Calculate the $100$th derivative of the function \[\frac{x^2+1}{x^3-x}\]

Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \] a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).

Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ , $(f(a)+b) f(a+f(b))=(a+f(b))^2$

Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying : $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

Every square on a $n\times n$ chessboard is colored with red, blue, or green. Each red square has at least one green square adjacent to it, each green square has at least one blue square adjacent to it, and each blue square has at least one red square adjacent to it. Let $R$ be the number of red squares. Prove that $\displaystyle \frac{n^2}{11} \le R \le \frac{2n^2}{3}$.

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

10) Call a string of letters $S$ an [i]almost-palindrome[/i] if $S$ and the reverse of $S$ differ in exactly $2$ places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost-palindrome. 11) Find all integers $n$, not necessarily positive, for which there exist positive integers ${a,b,c}$ satisfying $a^n + b^n = c^n$. 12) Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^2 + b^2} + \sqrt{a^2 + (b-1)^2} + \sqrt{(a-1)^2 + b^2} + \sqrt{(a-1)^2 + (b-1)^2}$. 13) Consider a $4$ x $4$ grid of squares, each originally colored red. Every minute, Piet can jump on any of the squares, changing the color of it and any adjacent squares to blue (two squares are adjacent if they share a side). What is the minimum number of minutes it will take Piet to change the entire grid to blue? 14) Let $ABC$ be an acute triangle with orthocenter $H$. Let ${D,E}$ be the feet of the ${A,B}$-altitudes, respectively. Given that $\overline{AH} = 20$ and $\overline{HD} =16$ and $\overline{BE} = 56$, find the length of $\overline{BH}$. 15) Find the smallest positive integer $b$ such that $1111 _b$ ($1111$ in base $b$) is a perfect square. If no such $b$ exists, write "No Solution" 16) For how many triples $( {x,y,z} )$ of integers between $-10$ and $10$, inclusive, do there exist reals ${a,b,c}$ that satisfy $ab = x$ $ac = y$ $bc = z$? 17) Unit squares $ABCD$ and $EFGH$ have centers $O_1$ and $O_2$, respectively, and are originally oriented so that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise around their centers at a rate of one revolution per hour. After $5$ minutes, what is the area of the intersection of the two squares? 18) A function $f$ satisfies, for all nonnegative integers $x$ and $y$, $f(x,0) = f(0,x) = x$ If $x \ge y \ge 0$, $f(x,y)=f(x-y,y)+1$ If $y \ge x \ge 0$, $f(x,y) = f(x,y-x)+1$ Find the maximum value of $f$ over $0 \le x,y \le 100$.

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

For real numbers $x$ and $y$, define \[\nabla(x,y)=x-\dfrac1y.\] If \[\underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \dfrac{m}{n}\] for relatively prime positive integers $m$, $n$, compute $100m + n$. [i] Proposed by David Altizio [/i]