Olympiad problems

1993 All-Russian Olympiad Regional Round, 11.7

Tags: geometry

Let $ABC$ be an equilateral triangle. For an arbitrary line $\ell$. through $B$, the orthogonal projections of $A$ and $C$ on $\ell$ are denoted by $D$ and $E$ respectively. If $D\ne E$, equilateral triangles $DEP$ and $DET$ are constructed on different sides of $\ell$. Find the loci of $P$ and $T$.

2017 Oral Moscow Geometry Olympiad, 4

Tags: geometry , circumcircle , circumradius , radius , minimum value

We consider triangles $ABC$, in which the point $M$ lies on the side $AB$, $AM = a$, $BM = b$, $CM = c$ ($c <a, c <b$). Find the smallest radius of the circumcircle of such triangles.

1969 IMO Longlists, 39

Tags: geometry , 3d geometry , cube , locus , locus problems

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

2006 Peru IMO TST, 4

Tags: circumcircle , parallelogram , perpendicular bisector , similar triangles , geometry

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 04[/b] In an actue-angled triangle $ABC$ draws up: its circumcircle $w$ with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on the straight lines $AQ$ and $AC$, respectively, in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the circumference $w_1.$ --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

2009 Today's Calculation Of Integral, 517

Tags: calculus , integration , geometry , search , calculus computations

Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.

2021 Science ON all problems, 2

Tags: geometry

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

2009 AMC 10, 18

Tags: geometry , rectangle , analytic geometry

Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$? $ \textbf{(A)}\ \frac{65}{8} \qquad \textbf{(B)}\ \frac{25}{3} \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ \frac{75}{8} \qquad \textbf{(E)}\ \frac{85}{8}$

MathLinks Contest 5th, 1.3

Tags: geometry

Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points. a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and $CC'$ lies on the common radical axis found above.

2007 Germany Team Selection Test, 3

Tags: ratio , geometry , projective geometry , homothety

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

2017 Purple Comet Problems, 26

Tags: geometry

The incircle of $\vartriangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$, and $\overline{AB}$ at $D, E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\vartriangle ABC$ have lengths $AB = 73, BC = 123$, and $AC = 120$. Find the length $EG$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png[/img]

2020-21 IOQM India, 9

Tags: geometry , angle bisector

Let A$BC$ be a triangle with $AB = 5, AC = 4, BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM\parallel AC$ and $DN \parallel BC$. If $(MN)^2 =\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?

2009 Germany Team Selection Test, 1

Tags: geometry

Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.

2024 Serbia JBMO TST, 4

Tags: geometry

Let $I$ be the incenter of a triangle $ABC$ with $AB \neq AC$. Let $M$ be the midpoint of $BC$, $M' \in BC$ be such that $IM'=IM$ and $K$ be the midpoint of the arc $BAC$. If $AK \cap BC=L$, show that $KLIM'$ is cyclic.

1999 National Olympiad First Round, 9

Tags: geometry , trigonometry , rectangle

Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$. $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$

2021 Belarusian National Olympiad, 9.6

Tags: ratio , geometry

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

2007 Peru Iberoamerican Team Selection Test, P3

Tags: geometry

We have an acute triangle $ABC$. Consider the square $A_1A_2A_3A_4$ which has one vertex in $AB$, one vertex in $AC$ and two vertices ($A_1$ and $A_2$) in $BC$ and let $x_A=\angle A_1AA_2$. Analogously we define $x_B$ and $x_C$. Prove that $x_A+x_B+x_C=90$

1996 Tournament Of Towns, (514) 1

Tags: geometry , 3d geometry , locus

Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$. (V Proizvolov,)

2002 Switzerland Team Selection Test, 7

Tags: geometry , linear algebra , area , triangle

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

1999 Switzerland Team Selection Test, 7

Tags: rectangle , ratio , square , combinatorial geometry , combinatorics , geometry

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

2022 Baltic Way, 13

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $AB < BC$ and $AD < DC$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $AB = BE$ and $AD = DF$. Let further M denote the midpoint of the segment $EF$. Prove that $\angle BMD = 90^o$.

2003 China Team Selection Test, 1

Tags: trigonometry , calculus , geometry

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

2005 Sharygin Geometry Olympiad, 10.5

Tags: tangent , distance , circles , geometry

Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.

1999 Romania National Olympiad, 4

Tags: geometry , ratio

In the triangle $ABC$, let $D \in (BC)$, $E \in (AB)$, $EF \parallel BC$, $F \in (AC)$, $EG\parallel AD$, $G\in (BC)$ and $M,N$ be the midpoints of $(AD)$ and $(BC)$, respectively. Prove that: a) $\frac{EF}{BC}+\frac{EG}{AD}=1$ b) the midpoint of $[FG]$ lies on the line $ MN$.

1986 India National Olympiad, 3

Tags: pythagorean theorem , geometry , algebra

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]

2012 Bogdan Stan, 4

Tags: geometric inequalities , inequalities , angle inequalities , geometry , rational geometry , ratio

Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following: $$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$