Olympiad problems

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Found problems: 30

Durer Math Competition CD 1st Round - geometry, 2016.D+3

Tags: geometry , perpendicular , Durer

Let $M$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. Let $P$ and $Q$ be the centroids of triangles $AMD$ and $BMC$ respectively. Let $R$ and $S$ are the orthocenters of triangles $AMB$ and $CMD$. Prove that the lines $P Q$ and $RS$ are perpendicular to each other.

Durer Math Competition CD 1st Round - geometry, 2013.C1

Tags: ratio , areas , Durer

Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle? [img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667caafee6dbf397632e57e0.png[/img]

Durer Math Competition CD 1st Round - geometry, 2015.C2

Tags: geometry , rectangle , equal angles , Durer

Given a rectangle $ABCD$, side $AB$ is longer than side $BC$. Find all the points $P$ of the side line $AB$ from which the sides $AD$ and $DC$ are seen from the point $P$ at an equal angle (i.e. $\angle APD = \angle DPC$)

Durer Math Competition CD 1st Round - geometry, 2008.D1

Tags: geometry , geometric inequality , Durer , inequalities , algebra

Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$ When does equality holds?

Durer Math Competition CD 1st Round - geometry, 2021.D4

Tags: geometry , incenter , Durer

In the triangle $ABC$ we have $30^o$ at the vertex $A$, and $50^o$ at the vertex $B$. Let $O$ be the center of inscribed circle. Show that $AC + OC = AB$.