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.
On a planet there are $3 \times 2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.
Suppose $M$ is a point on the side $AB$ of $\triangle ABC$ such that the incircles of $\triangle AMC$ and $\triangle BMC$ have the same radius. The two incircles, centered at $O_1$ and $O_2$, meet $AB$ at $P$ and $Q$ respectively. It is known that the area of $\triangle ABC$ is six times the area of the quadrilateral $PQO_2O_1$, determine the possible values of $\frac{AC+BC}{AB}$. Justify your claim.
Let $a$, $b$, $c$, $d$ be positive real numbers such that $a + b + c + d = 1$. Prove that
$$
6(a^3 + b^3 + c^3 + d^3) \ge (a^2 + b^2 + c^2 + d^2) + \frac{1}{8}.
$$
Let $AD$, $BE$ and $CF$ be respectively the altitudes of $\triangle ABC$. Let $P$, $Q$ and $R$ be points on the lines $BC$, $CA$ and $AB$ respectively such that $AP$ is perpendicular to $EF$, $BQ$ is perpendicular to $FD$ and $CR$ is perpendicular to $DE$. Show that $AP$, $BQ$ and $CR$ are concurrent.
Let $O$ and $H$ be respectively the circumcentre and orthocentre of $\triangle ABC$. Let $A'$, $B'$ and $C'$ be the midpoints of $BC$, $CA$ and $AB$ respectively and $D$, $E$ and $F$ be respectively the feet from $A$, $B$ and $C$ to the opposite sides. Show that $OA' \cdot HD = OB' \cdot HE = OC' \cdot HF$.
Let $r$ be the positive root of the equation $x^2 - 2004x - 1 = 0$. Define the sequence $\{a_n\}$ as follows:
$$
a_0 = 1, a_{n+1} = [ra_n], n \ge 0,
$$
where $[y]$ denotes the greatest integer not exceeding $y$. Find the remainder when $a_{2004}$ is divided by $2004$.
Let $M$ be the intersection point of the diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$. The bisector of $\angle ACD$ meets $BA$ extended at $K$. If $MA \cdot MC + MA \cdot CD = MB \cdot MD$, show that $\angle BKC = \angle CDB$.
Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations
$$
p x^{2}+2 q x+r=0, \quad q x^{2}+2 r x+p=0, \quad r x^{2}+2 p x+q=0
$$
have a common root, say $\alpha$. Prove that
Let $x_{1}$ be a given positive integer. A sequence $\langle x_{n}\rangle_{n=1}^{\infty} = \langle x_{1}, x_{2}, x_{3}, \cdots \rangle$ of positive integers is such that $x_{n}$, for $n \geq 2$, is obtained from $x_{n-1}$ by adding some nonzero digit of $x_{n-1}$. Prove that
(a) the sequence has an even number;
(b) the sequence has infinitely many even numbers.
Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that
$$
f\left(x^{2}+y f(z)\right)=x f(x)+z f(y)
$$
for all $x, y, z$ in $\mathbf{R}$. (Here $\mathbf{R}$ denotes the set of all real numbers.)
All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L$, $K$ being nearer to $A$ than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
Alberto and Barbara play with a dice. After a while they realize that the die is rigged, and that the number $1$ comes up more frequently than the other numbers $5$ (which instead remain equally probable). They then decide that when $1$ comes out, that roll is canceled and they roll again. If you continue to roll the dice until you get $2$ valid rolls, what is the probability that the sum of the $2$ valid numbers rolled is $8$?
Edward went on holiday to the city of Altanbulat. His outbound plane left Milan at 1pm and arrived in Altanbulat at 9am the next day (local time). The return flight instead left Altanbulat at 9:00 am and landed at 3:00 pm the same day in Milan (again, all times indicated are according to the local time zone). Assuming that the two trips had the same real duration, what is the time difference between Italy and Altanbulat?
Let $ABC$ be a right triangle in $A$, with $AB \gt AC$; let $AH$ be the height relative to the hypotenuse. On the straight line $BC$ take $D$ such that $H$ is the midpoint of $BD$; then let $E$ be the foot of the perpendicular conducted from $C$ to $AD$. Prove that $EH = AH$.