Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$

Let $A$ be a set of functions $f : R\to R$. For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$. Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\]

Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.) (E. Barabanov, I. Voronovich)

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A); E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F); G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$K$",K,W); label("$J$",J,S); [/asy] $ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $

A $9 \times 9$ chessboard with the standard checkered pattern has white squares at its four corners. What is the least number of rooks that can be placed on this board so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)

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 $