Olympiad problems

1972 All Soviet Union Mathematical Olympiad, 167

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 $$\angle A_1+ \angle A_2 + \angle A_3 < 450^o$$

2010 AMC 10, 20

Tags: ratio , geometry

Two circles lie outside regular hexagon $ ABCDEF$. The first is tangent to $ \overline{AB}$, and the second is tangent to $ \overline{DE}$. Both are tangent to lines $ BC$ and $ FA$. What is the ratio of the area of the second circle to that of the first circle? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 108$

1997 ITAMO, 4

Tags: geometry , 3d geometry , tetrahedron , projection

Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.

2007 Grigore Moisil Intercounty, 1

Tags: geometry

In a triangle $ ABC $ with $ AB\neq AC, $ let $ D $ be the midpoint of the side $ BC $ and denote with $ E $ the feet of the bisector of $ \angle BAC. $ Also, let $ M,N $ be two points situated in the exterior of $ ABC $ such that $ AMB\sim ANC. $ Prove the following propositions: $ \text{(a)} MN\perp AD\iff MA\perp AB $ $ \text{(b)} MN\perp AE \iff\angle MAN=180^{\circ } $

2011 AIME Problems, 6

Tags: parabola , analytic geometry , geometry , geometric transformation , number theory , relatively prime , conic

Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2020 Junior Balkan Team Selection Tests-Serbia, 1#

Tags: geometry

Given is triangle $ABC$ with arbitrary point $D$ on $AB$ and central of inscribed circle $I$. The perpendicular bisector of $AB$ intersects $AI$ and $BI$ at $P$ and $Q$, respectively. The circle $(ADP)$ intersects $CA$ at $E$, and the circle $(BDQ)$ intersects $BC$ at $F$ and $(ADP)$ intersects $(BDQ)$ at $K$. Prove that $E, F, K, I$ lie on one circle.

2001 Estonia Team Selection Test, 1

Tags: geometry , rectangle , combinatorial geometry , combinatorics

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

1998 IberoAmerican Olympiad For University Students, 4

Tags: rhombus , perpendicular bisector , geometry

Four circles of radius $1$ with centers $A,B,C,D$ are in the plane in such a way that each circle is tangent to two others. A fifth circle passes through the center of two of the circles and is tangent to the other two. Find the possible values of the area of the quadrilateral $ABCD$.

2004 Germany Team Selection Test, 1

Tags: vector , geometry , 3d geometry , tetrahedron , linear algebra , algebra

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Kyiv City MO 1984-93 - geometry, 1988.8.5

Tags: geometry , circumcircle

In the triangle $ABC$, the angle bisector $AK$ is drawn. The center of the circle inscribed in the triangle $AKC$ coincides with the center of the circle, circumscribed around the triangle $ABC$. Determine the angles of triangle $ABC$.

2018 239 Open Mathematical Olympiad, 8-9.2

Tags: geometry

On the hypotenuse $AB$ of a right-angled triangle $ABC$, point $R$ is chosen, on the cathetus $BC$ a point $T$, and on the segment $AT$ a point $S$ are chosen in such a way that the angles $\angle ART$ and $\angle ASC$ are right angles. Points $M$ and $N$ are the midpoints of the segments $CB$ and $CR$, respectively. Prove that points $M$, $T$, $S$, and $N$ lie on the same circle. [i]Proposed by S. Berlov[/i]

1996 Moldova Team Selection Test, 2

Tags: geometry

Circles $S_1{}$ and $S_2{}$ intersect in $M{}$ and $N{}$. Line $l$ intersects the circles in points $A,B\in S_1$ and $C,D\in S_2$. Prove that $\angle AMC=\angle BND$ and $\angle ANC=\angle BMD$ if the order of points on line $l$ is: [b]a)[/b] $A,C,B,D;\quad$ [b]b)[/b] $A,C,D,B.$

2018 Rio de Janeiro Mathematical Olympiad, 4

Tags: geometry

Let $ABC$ be an acute triangle inscribed on the circumference $\Gamma$. Let $D$ and $E$ be points on $\Gamma$ such that $AD$ is perpendicular to $BC$ and $AE$ is diameter. Let $F$ be the intersection between $AE$ and $BC$. Prove that, if $\angle DAC = 2 \angle DAB$, then $DE = CF$.

2010 Lithuania National Olympiad, 2

Tags: trapezoid , angle bisector , geometry

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

2002 Germany Team Selection Test, 2

Tags: geometry , triangle , square

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

2006 Moldova National Olympiad, 10.1

Tags: trigonometry , function , inequalities , geometry

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

2004 Uzbekistan National Olympiad, 2

Tags: geometry , quadratic , number theory

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

LMT Team Rounds 2021+, 4

Tags: geometry

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

2023 Sharygin Geometry Olympiad, 14

Tags: geometry , polygon , winding number

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$, and is symmetric with respect to $O$. Prove that the winding number of $\mathcal{L}$ around $O$ is odd. The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$: $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$

1975 Chisinau City MO, 90

Tags: geometry , right triangle , median , triangle construction

Construct a right-angled triangle along its two medians, starting from the acute angles.

2008 Cuba MO, 2

Tags: geometry , parallelogram

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

Novosibirsk Oral Geo Oly VII, 2019.7

Tags: geometry , acute , square

Cut a square into eight acute-angled triangles.

LMT Team Rounds 2021+, 9

Tags: geometry

Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .

Kyiv City MO 1984-93 - geometry, 1990.10.5

Tags: geometry , max , geometric inequality , analytic geometry , parabola , area

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.

2006 India IMO Training Camp, 1

Tags: geometry , inradius , circumcircle , inequalities

Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that \[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]