Olympiad problems

2015 Baltic Way, 19

Three pairwairs distinct positive integers $a,b,c,$ with $gcd(a,b,c)=1$, satisfy \[a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2\] Prove that there doesnt exist a non-degenerate triangle with side lengths $a,b,c.$

2005 Spain Mathematical Olympiad, 3

Tags: sides of a triangle , circumradius , inradius , geometry

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that: a) $0^o \le A \le 60^o$. b) The height relative to side $a$ is three times the inradius $r$. c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.

2017 Thailand Mathematical Olympiad, 8

Tags: algebra , sides of a triangle , minimum , right triangle , inequalities

Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.

2018 Dutch BxMO TST, 2

Tags: integer , sides of a triangle , coprime , geometry

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

2017 Adygea Teachers' Geometry Olympiad, 3

Tags: sides of a triangle , geometry

Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.

2015 Caucasus Mathematical Olympiad, 5

Tags: number theory , perfect square , sides of a triangle

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

2018 Dutch BxMO TST, 2

Tags: integer , sides of a triangle , coprime , geometry

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

2006 Junior Tuymaada Olympiad, 5

Tags: algebra , quadratic trinomial , trinomial , quadratic , constant , sides of a triangle

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

2011 Sharygin Geometry Olympiad, 24

Tags: geometry , minimal , sides of a triangle

Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

2024 Dutch IMO TST, 2

Tags: incenter , ratio , sides of a triangle , geometry

Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.

2004 Peru MO (ONEM), 4

Tags: minimum , area of a triangle , integer , sides of a triangle , geometry

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.