(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$. (b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.

Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$

Let $N$ be the smallest positive integer for which $x^2 + x + 1$ divides $166 -\sum_{d|N, d>0} x^d$. Find the remainder when $N$ is divided by $1000$.

Acute-angled $\triangle{ABC}$ triangle with condition $AB<AC<BC$ has cimcumcircle $C^,$ with center $O$ and radius $R$.And $BD$ and $CE$ diametrs drawn.Circle with center $O$ and radius $R$ intersects $AC$ at $K$.And circle with center $A$ and radius $AD$ intersects $BA$ at $L$.Prove that $EK$ and $DL$ lines intersects at circle $C^,$.

Let $a,b,c,d$ be integers such that $ad-bc$ is non zero. Suppose $b_1,b_2$ are integers both of which are multiples of $ad-bc$. Prove that there exist integers simultaneously satisfying both the equalities $ax+by=b_1, cx+dy=b_2$.

The points $ A,B,K,L,X$ lies of the circle $\Gamma$ in that order such that the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. The circle that passes through $A$ and tangent to $BK$ at $B$ intersects the line segment $KX$ at $P$ and $Q$. The circle that passes through $A$ and tangent to $BL$ at $B$ intersect the line segment $BX$ for the second time at $T$. Prove that $\angle{PTB} = \angle{XTQ}$

Let be given a triangle $ABC$. Points $P$ on side $AC$ and $Y$ on the production of $CB$ beyond $B$ are chosen so that $Y$ subtends equal angles with $AP$ and $PC$. Similarly, $Q$ on side $BC$ and $X$ on the production of $AC$ beyond $C$ are such that $X$ subtends equal angles with $BQ$ and $QC$. Lines $YP$ and $XB$ meet at $R$, $XQ$ and $YA$ meet at $S$, and $XB$ and $YA$ meet at $D$. Prove that $PQRS$ is a parallelogram if and only if $ACBD$ is a cyclic quadrilateral.

Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.

Three circles $\omega_1,\omega_2$ and $\omega_3$ pass through one point $D{}$. Let $A{}$ be the intersection of $\omega_1$ and $\omega_3$, and $E{}$ be the intersections of $\omega_3$ and $\omega_2$ and $F{}$ be the intersection of $\omega_2$ and $\omega_1$. It is known that $\omega_3$ passes through the center $B{}$ of the circle $\omega_2$. The line $EF$ intersects $\omega_1$ a second time at the point $G{}$. Prove that $\angle GAB=90^\circ$. [i]Proposed by K. Knop[/i]

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted: 1. Each person moves to the next seat clockwise. 2. Only the ones in seats 1 and $k$ exchange their seats. Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle. Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$.

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by $1$ run in the bottom of the $9\text{th}$ (last) inning, and the Reals are at bat. Each batter has a $\dfrac{1}{3}$ chance of hitting a single and a $\dfrac{2}{3}$ chance of making an out. If the Reals hit $5$ or more singles before they make $3$ outs, they will win. If the Reals hit exactly $4$ singles before they make $3$ outs, they will tie the game and send it into extra innings, and they will have a $\dfrac{3}{5}$ chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than $4$ singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

Let $\Gamma$ be the circumcircle of the triangle $ABC$ and let $M$ be the midpoint of the arc $\Gamma$ containing $A$ and bounded by $B$ and $C$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP = CQ$. Prove that $APQM$ is a cyclic quadrilateral.