Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

If x,y<0 prove that $\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8$. When do we have equality?

Let $a, b, c$ be positive numbers such that $abc \le 8$. Prove that $$\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1$$

Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \] for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \] for every $n \in \mathbb{N}$.

Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.

Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!\cdot b!\cdot c!\cdot d!=24!$$ [list=1] [*] 4 [*] 4! [*] $4^4$ [*] None of these [/list]

Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$

The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

Let $m$ be a positive integer. Find polynomials $P(x)$ with real coefficients such that $$(x-m)P(x+2023) = xP(x)$$ is satisfied for all real numbers $x.$

Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.

Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?

Consider the expression \[2013^2+2014^2+2015^2+ \cdots+n^2\] Prove that there exists a natural number $n > 2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal $9999$

If $x_0=x_1=1$, and for $n\geq1$ $x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$, find a formula for $x_n$ as a function of $n$.

Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ . (Folklore)

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

[b]p1.[/b] Trung has $2$ bells. One bell rings $6$ times per hour and the other bell rings $10$ times per hour. At the start of the hour both bells ring. After how much time will the bells ring again at the same time? Express your answer in hours. [b]p2.[/b] In a soccer tournament there are $n$ teams participating. Each team plays every other team once. The matches can end in a win for one team or in a draw. If the match ends with a win, the winner gets $3$ points and the loser gets $0$. If the match ends in a draw, each team gets $1$ point. At the end of the tournament the total number of points of all the teams is $21$. Let $p$ be the number of points of the team in the first place. Find $n + p$. [b]p3.[/b] What is the largest $3$ digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$? [b]p4.[/b] Let s(n) be the number of quadruplets $(x, y, z, t)$ of positive integers with the property that $n = x + y + z + t$. Find the smallest $n$ such that $s(n) > 2014$. [b]p5.[/b] Consider a decomposition of a $10 \times 10$ chessboard into p disjoint rectangles such that each rectangle contains an integral number of squares and each rectangle contains an equal number of white squares as black squares. Furthermore, each rectangle has different number of squares inside. What is the maximum of $p$? [b]p6.[/b] If two points are selected at random from a straight line segment of length $\pi$, what is the probability that the distance between them is at least $\pi- 1$? [b]p7.[/b] Find the length $n$ of the longest possible geometric progression $a_1, a_2,..,, a_n$ such that the $a_i$ are distinct positive integers between $100$ and $2014$ inclusive. [b]p8.[/b] Feng is standing in front of a $100$ story building with two identical crystal balls. A crystal ball will break if dropped from a certain floor $m$ of the building or higher, but it will not break if it is dropped from a floor lower than $m$. What is the minimum number of times Feng needs to drop a ball in order to guarantee he determined $m$ by the time all the crystal balls break? [b]p9.[/b] Let $A$ and $B$ be disjoint subsets of $\{1, 2,..., 10\}$ such that the product of the elements of $A$ is equal to the sum of the elements in $B$. Find how many such $A$ and $B$ exist. [b]p10.[/b] During the semester, the students in a math class are divided into groups of four such that every two groups have exactly $2$ students in common and no two students are in all the groups together. Find the maximum number of such groups. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

Find all real numbers $x$ such that the expression \[\log_2 |1 + \log_2 |2 + \log_2 |x| | |\] does not have a defined value.

Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.

Prove that: a polynomial is irreducible in $\mathbb{Z}[x]$ if and only if it is irreducible in $\mathbb{Q}[x].$