Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.

Express the sum of $99$ terms$$\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}$$ as an irreducible fraction.

The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$

Solve in the nonnegative real numbers the system of equations $$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\ x_0 = x_{1999} \end{cases}$$

Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]

Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

For every non-constant polynomial $p$, let $H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}$. Prove that if $H_p=H_q$ for some polynomials $p,q$, then there exists a polynomial $r$ such that $p=r^m$ and $q=\xi\cdot r^n$ for some positive integers $m,n$ and constant $|\xi|=1$.

A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the [i]load [/i] of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the [i]distance [/i] of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud. If n is the number of bifurcations that a certain plant of that type has, it is asks a) the number of branches of the plant, b) the number of buds, c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds. Hint: You can proceed by induction, showing that if some results are correct for a given plant, they remain correct for the plant that is obtained substituting a bud in it for a pair of branches ending in individual buds.

Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even. Lê Anh Vinh

Suppose $P(x)$ is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial $P(x)$ on a board. At every step, if the polynomial on the board is $f(x)$, he can replace it with 1. $f(x)+c$ for a real number $c$, or 2. the polynomial $P(f(x))$. Can he always find a finite sequence of steps so the final polynomial on the board has exactly $2020$ real roots? What about $2021$? [i]~Sutanay Bhattacharya[/i]

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations: \[ \begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases} \] Find the maximum value of \(x + y\).

Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions. 1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$. 2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$. Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following. For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.

Find all positive numbers $x$ such that $20\{x\}+0.5\lfloor x\rfloor = 2005$.

Determine a polynomial of non-negative real coefficients that satisfies the following two conditions: $$p(0) = 0, p(|z|) \le x^4 + y^4,$$ being $|z|$ the module of the complex number $z = x + iy$ .

The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

Let $ k$ be a positive integer. Find all polynomials \[ P(x) \equal{} a_0 \plus{} a_1 x \plus{} \cdots \plus{} a_n x^n,\] where the $ a_i$ are real, which satisfy the equation \[ P(P(x)) \equal{} \{ P(x) \}^k\]

Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\] for all positive real numbers $x$ and $y$. [i](Andrew Brahms, USA)[/i]

For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.