$(1+x^2)(1-x^3)$ equals $ \text{(A)}\ 1 - x^5\qquad\text{(B)}\ 1 - x^6\qquad\text{(C)}\ 1+ x^2 -x^3\qquad \\ \text{(D)}\ 1+x^2-x^3-x^5\qquad \text{(E)}\ 1+x^2-x^3-x^6 $

4. Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$: a) $f(n)$ < $f(n+1)$; b) $f(f(f(n)))$ = 4$n$ Find $f(2022)$.

The polynomial $$ Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2 $$ is represented as the sum of squares of four polynomials of four variables with integer coefficients. [b]a)[/b] Find at least one such representation [b]b)[/b] Prove that for any such representation at least one of the four polynomials isidentically zero. [i](A. Yuran)[/i]

The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*} $\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$

Prove that if $ a, b, c $ are non-negative numbers, then $$ a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$. a) Prove that: there exists integers $0\le i<j\le m^2$ such that $F_i\equiv F_j (\bmod m)$ and $F_{i+1}\equiv F_{j+1}(\bmod m)$. b) Prove that: there exists a positive integer $k$ such that $F_{n+k}\equiv F_n(\bmod m),$ for all natural numbers $n$. [i]*Denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$*[/i]. c) Prove that: $k(m)$ is the smallest positive integer such that $F_{k(m)}\equiv 0(\bmod m)$ and $F_{k(m)+1}\equiv 1(\bmod m)$. d) Given a positive integer $k$. Prove that: $F_{n+k}\equiv F_n(\bmod m)$ for all natural numbers $n$ iff $k\vdots k(m)$.

Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

If $\theta$ is the unique solution in $(0,\pi)$ to the equation $2\sin(x)+3\sin(\tfrac{3x}{2})+\sin(2x)+3\sin(\tfrac{5x}{2})=0,$ then $\cos(\theta)=\tfrac{a-\sqrt{b}}{c}$ for positive integers $a,b,c$ such that $a$ and $c$ are relatively prime. Find $a+b+c.$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ the following equation holds:$$1+f(xy)=f(x+f(y))+(y-1)f(x-1)$$ [i]M. Zorka[/i]

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.

Find all reals $(a, b, c)$ such that $$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$

Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

David has some pennies. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $3$ pennies left; is David spends all his money on cranberries, he will have $2$ pennies left. What is the least possible amount of pennies that David can originally have?

Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then $$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$ and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.

Solve the equation ("$2$" encounters $1985$ times): $$\dfrac{x}{2+ \dfrac{x}{2+\dfrac{x}{2+... \dfrac{x}{2+\sqrt {1+x}}}}}=1$$

Determine whether or not there exists a sequence of integers $a_1,a_2,\dots, a_{19}, a_{20}$ such that, the sum of all the terms is negative, and the sum of any three consecutive terms is positive.

Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that (i): For different reals $x,y$, $f(x) \not= f(y)$. (ii): For all reals $x,y$, $f(x+f(f(-y)))=f(x)+f(f(y))$

Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.