Solve the following system of equations in rational numbers: \[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]

Let $H$ be a regular hexagon with side length $1$. The sum of the areas of all triangles whose vertices are all vertices of $H$ can be expressed as $A\sqrt{B}$ for positive integers $A$ and $B$ such that $B$ is square-free. What is $1000A +B$?

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

Through the center of gravity of the acute-angled triangle $ ABC $, lines are drawn perpendicular to the sides $ BC $, $ CA $, $ AB $, intersecting them at the points $ P $, $ Q $, $ R $, respectively. Prove that if $ |BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB| $, then the triangle $ ABC $ is isosceles. Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines $ AP $, $ BQ $, $ CR $ have a common point.

Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $

Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Real numbers $a \neq 0, b, c$ are given. Prove that there is a polynomial $P(x)$ with real coefficients such that the polynomial $x^2+1$ divides the polynomial $aP(x)^2+bP(x)+c$. [i]Proposed by A. Golovanov[/i]

A function $ f$ has domain $ [0,2]$ and range $ [0,1]$. (The notation $ [a,b]$ denotes $ \{x: a\le x\le b\}$.) What are the domain and range, respectively, of the function $ g$ defined by $ g(x)\equal{}1\minus{}f(x\plus{}1)$? $ \textbf{(A)}\ [\minus{}1,1],[\minus{}1,0] \qquad \textbf{(B)}\ [\minus{}1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[\minus{}1,0] \qquad \textbf{(D)}\ [1,3],[\minus{}1,0] \qquad \textbf{(E)}\ [1,3],[0,1]$

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$. (Anton Trygub)

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$

$ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd(a,c) = 1$ and $b$ squarefree, find $a+b+c.$ [i]Proposed by Jason Wang[/i]

Let $ x_1,...,x_n$ be positive real numbers. Prove that: $ \displaystyle\sum_{k\equal{}1}^{n}x_k\plus{}\sqrt{\displaystyle\sum_{k\equal{}1}^{n}x_k^2} \le \frac{n\plus{}\sqrt{n}}{n^2} \left( \displaystyle\sum_{k\equal{}1}^{n} \frac{1}{x_k} \right) \left( \displaystyle\sum_{k\equal{}1}^{n} x_k^2 \right).$