Olympiad problems

1970 Miklós Schweitzer, 3

The traffic rules in a regular triangle allow one to move only along segments parallel to one of the altitudes of the triangle. We define the distance between two points of the triangle to be the length of the shortest such path between them. Put $ \binom{n\plus{}1}{2}$ points into the triangle in such a way that the minimum distance between pairs of points is maximal. [i]L. Fejes-Toth[/i]

2005 National Olympiad First Round, 25

Tags: geometry , rectangle

Let $E$, $F$, $G$ be points on sides $[AB]$, $[BC]$, $[CD]$ of the rectangle $ABCD$, respectively, such that $|BF|=|FQ|$, $m(\widehat{FGE})=90^\circ$, $|BC|=4\sqrt 3 / 5$, and $|EF|=\sqrt 5$. What is $|BF|$? $ \textbf{(A)}\ \dfrac{\sqrt{10} - \sqrt{2}}{2} \qquad\textbf{(B)}\ \sqrt 3 -1 \qquad\textbf{(C)}\ \sqrt 3 \qquad\textbf{(D)}\ \dfrac{\sqrt{11} - \sqrt{3}}{2} \qquad\textbf{(E)}\ 1 $

2012 Indonesia TST, 2

Tags: incenter , circumcircle , geometry

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

2011 Argentina National Olympiad Level 2, 3

Tags: geometry , altitude , incenter

Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$. Note: The incenter of a triangle is the intersection point of its angle bisectors.

1987 Tournament Of Towns, (153) 4

Tags: line , line construction , geometry , arc , broken line , bisects area

We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.

Kvant 2020, M2622

Tags: geometry , rhombus

The points $E, F, G$ and $H{}$ are located on the sides $DA, AB, BC$ and $CD$ of the rhombus $ABCD$ respectively, so that the segments $EF$ and $GH$ touch the circle inscribed in the rhombus. Prove that $FG\parallel HE$. [i]Proposed by V. Eisenstadt[/i]

2008 Pre-Preparation Course Examination, 3

Tags: geometry , 3d geometry , sphere , probability and stats

Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.

2008 JBMO Shortlist, 5

Tags: geometry

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)

2024 ELMO Shortlist, G7

Tags: geometry , rectangle

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$. [i]Linus Tang[/i]

Kyiv City MO Juniors 2003+ geometry, 2006.8.3

Tags: geometry , right triangle , equal segments , perpendicular , llllllllllllllllllllllllllllll

On the legs $AC, BC$ of a right triangle $\vartriangle ABC$ select points $M$ and $N$, respectively, so that $\angle MBC = \angle NAC$. The perpendiculars from points $M$ and $C$ on the line $AN$ intersect $AB$ at points $K$ and $L$, respectively. Prove that $KL=LB$. (O. Clurman)

1956 Poland - Second Round, 6

Tags: 3d geometry , geometry , tetrahedron

Prove that if in a tetrahedron $ ABCD $ the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then $$AB \cdot CD = AC \cdot BD = AD \cdot BC$$ and that the converse also holds.

2012 Today's Calculation Of Integral, 824

Tags: calculus , integration , geometry , limit , calculus computations , logarithm

In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

PEN H Problems, 27

Tags: diophantine equation , perimeter , inradius , modular arithmetic , geometry

Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.

2021 Indonesia TST, G

Tags: geometry , geometric transformation

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

2016 CCA Math Bonanza, T9

Tags: geometry , angle bisector

Let ABC be a triangle with $AB = 8$, $BC = 9$, and $CA = 10$. The line tangent to the circumcircle of $ABC$ at $A$ intersects the line $BC$ at $T$, and the circle centered at $T$ passing through $A$ intersects the line $AC$ for a second time at $S$. If the angle bisector of $\angle SBA$ intersects $SA$ at $P$, compute the length of segment $SP$. [i]2016 CCA Math Bonanza Team #9[/i]

1993 Greece National Olympiad, 15

Tags: number theory , relatively prime , pythagorean theorem , geometry

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

2002 Putnam, 2

Tags: geometry , 3d geometry , sphere , pigeonhole principle , putnam pigeonhole

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

2000 Greece National Olympiad, 1

Tags: rectangle , geometry

Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and suﬃcient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.

2018 Balkan MO Shortlist, G3

Tags: geometry , geometric inequality , inequalities , semiperimeter , maximum value , minimum

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

2001 Spain Mathematical Olympiad, Problem 5

Tags: geometry

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.

2015 JBMO Shortlist, 4

Tags: geometry

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

2018 Costa Rica - Final Round, G2

Tags: geometry , equal segments , angle bisector

Consider $\vartriangle ABC$, with $AD$ bisecting $\angle BAC$, $D$ on segment $BC$. Let $E$ be a point on $BC$, such that $BD = EC$. Through $E$ we draw the line $\ell$ parallel to $AD$ and consider a point $ P$ on it and inside the $\vartriangle ABC$. Let $G$ be the point where line $BP$ cuts side $AC$ and let F be the point where line $CP$ to side $AB$. Show that $BF = CG$.

2020 BMT Fall, 6

Tags: geometry

A tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. If the volume of the tetrahedron is $V$ , compute $V^2$ .

1997 All-Russian Olympiad Regional Round, 9.1

Tags: geometry , combinatorics , combinatorial geometry , acute

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

1996 Abels Math Contest (Norwegian MO), 1

Tags: locus , geometry

Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point. A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$. Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.