Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$?

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$ the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.

Let $ T$ be a surjective mapping of the hyperbolic plane onto itself which maps collinear points into collinear points. Prove that $ T$ must be an isometry. [i]M. Bognar[/i]

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.

Compute the largest integer $N \le 2012$ with four distinct digits. [i]Proposed by Evan Chen[/i]

The percent that $ M$ is greater than $ N$ is: $ \textbf{(A)}\ \frac {100(M \minus{} N)}{M} \qquad\textbf{(B)}\ \frac {100(M \minus{} N)}{N} \qquad\textbf{(C)}\ \frac {M \minus{} N}{N} \qquad\textbf{(D)}\ \frac {M \minus{} N}{M}$ $ \textbf{(E)}\ \frac {100(M \plus{} N)}{N}$

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality $$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$

Given a triangle $ABC$ and a point $O$ inside it, it is known that $AB\le BC\le CA$. Prove that $$OA+OB+OC<BC+CA.$$

Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd integer so that $f(m) = 2023$. Suppose that $\frac{M}{m}$ equals $p_1^{e_1}p_2^{e_2}... p_l^{e_l}$ , where the $e_i$ are all nonzero integers and the $p_i$ are primes. Find $\left| \sum^l_{i=1} (p_i + e_i) \right|$.

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.

А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$. [i]Alexey Tolpygo[/i]

Misha solved the equation $x^2 + ax + b = 0$ and told Dima the set of four numbers - two roots and two coefficients of this equation (but not said which of them are roots and which are coefficients). Will he be able to Dima, find out what equation Misha solved if all the numbers in the set turned out to be different?

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

Let $n \geq 2$ be a positive integer. At the center of a circular garden is a guard tower. On the outskirt of the garden there are $n$ garden dwarfs regularly spaced. In the tower are attentive supervisors. Each supervisor controls a portion of the garden delimited by two dwarfs. We say that the supervisor $A$ controls the supervisor $B$ if the region of $B$ is contained in that of $A$. Among the supervisors there are two groups: the apprentices and the teachers. Each apprentice is controlled by exactly one teachers, and controls no one, while the teachers are not controlled by anyone. The entire garden has the following maintenance costs: - One apprentice costs 1 gold per year. - One teacher costs 2 gold per year. - A garden dwarf costs 2 gold per year. Show that the garden dwarfs cost at least as much as the supervisors.

Find all pairs $(a,b)$ of positive rational numbers such that \[\sqrt{a}+\sqrt{b}=\sqrt{2+\sqrt{3}}. \]

In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the circumcircle of $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.

$ABCD$ is a convex quadrilateral such that $\angle A = \angle C < 90^{\circ}$ and $\angle ABD = 90^{\circ}$. $M$ is the midpoint of $AC$. Prove that $MB$ is perpendicular to $CD$.

Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$

Given natural numbers $n > k > 1$, find all real solutions $x_1,..., x_n$ of the system $$x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2$$ for 1 $\le i \le n$. Here $x_{n+i} = x_i$ for all$ i$.

2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

For positive real $x$, let $$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of $$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.