Circles $(K_1, r_1)$ and $(K_2, r_2)$ are tangent and also they are tangent to the sides of the rectangle $AB\Gamma\Delta$, with $AB = a$ and $B\Gamma = b$.
(i) Express the sides $a$ and $b$ with respect to the radius $r_1$ and $r_2$.
(ii) If the common internal tangent of the two circles passes through $\Delta$ compute the ratio $\frac{r_1}{r_2}$ and find the length of $\Delta K$.
Let $ABCD$ be a convex quadrilateral with $\angle DAC = \angle BDC = 36^\circ$, $\angle CBD = 18^\circ$ and $\angle BAC = 72^\circ$. The diagonals $AC$ and $BD$ intersect at point $P$. Determine the measure of $\angle APD$.
Given are 50 points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 130 scalene triangles with vertices of that color.
$ABCD$ is a convex cyclic quadrilateral. The diagonals $AC$ and $BD$ intersect at the point $E$. It is given that $AB = 39$, $AE = 45$, $AD = 60$ and $BC = 56$.
Determine the length of $CD$.
In the triangle $ABC$, the angle $\alpha = \hat{A}$ and the side $a = |BC|$ are given. It is known that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ is the circumradius. Determine all such triangles, that is, compute the sides $b$ and $c$ of all such triangles.
Let $x \gt 1$ a non integer number. Prove that
$$
\left( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \right) + \left( \frac{x+[x]}{\{x\}} - \frac{\{x\}}{x+[x]} \right) \gt \frac{9}{2},
$$
where $[x]$ and $\{x\}$ represents the integer and the fractional part of $x$.
Let $D$ be a point on the side $BC$ of triangle $ABC$ such that $AB + BD = AC + CD$. The line segment $AD$ cuts the incircle of triangle $ABC$ at $X$ and $Y$ with $X$ closer to $A$. Let $E$ be the point of contact of the incircle of triangle $ABC$ on the side $BC$. Show that
(i) $EY$ is perpendicular to $AD$,
(ii) $XD$ is $2IA'$, where $I$ is the incentre of the triangle $ABC$ and $A'$ is the midpoint of $BC$.
Is there a polynomial $f$ of degree $2007$ with integer coefficients, such that $f(n), f(f(n)), f(f(f(n))), \dots$ are pairwise relatively prime for every integer $n$? Justify your claim.
In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.
Determine if there exists a positive integer pair $(m, n)$, such that
(i) the greatest common divisor of $m$ and $n$ is $1$, and $m \le 2007$,
(ii) for any $k = 1, 2, \dots, 2007$, $\lfloor \frac{nk}{m} \rfloor = \lfloor \sqrt{2}k \rfloor$.
(Here $\lfloor x \rfloor$ stands for the greatest integer less than or equal to $x$.)
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B + |\cos A \sin B| = \sin A |\cos A| + |\sin B \cos B|$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C - \sin A)^2 + (\cot C - \cos B)^2$.
Let $ABC$ be a triangle with $AB = c$, $BC = a$ and $CA = b$. Show that
$$
\frac{\cos A}{a^3} + \frac{\cos B}{b^3} + \frac{\cos C}{c^3} \ge \frac{3}{2abc}.
$$
a. Given five points on a plane such that no three of the points are collinear, show that among the triangles which are drawn using any three of these five points as vertices, at least three of the triangles formed are not acute-angled triangles. (An acute-angled triangle is one in which all the three interior angles are acute angles.)
b. Given any 100 points on a plane such that no three of the points are collinear, show that among the triangles which are drawn using any three of these 100 points as vertices, at least 30% of the triangles are not acute-angled triangles.
The sequence $\{a_n\}$ is defined by $a_1 = 0$ and $(n+1)^3 a_{n+1} = 2n^2(2n+1)a_n + 2(3n+1)$ for all integers $n \ge 1$. Show that infinitely many members of the sequence are positive integers.
Let $ABC$ be an equilateral triangle with circumcentre $O$. Let $P$ be a point inside the triangle such that $\angle BPC = 120^\circ$ and $P \ne O$. Let $B'$ and $C'$ be the images of $B$ and $C$ respectively under the reflection in the line $PO$. Furthermore $B'C'$ meets $AP$ at $D$. Determine $AD : DP$. Justify your claim.
Let $ABCD$ be a cyclic quadrilateral such that $AD + BC = AB$. Show that the intersection of the bisectors of $\angle ADC$ and $\angle BCD$ is on line $AB$.