Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$. (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$. (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$. (c) Show, in (b) that $s=0$ implies $a_2=0$.

Find the least positive integer $n$ such that $15$ divides the product \[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\] , for every positive integers $a_1, a_2, \dots, a_{15}$.

A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n\leq 1000$. Given that \[p(1001) = a\cdot 2^b - c,\] where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that: [list] [*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and [*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. [/list] [i]Proposed by Ricky Liu[/i]

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$. The tangent line to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. The point $M$ is on the side $AB$ such that $\frac{AK}{BL} = \frac{AM}{BM}$. Let the perpendiculars from the point $M$ to the straight lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$ respectively. Prove that $2\angle CQP=\angle ACB$

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m \minus{} \log k| < \log n$. Find the sum of all possible values of the product $ mn$.

In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$. A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$. Points $C$, $A$, and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE = 6$ and $FG = 9$. Find $AG$. [asy] unitsize(5); pair A = (-9 sqrt(3), 0); pair B = (9 sqrt(3), 0); pair C = (-18 sqrt(3), 0); pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3); pair E = (2 sqrt(3), 4 sqrt(6)); pair F = (7 sqrt(3), 5 sqrt(6)); pair G = (12 sqrt(3), 6 sqrt(6)); real r = 9sqrt(3); draw(circle(A, r)); draw(circle(B, r)); draw(circle((B + C) / 2, 3r / 2)); draw(C -- D); draw("$6$", E -- D); draw(E -- F); draw("$9$", F -- G); dot(A); dot(B); label("$A$", A, plain.E); label("$B$", B, plain.E); label("$C$", C, W); label("$D$", D, dir(160)); label("$E$", E, S); label("$F$", F, SSW); label("$G$", G, N); [/asy]

Prove that if $a$, $b$ and $c$ are integers and the sums $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \,\,\,\, and \,\,\,\, \frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$ are also integers, then we have $|a| = |v| = |c|$. (A Gribalko)