Olympiad problems

2023 Romanian Master of Mathematics Shortlist, G2

Let $ABCD$ be a cyclic quadrilateral. Let $DA$ and $BC$ intersect at $E$ and let $AB$ and $CD$ intersect at $F$. Assume that $A, E, F$ all lie on the same side of $BD$. Let $P$ be on segment $DA$ such that $\angle CPD = \angle CBP$, and let $Q$ be on segment $CD$ such that $\angle DQA = \angle QBA$. Let $AC$ and $PQ$ meet at $X$. Prove that, if $EX = EP$, then $EF$ is perpendicular to $AC$.

2021 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , rectangle

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

2013 China Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , geometry , parallelogram

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

Indonesia MO Shortlist - geometry, g5

Tags: circles , tangent , geometry , collinear

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

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.

2021 Indonesia TST, G

Tags: geometry , trapezoid

Given points $A$, $B$, $C$, and $D$ on circle $\omega$ such that lines $AB$ and $CD$ intersect on point $T$ where $A$ is between $B$ and $T$, moreover $D$ is between $C$ and $T$. It is known that the line passing through $D$ which is parallel to $AB$ intersects $\omega$ again on point $E$ and line $ET$ intersects $\omega$ again on point $F$. Let $CF$ and $AB$ intersect on point $G$, $X$ be the midpoint of segment $AB$, and $Y$ be the reflection of point $T$ to $G$. Prove that $X$, $Y$, $C$, and $D$ are concyclic.

1965 AMC 12/AHSME, 35

Tags: rectangle , similar triangles , geometry

The length of a rectangle is $ 5$ inches and its width is less than $ 4$ inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is $ \sqrt {6}$, then the width is: $ \textbf{(A)}\ \sqrt {2} \qquad \textbf{(B)}\ \sqrt {3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt {5} \qquad \textbf{(E)}\ \sqrt {\frac {11}{2}}$

2022 All-Russian Olympiad, 2

Tags: geometry

On side $BC$ of an acute triangle $ABC$ are marked points $D$ and $E$ so that $BD = CE$. On the arc $DE$ of the circumscribed circle of triangle $ADE$ that does not contain the point $A$, there are points $P$ and $Q$ such that $AB = PC$ and $AC = BQ$. Prove that $AP=AQ$.

2010 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , cyclic , quadrilateral

In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic

2014 Contests, 2

Tags: equal segments , parallelogram , geometry

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

2005 Turkey MO (2nd round), 2

Tags: geometry , radical axis , power of a point , parallelogram , circumcircle , modular arithmetic

In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.

1998 IMC, 2

Tags: rotation , superior algebra , geometric transformation , geometry

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

ICMC 7, 5

Tags: geometry

Is it possible to dissect an equilateral triangle into three congruent polygonal pieces (not necessarily convex), one of which contains the triangle’s centre in its interior? [i]Note:[/i] The interior of a polygon is the polygon without its boundary. [i]Proposed by Dylan Toh[/i]

1993 Greece National Olympiad, 14

Tags: geometric transformation , geometry , circumcircle , rotation , perimeter , analytic geometry , rectangle

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

2019 Polish Junior MO First Round, 2

Tags: angle , geometry

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$. [img]https://cdn.artofproblemsolving.com/attachments/1/2/046e31a628b3df4d23d3162cb570e1b9cb71e2.png[/img]

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , circles , geometry

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

2017 Korea Winter Program Practice Test, 2

Tags: geometry

$ABC$ is an obtuse triangle satisfying $\angle A>90^\circ$, and its circumcenter $O$ and circumcircle $\omega_1$. Let $\omega_2$ be a circle passing $C$ with center $B$. $\omega_2$ meets $BC$ at $D$. $\omega_1$ meets $AD$ and $\omega_2$ at $E$ and $F(\neq C)$, respectively. $AF$ meets $\omega_2$ at $G(\neq F)$. Prove that the intersection of $CE$ and $BG$ lies on the circumcircle of $AOB$.

2020 Korean MO winter camp, #5

Tags: geometry

$\square ABCD$ is a quadrilateral with $\angle A=2\angle C <90^\circ$. $I$ is the incenter of $\triangle BAD$, and the line passing $I$ and perpendicular to $AI$ meets rays $CB$ and $CD$ at $E,F$ respectively. Denote $O$ as the circumcenter of $\triangle CEF$. The line passing $E$ and perpendicular to $OE$ meets ray $OF$ at $Q$, and the line passing $F$ and perpendicular to $OF$ meets ray $OE$ at $P$. Prove that the circle with diameter $PQ$ is tangent to the circumcircle of $\triangle BCD$.

1994 Korea National Olympiad, Problem 3

Tags: circumradius , excenter , incenter , geometry

In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that: (i) $R'= 2R $ (ii) $IO' = 2IO$

2016 Oral Moscow Geometry Olympiad, 6

Tags: paper , segment , folding , square , geometry

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

Brazil L2 Finals (OBM) - geometry, 2008.5

Tags: angle , circumcenter , orthocenter , geometry

Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .

1997 Tournament Of Towns, (534) 6

Tags: isosceles , angle , isosceles triangle , geometry

Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

2007 Cuba MO, 9

Tags: triangle inequality , circumcircle , inequalities , geometry

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.

2023 Junior Macedonian Mathematical Olympiad, 4

Tags: geometry

We are given an acute $\triangle ABC$ with circumcenter $O$ such that $BC<AB$. The bisector of $\angle ACB$ meets the circumcircle of $\triangle ABC$ at a second point $D$. The perpendicular bisector of $AC$ meets the circumcircle of $\triangle BOD$ for the second time at $E$. The line $DE$ meets the circumcircle of $\triangle ABC$ for the second time at $F$. Prove that the lines $CF$, $OE$ and $AB$ are concurrent. [i]Authored by Petar Filipovski[/i]

2010 Contests, 3

Tags: ratio , trigonometry , homothety , projective geometry , trig identities , geometric transformation , geometry

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]