There are $n\geq1$ cities on a horizontal line. Each city is guarded by a pair of stationary elephants, one just to the left and one just ot the right of the city, and facing away from it. The $2n$ elephants are of different sizes. If an elephant walks forward, it will knock aside any elephant that it approaches from behind, and in face-to-face meeting, the smaller elephant will be knocked aside. A moving elephant will keep walking in the same direction until it is knocked aside. Show that there is a unique city with the property that if any of the other cities orders its elephants to walk, then that city will not be invaded by an elephant. [url=https://artofproblemsolving.com/community/c6h1268873p6622370]IMO 2015, Shortlist C1[/url], modified by G. Smith

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

Let $a\geq 0$ be constant. Find the number of Intersection points of the graph of the function $y=x^3-3a^2x$ and the figure expressed by the equation $|x|+|y|=2$.

Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$

Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.

A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles? [asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B = (0,1); pair A = (0,-1); label("$B$",B,NW);label("$A$",A,2S); draw(Circle(A,2));draw(Circle(B,2)); fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray); fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray); draw((-sqrt(3),0)..B..(sqrt(3),0)); draw((-sqrt(3),0)..A..(sqrt(3),0)); path circ = Circle(origin,1); fill(circ,white); draw(circ); dot(A);dot(B); pair A1 = B + dir(45)*2; pair A2 = dir(45); pair A3 = dir(-135)*2 + A; draw(B--A1,EndArrow(HookHead,2)); draw(origin--A2,EndArrow(HookHead,2)); draw(A--A3,EndArrow(HookHead,2)); label("$2$",midpoint(B--A1),NW); label("$1$",midpoint(origin--A2),NW); label("$2$",midpoint(A--A3),NW);[/asy]$ \textbf{(A)}\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad \textbf{(B)}\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad \textbf{(C)}\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad\textbf{(D)}\ \frac {8}{3}\pi \minus{} 3\sqrt {2}$ $ \textbf{(E)}\ \frac {8}{3}\pi \minus{} 2\sqrt {3}$

$s,t\in\mathbb{R}$, then the minumum value of $(s+5-3|\cos t|)^2+(s-2|\sin t|)^2$ is________.

The factors of the expression $ x^2\plus{}y^2$ are: $ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\ \textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$

Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$

How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable) $\textbf{(A)}\ 210\qquad\textbf{(B)}\ 420\qquad\textbf{(C)}\ 630\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 1050$

Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

Prove that: $ \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6}$ For all $ n \in N$.

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N_{21}$ be the answer to question 21. Suppose a jar has $3N_{21}$ colored balls in it: $N_{21}$ red, $N_{21}$ green, and $N_{21}$ blue balls. Jonathan takes one ball at a time out of the jar uniformly at random without replacement until all the balls left in the jar are the same color. Compute the expected number of balls left in the jar after all balls are the same color. [b]p20.[/b] Let $N_{19}$ be the answer to question 19. For every non-negative integer $k$, define $$f_k(x) = x(x - 1) + (x + 1)(x - 2) + ...+ (x + k)(x - k - 1),$$ and let $r_k$ and $s_k$ be the two roots of $f_k(x)$. Compute the smallest positive integer $m$ such that $|r_m - s_m| > 10N_{19}$. [b]p21.[/b] Let $N_{20}$ be the answer to question 20. In isosceles trapezoid $ABCD$ (where $\overline{BC}$ and $\overline{AD}$ are parallel to each other), the angle bisectors of $A$ and $D$ intersect at $F$, and the angle bisectors of points $B$ and $C$ intersect at $H$. Let $\overline{BH}$ and $\overline{AF}$ intersect at $E$, and let $\overline{CH}$ and $\overline{DF}$ intersect at $G$. If $CG = 3$, $AE = 15$, and $EG = N_{20}$, compute the area of the quadrilateral formed by the four tangency points of the largest circle that can fit inside quadrilateral $EFGH$.

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.

Let $ ABC$ be an equilateral triangle and $ \Gamma$ the semicircle drawn exteriorly to the triangle, having $ BC$ as diameter. Show that if a line passing through $ A$ trisects $ BC,$ it also trisects the arc $ \Gamma.$

Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives. [i]Proposed by Nathan Ramesh

The diagram below shows an $8$x$7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find m + n. For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

The perimeter of an isosceles right triangle is $ 2p$. Its area is: $ \textbf{(A)}\ (2\plus{}\sqrt{2})p \qquad\textbf{(B)}\ (2\minus{}\sqrt{2})p \qquad\textbf{(C)}\ (3\minus{}2\sqrt{2})p^2\\ \textbf{(D)}\ (1\minus{}2\sqrt{2})p^2 \qquad\textbf{(E)}\ (3\plus{}2\sqrt{2})p^2$

Which of the following is the largest? $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3}{8} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{7}{24}$