To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

Let the altitude from $A$ and $B$ of triangle $ABC$ meet the circumcircle of $ABC$ again at $D$ and $E$ respectively. Let $DE$ meet $AC$ and $BC$ at $P$ and $Q$ respectively. Show that $ABQP$ is cyclic

Let $ ABC $ be a triangle. The perpendicular in $ B $ of the bisector of the angle $ \angle ABC $ intersects the bisector of the angle $ \angle BAC $ in $ M. $ Show that $ MC $ is perpendicular to the bisector of $ \angle BCA. $

Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

In triangle $ABC$ point $M$ is on side $AB$ such that $AM:AB=3:4$ and point $P$ is on side $BC$ such that $CP:CB=3:8$. Point $N$ is symmetric to $A$ with respect to point $P$. Prove that lines $MN$ and $AC$ are parallel.

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.

Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$ [i]P. Turan[/i]

There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Let $a>b>0$. Prove that $\sqrt2 a^3+ \frac{3}{ab-b^2}\ge 10$ When does equality hold?

Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$ If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.

We have a $m\times n$ table and $m\geq{4}$ and we call a $1\times 1$ square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What's the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove: $ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

Of all functions $h : Z_{>0} \to Z_{\ge 0}$, choose one satisfying $h(ab) = ah(b) + bh(a)$ for all $a, b \in Z_{>0}$ and $h(p) = p$ for all prime numbers $p$. Find the sum of all positive integers $n\le 100$ such that $h(n) = 4n$.

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

For every positive integer $n$ there exist unambiguously determined non-negative integers $a(n)$ and $b(n)$ such that $$n = 2^{a(n)}(2b(n)+1),$$ For positive integer $k$ we define $S(k)$ by: $$a(1) + a(2) + ... + a(2^k) = S(k)$$ Express $S(k)$ in terms of $k$.

Solve the equation $ 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}$, where $ x\in R$

Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent. [i]Pablo Jaén, Panama[/i]

Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.

A club provides 30 snacks to 18 members, and each member orders 3 different snacks. It is known that every snack is ordered by at least one member, and that any two members order at most one same snack. Is it possible to find 12 snacks, such that the snacks ordered by any member is not completely in these 12 snacks?