Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality $$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$ holds.

Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

Determine all functions $f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}$ such that $$f(x^2yf(x))+f(1)=x^2f(x)+f(y)$$ holds for all nonzero real numbers $x$ and $y$.

Consider the grid of points $X = \{(m,n) | 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $( c = a \pm 2$ and $d = b \pm 1)$ or $( c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:

Prove that if a natural number $ N $ can be represented in the form the sum of three squares of integers divisible by $3$, then it is also is represented as the sum of three squares of integers that are not divisible by $3$.

Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

Clara computed the product of the first $n$ positive integers, and Valerie computed the product of the first $m$ even positive integers, where $m\ge2$. They got the same answer. Prove that one of them had made a mistake.

It is not possible to construct a segment of length $\pi$ using a straightedge, compass, and a given segment of length $1$. The following construction, given in $1685$ by Adam Kochansky, yields a segment whose length agrees with $\pi$ to five decimal places: Construct a circle of radius $1$ and call its center $O$. Construct a diameter $AB$ of this circle and a line $\ell$ tangent to the circle at $A$. Next, draw a circle with radius $1$ centered at $A$, and call one of the intersections with the original circle $C$. Now from C draw an arc of radius $1$ intersecting the circle around $A$ at $D$, where $D$ lies outside of the circle centered at $O$. Draw $OD$ and let $E$ be its point of intersection with $\ell$ . Construct $H$ on $AE$ such that $A$ is between $H$ and $E$, and $HE=3$. The distance between $B$ and $H$ is then close to $\pi$; calculate its exact value.

Let $A_1A_2\ldots A_6$ be a regular hexagon with side length $11\sqrt{3},$ and let $B_1B_2\ldots B_6$ be another regular hexagon completely inside $A_1A_2\ldots A_6$ such that for all $i \in \{1, 2, \ldots, 5\}$ $A_iA_{i+1}$ is parallel to $B_iB_{i+1}.$ Suppose that the distance between lines $A_1A_2$ and $B_1B_2$ is $7,$ the distance between lines $A_2A_3$ and $B_2B_3$ is $3,$ and the distance between lines $A_3A_4$ and $B_3B_4$ is $8.$ Compute the side length of $B_1B_2\ldots B_6.$

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

Written in mm/dd format, a date is called [i]cute[/i] if the month is divisible by the day. For example, the date [i]cute[/i] is a [i]cute[/i] date because $8$ is divisible by $2$. Find the number of [i]cute[/i] dates in a year. [i]Proposed by Andy Xu[/i]

Set $M=\{z\in\mathbb{C}|(z-1)^2=|z-1|^2\}$, then $\text{(A)}M=\{\text{pure imaginary number}\}$ $\text{(B)}M=\{\text{real number}\}$ $\text{(C)}M=\{\text{real number}\}\subset M\subset\{\text{complex number}\}$ $\text{(D)}M=\{\text{complex number}\}$

What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$? $\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$