Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

A base-$ 10$ three-digit number $ n$ is selected at random. Which of the following is closest to the probability that the base-$ 9$ representation and the base-$ 11$ representation of $ n$ are both three-digit numerals? $ \textbf{(A)}\ 0.3 \qquad \textbf{(B)}\ 0.4 \qquad \textbf{(C)}\ 0.5 \qquad \textbf{(D)}\ 0.6 \qquad \textbf{(E)}\ 0.7$

Let $a_{1}, a_{2}, \ldots, a_{n}\in [0;1]$. If $S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}$ then prove that \[\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}\]

Suppose $a$, $b$, $c$, and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, \\ ab+cd &= 260, \\ ac+bd &= 365, \\ ad+bc &= 244. \end{aligned}\] Compute $abcd.$ [i]Proposed by Michael Tang[/i]

Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.

Find all real numbers $ a $ such that $ x,y>a\implies x+y+xy>a. $ [i]Gheorghe Iurea[/i]

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$. (i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$. (ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$. (iii) $f(1)=1$.

What is the largest number of knights that can be put on a $5 \times 5$ chess board so that each knight attacks exactly two other knights? (M Gorelov)

What is the sum of all of the distinct prime factors of $25^3 - 27^2$?

Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$. [hide="Variants"]Variants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$[/hide]

Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.

At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$. Note: The tiles must completely cover all the board, with no overlappings.

Let $b>a\geq 2$ be positive integers. Prove that if number $a+k$ is coprime to number $b+k$, for all $k=1,2,...,b-a$, then $a,b$ are consecutive numbers

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.