Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.

Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$.

Consider the regular $1987$-gon $A_1A_2 . . . A_{1987}$ with center $O$. Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.

Find all positive integer $n$ such that there exists a permutation $(a_1, a_2,...,a_n)$ of $(1, 2,3,..., n)$ satisfying the condition: $a_1 + a_2 +... + a_k$ is divisible by $k$ for each $k = 1, 2,3,..., n$.

Given triangle $ABC$, let $M$ be the midpoint of side $AB$ and $N$ be the midpoint of side $AC$. A circle is inscribed inside quadrilateral $NMBC$, tangent to all four sides, and that circle touches $MN$ at point $X.$ The circle inscribed in triangle $AMN$ touches $MN$ at point $Y$, with $Y$ between $X$ and $N$. If $XY=1$ and $BC=12$, find, with proof, the lengths of the sides $AB$ and $AC$.

Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.

In the segment $AC$ of an isosceles triangle $\triangle ABC$ with base $BC$ is chosen a point $D$. On the smaller arc $CD$ of the circumcircle of $\triangle BCD$ is chosen a point $K$. Line $CK$ intersects the line through $A$ parallel to $BC$ at $T$. $M$ is the midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$. [i](A.Kuznetsov)[/i]

Given pairs $(a_1,b_1)$, $(a_2,b_2),\ldots, (a_n,b_n)$ of non-negative real numbers such that for any real $x$ and $y$ the equality $$\sqrt{a_1x^2+b_1y^2}+\sqrt{a_2x^2+b_2y^2}+\ldots+\sqrt{a_nx^2+b_ny^2}=\sqrt{x^2+y^2}$$ Prove that $a_1=b_1,a_2=b_2,\ldots$,$a_n=b_n$ [i]A. Vaidzelevich[/i]

As shown in the figure, in $\vartriangle ABC$, $AB>AC$, the inscribed circle $I$ is tangent to the sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$ respectively, and the straight lines $BC$ and $EF$ intersect at point $K$, $DG \perp EF$ at point $G$, ray $IG$ intersects the circumscribed circle of $\vartriangle ABC$ at point $H$. Prove that points $H$, $G$, $D$, $K$ lie on a circle. [img]https://cdn.artofproblemsolving.com/attachments/5/e/804fb919e9c2f9cf612099e44bad9c75699b2e.png[/img]

Let $x$ and $y$ be positive real numbers such that $x+y=1$. Show that \[ \left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9. \]

In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$? [asy] unitsize(1.5cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label("$A$",A,N); dot(B); label("$B$", B,SW);dot(C); label("$C$",C,SE); dot(DD); label("$D$",DD,NE); dot(EE); label("$E$",EE,NW); dot(FF); label("$F$",FF,S); [/asy] $\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40$

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Determine the minimum value of the expression $$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$ for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.

If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)? (A) at B (B) at C (C) at D (D) at both B and D (E) From C to D

Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $\angle MAB$ is $15 ^o$ larger than the angle $\angle MAC$ , and the angle $\angle MCB$ is $15^o$ larger than the angle $\angle MBC$. Find the angle $\angle BMC$ .

The value of which property decreases with an increase in the strength of intermolecular forces? $ \textbf{(A)}\hspace{.05in}\text{viscosity}\qquad\textbf{(B)}\hspace{.05in}\text{boiling point}\qquad\textbf{(C)}\hspace{.05in}\text{surface tension}\qquad\textbf{(D)}\hspace{.05in}\text{vapor pressure}\qquad $

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]

In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$.

At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of $2019$ other selected numbers and a) the selected numbers are not necessarily different? b) the selected numbers are pairwise different?

let $a_1,a_2,...,a_n$,$b_1,b_2,...,b_n$,$c_1,c_2,...,c_n$ be real numbers. prove that $$ \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}$$

Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.

Prove that if $ E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $ u < 1/2$, a point $ x\equal{}x(u)$ can be found so that \[ |(x\minus{}h,x\plus{}h) \cap E| \geq uh\] and \[ |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh\] for all sufficiently small positive values of $ h$. [i]K. I. Koljada[/i]

Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$