Olympiad problems

2002 Estonia National Olympiad, 3

Prove that for positive real numbers $a, b$ and $c$ the inequality $2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2$ holds if and only if $a,b,c$ are the sides of a triangle.

1995 Romania Team Selection Test, 3

Tags: geometry , sidelenghts , altitude , inradius

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

1995 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , parameter , algebra , polynomial

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

1992 Romania Team Selection Test, 5

Tags: circumradius , sidelenghts , geometry

Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.

2008 Postal Coaching, 2

Tags: altitude , sidelenghts , rational , geometry

Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

1998 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , system of equations , algebra

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

2015 Singapore Junior Math Olympiad, 4

Tags: combinatorics , sidelenghts , isosceles , combinatorial geometry

Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

2017 Singapore Junior Math Olympiad, 1

Tags: geometry , rectangle , ratio , geometric inequalities , square , sidelenghts , inequalities

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

2000 Estonia National Olympiad, 4

Tags: trigonometry , angle , sidelenghts

Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$, where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.

1999 Moldova Team Selection Test, 2

Tags: system of equations , algebra , sidelenghts

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

1998 North Macedonia National Olympiad, 4

Tags: inequalities , geometry , geometric inequalities , sidelenghts , triangle area

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

1996 Singapore MO Open, 1

Tags: combinatorics , probability , sidelenghts

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?