Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with $$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$ for all polygons $P$, where the double integral is taken over $H(P).$

Let $ABCD$ be a square with side length $6$. Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$, $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$, and $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$ . The radius of the circle $X$ can be written $m -\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.

Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$, erases them, and replaces them with $a+b-1$. She repeats this process until only one number, $x$, remains. What are all the possible values of $x$?

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.

Positive integers are written on the blackboard one after another. The next integer $a_{n+1}$ (to be written after $a_1$,$a_2$,$...$,$a_n$) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e. $a_{n+1}$ is not of the form $k_1 *a_i + k_2 *a_2 + ... + k_n *a_n$ where$ k_1$, $k_2$,$..$, $k_n$ are non-negative integers). Prove that the process of writing cannot be infinite. (A Belov)

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

Shaft section of a circular cone with vertex $P$ is an isosceles right triangle. $A$ is a point on the circle of the bottom surface, while $B$ is a point inside the circle, $O$ is the center of the circle. If $AB\perp OB$ at $B$, $OH\perp PB$ at $H$, $PA=4$, $C$ is the midpoint of $PA$, then when the volume of triangular pyramid $O-HPC$ takes its maximum value, the length of $OB$ is $\text{(A)}\frac{\sqrt5}{3}\qquad\text{(B)}\frac{2\sqrt5}{3}\qquad\text{(C)}\frac{\sqrt6}{3}\qquad\text{(D)}\frac{2\sqrt6}{3}\qquad$

Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way: \[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&2008\\2009,&2010,&2011,&\ldots,&4026\\4017,&4018,&4019,&\ldots,&6024\\\vdots&&&&\vdots\\2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\] She picks one number from each row so that no two numbers she picks are in the same column. She them proceeds to add them together and finds that $S$ is the sum. Next, she picks $2008$ of the numbers that are distinct from the $2008$ she picked the first time. Again she picks exactly one number from each row and column, and again the sum of all $2008$ numbers is $S$. Find the remainder when $S$ is divided by $2008$.

Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality: $ 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD)$.

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]