Olympiad problems

2004 Romania National Olympiad, 3

Tags: geometry , trapezoid

Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$, $\left( OE \right.$ is the bisector of $\measuredangle AOD$, $E \in \left( AD \right)$ and $EF \| AB$, $F \in \left( BC \right)$. Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$. Prove that: (a) $EP=QF$; (b) $EF=AD$. [i]Claudiu-Stefan Popa[/i]

2021 BMT, T2

Tags: geometry , combinatorics

A gradian is a unit of measurement of angles much like degrees, except that there are $100$ gradians in a right angle. Suppose that the number of gradians in an interior angle of a regular polygon with $m$ sides equals the number of degrees in an interior angle of a regular polygon with $n$ sides. Compute the number of possible distinct ordered pairs $(m, n)$.

1993 IMO Shortlist, 4

Tags: geometry , trigonometry , incenter , law of sines , inradius

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

2001 Estonia National Olympiad, 3

Tags: geometry , ratio

Points $D,E$ and $F$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$ respectively so that the segments $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB}$ .

1983 IMO Shortlist, 23

Tags: circles , geometry , angle , midpoint

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

2010 Cono Sur Olympiad, 5

Tags: geometry

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.

2001 China Team Selection Test, 1

Tags: geometry

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

2008 Iran Team Selection Test, 4

Tags: geometry , function , 3d geometry , tetrahedron , vector , inequalities

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

VI Soros Olympiad 1999 - 2000 (Russia), 9.7

Tags: geometry , ratio

Points $A, B, C$ and $D$ are located on line $\ell$ so that $\frac{AB}{BC}=\frac{AC}{CD}=\lambda $. A certain circle is tangent to line $\ell$ at point $C$. A line is drawn through $A$ that intersects this circle at points $M$ and $N$ such that the bisector perpendiculars to segments $BM$ and $DN$ intersect at point $Q$ on line $\ell$ . In what ratio does point $Q$ divide segment $AD$?

2007 Tuymaada Olympiad, 2

Tags: geometry , perpendicular bisector

Point $ D$ is chosen on the side $ AB$ of triangle $ ABC$. Point $ L$ inside the triangle $ ABC$ is such that $ BD=LD$ and $ \angle LAB=\angle LCA=\angle DCB$. It is known that $ \angle ALD+\angle ABC=180^\circ$. Prove that $ \angle BLC=90^\circ$.

1999 Portugal MO, 6

Tags: geometry , angle , equal angles

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$. If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .

2024 Iberoamerican, 2

Tags: geometry , circumcircle

Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

2022 Harvard-MIT Mathematics Tournament, 8

Tags: combinatorics , geometry

Let $P_1P_2...P_n$ be a regular $n$-gon in the plane and $a_1, . . . , a_n$ be nonnegative integers. It is possible to draw $m$ circles so that for each $1 \le i \le n$, there are exactly $a_i$ circles that contain $P_i$ on their interior. Find, with proof, the minimum possible value of $m$ in terms of the $a_i$. .

1986 AMC 8, 18

Tags: geometry

A rectangular grazing area is to be fenced off on three sides using part of a $ 100$ meter rock wall as the fourth side. Fence posts are to be placed every $ 12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $ 36$ m by $ 60$ m? \[ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16 \]

2020-21 KVS IOQM India, 22

Tags: geometry , right triangle , inradius

Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.

Russian TST 2017, P1

Tags: geometry , nine point center

The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.

1953 Kurschak Competition, 3

Tags: geometry , equal segments , equal angles , hexagon

$ABCDEF$ is a convex hexagon with all its sides equal. Also $\angle A + \angle C + \angle E = \angle B + \angle D + \angle F$. Show that $\angle A = \angle D$, $\angle B = \angle E$ and $\angle C = \angle F$.

1983 Putnam, B1

Tags: geometry , calculus , 3d geometry , volume , integration

Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.

1972 All Soviet Union Mathematical Olympiad, 167

Tags: geometry , heptagon , cyclic

The $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle. Prove that if the centre of the circle is inside the $7$-gon , than $$\angle A_1+ \angle A_2 + \angle A_3 < 450^o$$

2021 Saudi Arabia BMO TST, 2

Tags: geometry , circumcircle , parallel

Let $ABC$ be an acute triangle with $AB < AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D$, $E$ as the intersections of $AI$ with $BC$, $(O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^o$ and $KA$, $KE$ meet $(O)$ again at M,N respectively. The rays $ND$, $NI$ meet the circle $(O)$ at $Q$,$P$. 1. Prove that the quadrilateral $MPQE$ is a kite. 2. Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$ ,cuts $EK$ at $S$ .Prove that $OR \parallel JS$.

2012 Princeton University Math Competition, B1

Tags: geometry

During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side? [img]https://cdn.artofproblemsolving.com/attachments/d/2/f8e3a7afb606dfd6fad277f547b116566a4a91.png[/img]

Russian TST 2014, P1

Tags: geometry , inequality

Let $R{}$ and $r{}$ be the radii of the circumscribed and inscribed circles of the acute-angled triangle $ABC{}$ respectively. The point $M{}$ is the midpoint of its largest side $BC.$ The tangents to its circumscribed circle at $B{}$ and $C{}$ intersect at $X{}$. Prove that \[\frac{r}{R}\geqslant\frac{AM}{AX}.\]

2018 China Second Round Olympiad, 2

Tags: geometry

In triangle $\triangle ABC, AB=AC.$ Let $D$ be on segment $AC$ and $E$ be a point on the extended line $BC$ such that $C$ is located between $B$ and $E$ and $\frac{AD}{DC}=\frac{BC}{2CE}$. Let $\omega$ be the circle with diameter $AB,$ and $\omega$ intersects segment $DE$ at $F.$ Prove that $B,C,F,D$ are concyclic.

2016 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle , ratio

A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$. by E.Bakaev

2011 Morocco National Olympiad, 4

Tags: perpendicular bisector , geometry

$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'. Prove that the bisector of $[PP']$ passes through a non-moving point.