Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

Prove that for all positive reals $a, b, c$ the following double inequality holds: $$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$

Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$

Find the number of ordered quadruplets $(a_1,a_2,a_3,a_4)$ of integers, for which $a_1\geq 1$, $a_2\geq 2$, $a_3\geq 3$, and $-10\leq a_4\leq 10$ and $a_1+a_2+a_3+a_4=2011$ .

Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$.

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

[b]p10.[/b] Three rectangles of dimension $X \times 2$ and four rectangles of dimension $Y \times 1$ are the pieces that form a rectangle of area $3XY$ where $X$ and $Y$ are positive, integer values. What is the sum of all possible values of $X$? [b]p11.[/b] Suppose we have a polynomial $p(x) = x^2 + ax + b$ with real coefficients $a + b = 1000$ and $b > 0$. Find the smallest possible value of $b$ such that $p(x)$ has two integer roots. [b]p12.[/b] Ten square slips of paper of the same size, numbered $0, 1, 2, ..., 9$, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$ When does the equality hold? (L. Parts)

Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.

For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.

Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that \[ f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y) \] for arbitrary positive real numbers \(x\) and \(y\).

Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and \[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\] find all integers $k$ such that every term of the sequence is a perfect square.

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.

The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$, \[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \] Find $a_{2004}$.