Olympiad problems

Estonia Open Senior - geometry, 2004.1.3

a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions (1) $ABCD$ is not cyclic; (2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths; (3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal? b) Does there exist such a non-convex quadrangle?

2011 QEDMO 9th, 5

Tags: convex polygon , convex , perpendicular , geometry

Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.

1982 Austrian-Polish Competition, 2

Tags: tangent , locus , convex , circles , geometry

Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.

Durer Math Competition CD 1st Round - geometry, 2018.C5

Tags: geometry , convex polygon , convex

A convex $n$-gon is called [i]nice[/i] if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is $1$. Find all integers $n \ge 4$ such that a nice $n$-gon exists .

Ukrainian TYM Qualifying - geometry, VI.18

Tags: geometry , convex , convex polygon , max , area , geometric inequality

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

1962 Kurschak Competition, 2

Tags: diagonal , convex , geometry

Show that given any $n+1$ diagonals of a convex $n$-gon, one can always find two which have no common point.

1948 Moscow Mathematical Olympiad, 151

Tags: midpoint , parallel , distance , geometry , convex

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

1978 Swedish Mathematical Competition, 4

Tags: algebra , convex , logarithm , sequence

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

1974 All Soviet Union Mathematical Olympiad, 191

Tags: hexagon , diagonal , geometry , convex

a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?

1951 Moscow Mathematical Olympiad, 189

Tags: equal segments , convex , inequalities , angle

Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.