Solve the following system of equations in positive integers \[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]

Let the polynomial $P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ has at least one real root and $a_0 \ne 0$. Prove that, consequently crossing out the monomials in the notation $P(x)$ in some order, we can obtain the number $a_0$ from it so that each intermediate polynomial also has at least one real root.

Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.

Let $x$ and $y$ be nonnegative real numbers. Prove that $(x +y^3) (x^3 +y) \ge 4x^2y^2$. When does equality holds? (Task committee)

Consider the polynomial with real coefficients: \[ p(x) = a_{2008}x^{2008} + a_{2007}x^{2007} + \dots + a_1x + a_0 \] and it is given that its coefficients satisfy: \[ a_i + a_{i+1} = a_{i+2}, \quad i \in \{0,1,2,\dots,2006\} \] If $p(1) = 2008$ and $p(-1) = 0$, compute $a_{2008} - a_0$.

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

Let $\mathbb{R}_{>0}$ be the set of the positive real numbers. Find all functions $f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that $$f(xy+f(x))=f(f(x)f(y))+x$$ for all positive real numbers $x$ and $y$.

How many solutions satisfying the condition $1 < x < 5$ does the equation $\{x[x]\} = 0.5$ have? (Here $[x]$ is the integer part of the number $x$, $\{x\} = x - [x]$ is the fractional part of the number $x$.)

Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following $f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$

Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers. (a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not. (b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.

Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies \[P(x_1)=P(x_2)= \cdots = P(x_k)=54\] \[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\] Determine the maximal value of $kn$.

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers. (L Borisov)

Prove that if the numbers $a, b, c, d$ satisfy the system of equations $$\begin{cases} a^2+b^2=2cd \\ b^2+c^2=2da \\ c^2+d^2=2ab \end{cases}$$ then $a=b=c=d$.

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

Let $G$ be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\] where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$? ${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $

In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.

Find all functions $f(x) = ax^2 + bx + c$, with $a \ne 0$, such that $f(f(1)) = f(f(0)) = f(f(-1))$ .

p1. The parabola $y = ax^2 + bx$, $a < 0$, has a vertex $C$ and intersects the $x$-axis at different points $A$ and $B$. The line $y = ax$ intersects the parabola at different points $A$ and $D$. If the area of triangle $ABC$ is equal to $|ab|$ times the area of ​​triangle $ABD$, find the value of $ b$ in terms of $a$ without use the absolute value sign. p2. It is known that $a$ is a prime number and $k$ is a positive integer. If $\sqrt{k^2-ak}$ is a positive integer, find the value of $k$ in terms of $a$. p3. There are five distinct points, $T_1$, $T_2$, $T_3$, $T_4$, and $T$ on a circle $\Omega$. Let $t_{ij}$ be the distance from the point $T$ to the line $T_iT_j$ or its extension. Prove that $\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}$ and $\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}$ [img]https://cdn.artofproblemsolving.com/attachments/2/8/07fff0a36a80708d6f6ec6708f609d080b44a2.png[/img] p4. Given a $7$-digit positive integer sequence $a_1, a_2, a_3, ..., a_{2017}$ with $a_1 < a_2 < a_3 < ...<a_{2017}$. Each of these terms has constituent numbers in non-increasing order. Is known that $a_1 = 1000000$ and $a_{n+1}$ is the smallest possible number that is greater than $a_n$. As For example, we get $a_2 = 1100000$ and $a_3 = 1110000$. Determine $a_{2017}$. p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume $V$. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $ b$ hours and continues again using only pump-2 for $c$ hours. If the operating cost of pump-1 is $15(a + b)$ thousand per hour and pump-2 operating cost is $4(a + c)$ thousand per hour, determine $ b$ and $c$ so that the operating costs of all pumps are minimum (express $b$ and $c$ in terms of $a$). Also determine the possible values ​​of $a$.

Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1, x_2,...,, x_n$. Find all solutions for each such $n$. $$\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}$$

Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.

Kostya marked the points $A(0, 1), B(1, 0), C(0, 0)$ in the coordinate plane. On the legs of the triangle ABC he marked the points with coordinates $(\frac{1}{2},0), (\frac{1}{3},0), \cdots, (\frac{1}{n+1},0)$ and $(0,\frac{1}{2}), (0,\frac{1}{3}), \cdots, (0,\frac{1}{n+1}).$ Then Kostya joined each pair of marked points with a segment. Sasha drew a $1 \times n$ rectangle and joined with a segment each pair of integer points on its border. As a result both the triangle and the rectangle are divided into polygons by the segments drawn. Who has the greater number of polygons: Sasha or Kostya? [i](M. Alekseyev )[/i]

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? $\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$