Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle. by A.Khachaturyan

If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$

Infinitesimal Randall Munroe is glued to the center of a pentagon with side length $1$. At each corner of the pentagon is a confused infinitesimal velociraptor. At any time, each raptor is running at one unit per second directly towards the next raptor in the pentagon (in counterclockwise order). How far does each confused raptor travel before it reaches Randall Munroe?

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Solve for $x$ if $4x + 1 = 37$. $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.

Determine all real solutions to the system of equations: $$x^2 - y = z^2$$ $$y^2 - z = x^2$$ $$z^2 - x = y^2$$

Let \[ I(R)=\iint\limits_{x^2+y^2 \le R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right) dx dy. \] Find \[ \lim_{R \to \infty}I(R), \] or show that this limit does not exist.

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \leq yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$. [i]Proposed by Igor Voronovich, Belarus[/i]

Prove that for any distinct rational numbers $a, b, c$, the number \[\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}+\frac{1}{(a-b)^{2}}\] is the square of some rational number.

In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? $\textbf{(A) } 1 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 92 \qquad \textbf{(E) } 106$

$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible?

[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$.

Consider the set $S = \{1,2,3, ...,1441\}$. 1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania. 2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$. 3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.

Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$. If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.

It is known about real $a$ and $b$ that the inequality $$a \cos x + b \cos (3x) > 1$$ has no real solutions. Prove that $|b|\le 1$.

A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

Solve the equation $\{(x + 1)^3\} = x^3$, where $\{z\}$ is the fractional part of the number z, i.e. $\{z\} = z - [z]$.

Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b \times 5^c \times 7^d \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.