Olympiad problems

1958 AMC 12/AHSME, 42

In a circle with center $ O$, chord $ \overline{AB}$ equals chord $ \overline{AC}$. Chord $ \overline{AD}$ cuts $ \overline{BC}$ in $ E$. If $ AC \equal{} 12$ and $ AE \equal{} 8$, then $ AD$ equals: $ \textbf{(A)}\ 27\qquad \textbf{(B)}\ 24\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 18$

2004 China Team Selection Test, 1

Tags: geometry , geometric transformation

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

2020 Brazil National Olympiad, 1

Tags: geometry , angle bisector

Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.

1955 Moscow Mathematical Olympiad, 295

Tags: convex , line , geometry , combinatorial geometry

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

2009 Sharygin Geometry Olympiad, 5

Tags: rhombus , circumcenter , geometry , collinear

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic. (D.Prokopenko)

2008 IMO Shortlist, 1

Tags: circumcircle , radical axis , geometry , trigonometry

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

2009 AMC 10, 23

Tags: trapezoid , ratio , geometry , trigonometry , parallelogram

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

2010 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

You are standing in an infinitely long hallway with sides given by the lines $x=0$ and $x=6$. You start at $(3,0)$ and want to get to $(3,6)$. Furthermore, at each instant you want your distance to $(3,6)$ to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from $(3,0)$ to $(3,6)$?

2019 CMIMC, 7

Tags: geometry

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Denote by $\omega$ its incircle. A line $\ell$ tangent to $\omega$ intersects $\overline{AB}$ and $\overline{AC}$ at $X$ and $Y$ respectively. Suppose $XY=5$. Compute the positive difference between the lengths of $\overline{AX}$ and $\overline{AY}$.

1998 Federal Competition For Advanced Students, Part 2, 3

Tags: ratio , geometry , parallelogram

In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?

2010 AMC 12/AHSME, 14

Tags: perimeter , geometry

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

2023 Oral Moscow Geometry Olympiad, 2

Tags: geometry

Points $X_1$ and $X_2$ move along fixed circles with centers $O_1$ and $O_2$, respectively, so that $O_1X_1 \parallel O_2X_2$. Find the locus of the intersection point of lines $O_1X_2$ and $O_2X_1$.

1977 Czech and Slovak Olympiad III A, 6

Tags: cube , 3d geometry , volume , geometry , tetrahedron

A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.

2000 Baltic Way, 3

Tags: geometry , circumcircle , trigonometry

Given a triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.

2011 JBMO Shortlist, 6

Tags: geometry , rhombus , symmetry , ratio , combinatorics

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

2019 Adygea Teachers' Geometry Olympiad, 4

Tags: tangential , right angle , trapezium , geometry , trapezoid

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

2014 Czech-Polish-Slovak Match, 4

Tags: geometry , concurrency

Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$. A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$. Let $S$ be the midpoint of $KL$. Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$. (Jan Mazák)

2016 Japan MO Preliminary, 8

Tags: geometry

Let $\omega$ be an incircle of triangle $ABC$. Let $D$ be a point on segment $BC$, which is tangent to $\omega$. Let $X$ be an intersection of $AD$ and $\omega$ against $D$. If $AX : XD : BC = 1 : 3 : 10$, a radius of $\omega$ is $1$, find the length of segment $XD$. Note that $YZ$ expresses the length of segment $YZ$.

2015 Romanian Master of Mathematics, 4

Tags: geometry

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

2020 CMIMC Geometry, 10

Tags: geometry

Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$. Compute the largest possible value for the circumradius of $\mathcal T$.

Indonesia Regional MO OSP SMA - geometry, 2016.4

Tags: geometry

Let $PA$ and $PB$ be the tangent of a circle $\omega$ from a point $P$ outside the circle. Let $M$ be any point on $AP$ and $N$ is the midpoint of segment $AB$. $MN$ cuts $\omega$ at $C$ such that $N$ is between $M$ and $C$. Suppose $PC$ cuts $\omega$ at $D$ and $ND$ cuts $PB$ at $Q$. Prove $MQ$ is parallel to $AB$.

1998 Singapore MO Open, 1

Tags: equal segments , tangent , geometry , circles

In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$. [img]https://cdn.artofproblemsolving.com/attachments/6/f/f21c0c70b37768f3e80e9ee909ef34c57635d5.png[/img]

Novosibirsk Oral Geo Oly VII, 2022.2

Tags: perimeter , geometry

A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

2017 Sharygin Geometry Olympiad, P19

Tags: midpoint , concurrency , circumcircle , cevian , geometry

Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.

2024 Sharygin Geometry Olympiad, 8

Tags: quadrilateral , geometry

Let $ABCD$ be a quadrilateral $\angle B = \angle D$ and $AD = CD$. The incircle of triangle $ABC$ touches the sides $BC$ and $AB$ at points $E$ and $F$ respectively. Prove that the midpoints of segments $AC, BD, AE,$ and $CF$ are concyclic.