Let @ denote the "averaged with" operation: $a$ @ $b$ = $\frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x,y$ and $z$? I. x @ (y+z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z) $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{III only} \qquad \textbf{(D)}\ \text{I and III only} \qquad \textbf{(E)}\ \text{II and III only} $

Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]

In an excavation in ancient Rome an unusual clock with $18$ divisions marked with Roman numerals (see figure). Unfortunately the watch was broken into $5$ pieces. The sum of the numbers on each piece was the same. Show how he could be broken the clock. [img]https://cdn.artofproblemsolving.com/attachments/7/a/6e83df1bb7adb13305239a152ac95a4a96f556.png[/img]

Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ D$ is inside $ \triangle ABC$, $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Find the value of the following expression: $\frac{1}{2} + (\frac{1}{3} + \frac{2}{3}) + (\frac{1}{4} + \frac{2}{4} + \frac{3}{4}) + \ldots + (\frac{1}{1000} + \frac{2}{1000} + \ldots + \frac{999}{1000})$

Find all positive integers $n$ such that the number $$n^6 + 5n^3 + 4n + 116$$ is the product of two or more consecutive numbers.

Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]

Let $a, b, c$ be positive reals so that $a^2+b^2+c^2=1$. Find the minimum value of $S=1/a^2+1/b^2+1/c^2-2(a^3+b^3+c^3)/abc$

Let $ABC$ be a acute angled triangle and let $AD,BE,CF$ be its altitudes. $X \not=A,B$ and $Y \not=A,C$ lie on sides $AB$ and $AC$, respectively, so that $ADXY$ is a cyclic quadrilateral. Let $H$ be the orthocenter of triangle $AXY$. Prove that $H$ lies on line $EF$.

Solve the equation $1 + x + x^2 + x^3 + ... + x^{2011} = 0$.

Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

Find the sum of squares of the digits of the least positive integer having $72$ positive divisors. $\textbf{(A)}\ 41 \qquad\textbf{(B)}\ 65 \qquad\textbf{(C)}\ 110 \qquad\textbf{(D)}\ 123 \qquad\textbf{(E)}\ \text{None}$

Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$. Find \[\frac{S^2}{m_c^3}\]

For all $n\ge2$ positive integer, let $f(n)$ denote the product of all distinct prime divisors of $n$. For example, $f(5)=5$, $f(8)=2$, and $f(12)=6$. Given a sequence ${a_n}$, where $a_1\ge2$, defined as follows: $$a_{n+1}=a_n+f(a_n)$$ Show that for any prime $p$, there exists a term $a_k$ in the sequence such that $p|a_k$.

Let $ABCD$ be a quadrilateral such that $\angle A$ and $\angle D$ are acute and $\overline{AB} = \overline{BC} = \overline{CD}$. Suppose that $\angle BDA = 30^\circ$, prove that $\angle DAC= 30^\circ$.

Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half?

If $x$ is a real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is: $\text{(A)} \ x\le -2~\text{or}~x\ge3 \qquad \text{(B)} \ x\le 2~\text{or}~x\ge3 \qquad \text{(C)} \ x\le -3 ~\text{or}~x\ge 2$ $\text{(D)} \ -3\le x \le 2\qquad \text{(E)} \ \-2\le x \le 3$

$N$ cards are arranged in a circle, with exactly one card face up and the rest face-down. In a turn, choose a proper divisor $k$ of $N$. You may begin at any card on the circle and flip every $k$-th card, counting clockwise, if and only if every $k$-th card begins the turn in the same orientation (either all face-up or all face-down). For example, with $15$ cards, you may start at any position and flip the $3$rd, $6$th, $9$th, $12$th, and $15$th cards around the circle if they all begin the turn face up (or all face-down). For what values of $N$ can all of the cards be flipped face-up in a finite number of turns?

Robot "Mag-o-matic" manipulates $101$ glasses, displaced in a row whose positions are numbered from $1$ to $101$. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form $(a;b,c)$, which it interprets as "consider the glass in position $a$: if it contains a ball, then switch the glasses in positions $b$ and $c$ (together with their own content), otherwise move on to the following instruction" (it means that $a,\,b,\,c$ are integers between $1$ and $101$, with $b$ and $c$ different from each other but not necessarily different from $a$). A $\emph{programme}$ is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one. A subset $S\subseteq \{0,\,1,\,2,\dots,\,101\}$ is said to be $\emph{identifiable}$ if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position $1$ contains a ball if and only if the number of glasses containing a ball is an element of $S$. (a) Prove that the subset $S$ of the odd numbers is identifiable. (b) Determine all subsets $S$ that are identifiable.

A mysterious machine contains a secret combination of $2016$ integer numbers $x_1,x_2,\ldots,x_{2016}$. It is known that all the numbers in the combination are equal but one. One may ask questions to the machine by giving to it a sequence of $2016$ integer numbers $y_1,\ldots,y_{2016}$, and the machine answers by telling the value of the sum \[ x_1y_1+\dots+x_{2016}y_{2016}. \] After answering the first question, the machine accepts a second question and then a third one, and so on. Determine how many questions are necessary to determine the combination: (a) knowing that the number which is different from the others is equal to zero; (b) not knowing what the number different from the others is.

Prove that the family of curves $$\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$$ satisfies $$\frac{dy}{dx}(a^2-b^2)=\left(x+y\frac{dy}{dx}\right)\left(x\frac{dy}{dx}-y\right)$$

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]