Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$. [i]Proposed by Nathan Xiong[/i]

The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?

Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

Find all positive integers $d$ for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0),P(1),P(2),\cdots,P(d^2-d)$ .

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f \colon \mathbb{N} \to \mathbb{Z}$ such that \[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\] for all positive integers $m,n$. [i]Merlijn Staps[/i]

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Problem 4. Let$\left( A,+,\cdot \right)$ be a ring with the property that $x=0$ is the only solution of the ${{x}^{2}}=0,x\in A$ecuation. Let $B=\left\{ a\in A|{{a}^{2}}=1 \right\}$. Prove that: (a) $ab-ba=bab-a$, whatever would be $a\in A$ and $b\in B$. (b) $\left( B,\cdot \right)$ is a group

A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.

The roots of $ (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0$ are: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 0\text{ and }4 \qquad \textbf{(C)}\ 1\text{ and }2 \qquad \textbf{(D)}\ 0,1,2\text{ and }4\qquad \textbf{(E)}\ 1,2\text{ and }4$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$. a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously. b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

Show that \[ \cos x^2 + \cos y^2 - \cos xy < 3 \] for reals $x$, $y$.

Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$. Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that: (a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points. (b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.

Find the number of integers $1\leq k\leq1336$ such that $\binom{1337}{k}$ divides $\binom{1337}{k-1}\binom{1337}{k+1}$. [i]Proposed by Tristan Shin[/i]

In the following expression, Melanie changed some of the plus signs to minus signs: $$ 1 + 3+5+7+\cdots+97+99$$ When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs? $ \textbf{(A) }14 \qquad \textbf{(B) }15 \qquad \textbf{(C) }16 \qquad \textbf{(D) }17 \qquad \textbf{(E) }18 \qquad $

Evaluate \[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]

Define alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number $S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$) Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them. What is the last digit of the sum obtained?

Let $ x,y$ be reals s.t. $ x^2\plus{}y^2\leq1$ and $ n$ a natural number.Prove that: $ (x^n\plus{}y)^2\plus{}y^2\geq\dfrac{1}{n\plus{}2}(x^2\plus{}y^2)^n$

Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.

[asy] real h = 7; real t = asin(6/h)/2; real x = 6-h*tan(t); real y = x*tan(2*t); draw((0,0)--(0,h)--(6,h)--(x,0)--cycle); draw((x,0)--(0,y)--(6,h)); draw((6,h)--(6,0)--(x,0),dotted); label("L",(3.75,h/2),W); label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm)); label("6''",(3,0),S); draw((2.5,-.5)--(0,-.5),Arrow(2mm)); draw((3.5,-.5)--(6,-.5),Arrow(2mm)); draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75)); //Credit to Zimbalono for the diagram[/asy] A rectangular piece of paper $6$ inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease $L$ in terms of angle $\theta$ is $\textbf{(A) }3\sec ^2\theta\csc\theta\qquad\textbf{(B) }6\sin\theta\sec\theta\qquad\textbf{(C) }3\sec\theta\csc\theta\qquad\textbf{(D) }6\sec\theta\csc ^2\theta\qquad \textbf{(E) }\text{None of these}$

If $x^4+8x^3+18x^2+8x+a = 0$ has four distinct real roots, then the real set of $a$ is $ \textbf{(A)}\ (-9,2) \qquad \textbf{(B)}\ (-9,0) \qquad \textbf{(C)}\ [-9,0) \qquad \textbf{(D)}\ [-8,1) \qquad \textbf{(E)}\ (-8,1)$

Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that \[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that $$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$ [i]Borislav Boyanov[/i]