Find all relatively distinct integers $m, n, p\in \mathbb{Z}_{\ne 0}$ such that the polynomial $$F(x) = x(x - m)(x - n)(x - p) + 1$$is reducible in $\mathbb{Z}[x].$

Determine all real numbers $x>1$ such that \[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \] for any positive integer $n$.

A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]

Let $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that \[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]

Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawig the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies on between the parallel lines (excluding the intersection points on the lines) ? $ \textbf{(A)}\ 5600 \qquad\textbf{(B)}\ 5650 \qquad\textbf{(C)}\ 6500 \qquad\textbf{(D)}\ 6560 \qquad\textbf{(E)}\ 6600 $

Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

Source: 1976 Euclid Part A Problem 8 ----- Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is $\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E) } \text{none of these}$

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively. a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$ b. Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.

Compute the remainder when $98!$ is divided by $101$.

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

A league consists of $2024$ players. A [i]round[/i] involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called [i]balanced[/i] if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds. [i]Proposed by Anant Mudgal and Navilarekallu Tejaswi[/i]

Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$. Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$. Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$

On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?

The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$: [list][*]$97$ is the largest two-digit prime. [*]Reversing the order of its digits results in another prime. [*]There is only one way in which $97$ can be written as a difference of two perfect squares. [*]There is only one way in which $97$ can be written as a sum of two perfect squares. [*]$\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion. [*]Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.[/list] A repunit is a natural number whose digits are all $1$. For instance, \begin{align*}&1,\\&11,\\&111,\\&1111,\\&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$

Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

Find all polynomials $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties (a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and (b) all the roots of the equation $f(x)=0$ are real.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.

Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .