Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

(a) Prove that for any triangle $H_1$, there exists a triangle $H_2$ whose side lengths are equal to the lengths of the medians of $H_1$. (b) If $H_3$ is the triangle whose side lengths are equal to the lengths of the medians of $H_2$, prove that $H_1$ and $H_3$ are similar.

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.

Four distinct points are arranged in a plane so that the segments connecting them has lengths $a,a,a,a,2a,$ and $b$. What is the ratio of $b$ to $a$? $ \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi $

$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $ [b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle. [b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $ [i]Vasile Pop[/i]

A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that $|XY|=|XZ|+|XW|$ Prove that all four points lie on a line.

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

Let $\vartriangle ABC$ be a triangle with orthocenter $H$ and altitudes $AD$, $BE$ and $CF$. Let $D'$, $E' $ and $F'$ be the intersections of the heights $AD$, $BE$ and $CF$, respectively, with the circumcircle of $\vartriangle ABC $, so that they are different points from the vertices of triangle $\vartriangle ABC$. Let $L$, $M$ and $N$ be the midpoints of $BC$, $AC$ and $AB$, respectively. Let $ P$, $Q$ and $R$ be the intersections of the circumcircle with $LH$, $MH$ and $NH$, respectively, such that $ P$ and $ A$ are on opposite sides of $BC$, $Q$ and $A$ are on opposite sides of $AC$ and $R$ and $C$ are on opposite sides of $AB$. Show that there exists a triangle whose sides have the lengths of the segments $D' P$, $E'Q$, and $F'R$.

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

Let $ABCD$ be a convex quadrilateral for which \[ AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.\] Prove that $ABCD$ is a parallelogram.

A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?

Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.

$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.

If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$ is also a rational number.

For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality: $3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.

$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.

Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. [b](a)[/b] For what values of $a/b$ can these segments be sides of a triangle ? [b](b)[/b] For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude? $ \textbf{a)}\ 4 \qquad\textbf{b)}\ 7 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ 23 $