Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.

Which operation makes the following expression true: $(4 \underline{~~~~} 1) \times (3 \underline{~~~~} 2 - 1) = 2$? $\textbf{(A) } +\qquad\textbf{(B) } -\qquad\textbf{(C) } \times\qquad\textbf{(D) } \div\qquad\textbf{(E) } \text{There is no such operation}$

Let $\Omega$ be a unit circle with diameter $AB$ and center $O$. Let $C$, $D$ be on $\Omega$ and lie on the same side of $AB$ such that $\angle CAB = 50^\circ$ and $\angle DBA = 70^\circ$. Suppose $AD$ intersects $BC$ at $E$. Let the perpendicular from $O$ to $CD$ intersect the perpendicular from $E$ to $AB$ at $F$. Find the length of $OF$. [i]Proposed by Puhua Cheng[/i]

Worms grow at the rate of $1$ metre per hour. When they reach their maximal length of $1$ metre, they stop growing. A full-grown worm may be dissected into two not necessarily equal parts. Each new worm grows at the rate of $1$ metre per hour. Starting with $1$ full-grown worm, can one obtain $10$ full-grown worms in less than $1$ hour?

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? $ \textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}$

Let $a_1, a_2, \cdots a_{100}$ be a permutation of $1,2,\cdots 100$. Define $l(k)$ as the maximum $m$ such that there exists $i_1, i_2 \cdots i_m$ such that $a_{i_1} > a_{i_2} > \cdots > a_{i_m}$ or $a_{i_1} < a_{i_2} < \cdots < a_{i_m}$, where $i_1=k$ and $i_1<i_2< \cdots <i_m$ Find the minimum possible value for $\sum_{i=1}^{100} l(i)$.

Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.

Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$. A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$. If $AD=CE$ prove that $BE$ is parallel to $AD$.

Find all integers $ b$ and $ c$ such that the equation $ x^2 - bx + c = 0$ has two real roots $ x_1, x_2$ satisfying $ x_1^2 + x_2^2 = 5$.

On a semicircle of diameter $AB$ and center $C$, consider vari­able points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value.

In a finite, simple, connected graph $G$ we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color. Knowing graph $G$ find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible? [i]Proposed by E. Bakaev[/i]

Let $m$ be a positive integer. Define the sequence $\{a_{n}\}_{n \ge 0}$ by \[a_{0}=0, \; a_{1}=m, \; a_{n+1}=m^{2}a_{n}-a_{n-1}.\] Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \le b$, gives a solution to the equation \[\frac{a^{2}+b^{2}}{ab+1}= m^{2}\] if and only if $(a, b)$ is of the form $(a_{n}, a_{n+1})$ for some $n \ge 0$.

Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

Real positive numbers $a, b, c$ satisfy $abc=1$. Prove the inequality $$\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.$$

Given 11 natural numbers under 21, show that you can choose two such that one divides the other.

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

How many functions $f:\{1,2,3, \ldots, 7\} \rightarrow \{1,2,3, \ldots, 7\}$ are there such that the set $\mathcal{F} = \{f(i) : i\in\{1,\ldots, 7\}\}$ has cardinality four, while the set $\mathcal{G} = \{f(f(f(i))) : i\in\{1,\ldots, 7\}\}$ consists of a single element? [i]Proposed by Sam Delatore[/i]

Let $ABC$ a triangle, $I$ the incenter, $D$ the contact point of the incircle with the side $BC$ and $E$ the foot of the bisector of the angle $A$. If $M$ is the midpoint of the arc $BC$ which contains the point $A$ of the circumcircle of the triangle $ABC$ and $\{F\} = DI \cap AM$, prove that $MI$ passes through the midpoint of $[EF]$.

Prove that \[\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\geq \frac{a+b+c}{3}\] for positive reals $a,b,c$ (S Tokarev)

An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same. An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino. Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$. (a) Prove that for any natural $n$, $S_n \geq M_{n+1}$. (b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$. A [b]grid segment[/b] is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called [b]wise[/b] if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called [b]unwise[/b]. (c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks. (d) Prove that any polystick which is in form of a path only going up and right is wise. (e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$ Time allowed for this exam was 2 hours.

Let $K, L, M$ be the feet of the altitudes drawn from the vertices $A, B, C$ of triangle $ABC$, respectively. Prove that $\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O}$ if and only if $ABC$ is equilateral.

Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ($1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.