Find all the possible values of a positive integer $n$ for which the expression $S_n=x^n+y^n+z^n$ is constant for all real $x,y,z$ with $xyz=1$ and $x+y+z=0$.

Luke chose a set of three different dates $a,b,c$ in the month of May, where in any year, if one makes a calendar with a sheet of grid paper the centers of the cells with dates $a,b,c$ would form an isosceles right triangle or a straight line. How many sets can be chosen? [img]https://cdn.artofproblemsolving.com/attachments/7/3/dbf90fdc81fc0f0d14c32020b69df53b67b397.png[/img]

An $\textit{access pattern}$ $\pi$ is a permutation of $\{1,2,\dots,50\}$ describing the order in which some $50$ memory addresses are accessed. We define the $\textit{locality}$ of $\pi$ to be how much the program jumps around the memory, or numerically, \[\sum_{i=2}^{50}\left\lvert\pi(i)-\pi(i-1)\right\rvert.\] If $\pi$ is a uniformly randomly chosen access pattern, what is the expected value of its locality?

Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.

Let $s$ and $t$ be the solutions to $x^2-10x+10=0$. Compute $\tfrac{1}{s^5}+\tfrac{1}{t^5}$.

Find with proof all solutions in nonnegative integers $a,b,c,d$ of the equation $$11^a5^b-3^c2^d=1$$

You have $100$ equal rods. It is allowed to split each rod into two or three shorter rods, not necessarily the same. The objective is that by rearranging the pieces (and using them all) $q>200$ can be assembled new rods, all of equal length. Find the values ​​of $q$ for whom this can be done.

The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, find in terms of $n$ the total number of $B$-squares of this chessboard.

If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$. [i]Proposed by Muztaba Syed[/i] [hide=Solution] [i]Solution.[/i] $\boxed{9}$ $a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$. [/hide]

What is the greatest integer less than or equal to $$\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$$ $ \textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad $

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

Sketch the phase portrait and investigate its variation under variation of the small complex parameter $\epsilon$: \[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]

Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$. [i]Proposed by Lewis Chen[/i]

In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$. Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$. Find the length of $BD$.

Take a point $P\left(\frac 12,\ \frac 14\right)$ on the coordinate plane. Let two points $Q(\alpha ,\ \alpha ^ 2),\ R(\beta ,\ \beta ^2)$ move in such a way that 3 points $P,\ Q,\ R$ form an isosceles triangle with the base $QR$, find the locus of the barycenter $G(X,\ Y)$ of $\triangle{PQR}$. [i]2011 Tokyo University entrance exam[/i]

Seven equally-spaced points are drawn on a circle of radius $1$. Three distinct points are chosen uniformly at random. What is the probability that the center of the circle lies in the triangle formed by the three points?

If $\sin\ x\ =\ 3\ \cos\ x$ then what is $\sin\ x\ \cos\ x$? $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ \frac{3}{10} $

Given real numbers $x, y, z, w$ such that $(x + y + 2z)(x + z + 3w) = 1$, what is the minimum possible value of $x^2 + y^2 + z^2 + w^2$?

Let $BL$ be angle bisector of acute triangle $ABC$.Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$.point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$.Point $T$ is diametrically opposite the point $K$ on the circumcircle of $\triangle AKC$.Prove that $ST$ passes through midpoint of arc $ABC$.(S. Berlov) [hide] :trampoline: my 100th post :trampoline: [/hide]

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$ prove that: ($\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a})^5 \geq 5^5(\frac{ac}{27})^2$

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

Given $r\in\mathbb{R}$. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below: $$S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box|$$ Every round, Alice choose a real number from $[0,1]$ (not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and $S$ is determined at the same time. If $S\ge r$ then Alice wins, otherwise Bob wins. Find all $r$ such that Alice can guarantee her victory.

Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. \begin{align*}\text{I.}&\text{ Bill is the oldest.}\\ \text{II.}&\text{ Amy is not the oldest.}\\ \text{III.}&\text{ Celine is not the youngest.}\end{align*} Rank the friends from the oldest to the youngest. $\textbf{(A)}\ \text{Bill, Amy, Celine}\qquad \textbf{(B)}\ \text{Amy, Bill, Celine}\qquad \textbf{(C)}\ \text{Celine, Amy, Bill}\qquad \\ \textbf{(D)}\ \text{Celine, Bill, Amy}\qquad \textbf{(E)}\ \text{Amy, Celine, Bill}$

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon. [b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio. [b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon. [b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon. [b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). [i]Proposed by Elliott Liu[/i]