurrent.

2007 IMO Shortlist, 2

Tags: geometry , rectangle , dissection , combinatorics

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

1998 Greece JBMO TST, 2

Tags: trapezoid , geometry

Let $ABCD$ be a trapezoid with parallel sides $AB, CD$. $M,N$ lie on lines $AD, BC$ respectively such that $MN || AB$. Prove that $DC \cdot MA + AB \cdot MD = MN \cdot AD$.

2019 Belarusian National Olympiad, 10.6

Tags: circumcircle , inequalities , geometry

The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively. Prove the inequality $AM+AL\ge 2AN$. [i](V. Karamzin)[/i]