Find consecutive integers bounding the expression \[\frac{1}{x_1 + 1}+\frac{1}{x_2 + 1}+\frac{1}{x_3 + 1}+...
+\frac{1}{x_{2001} + 1}+\frac{1}{x_{2002} + 1}\] where $x_1 = 1/3$ and $x_{n+1} = x_n^2 + x_n.$

Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic. [i]Florian Dumitrel[/i]

Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Then $P(0)=$ [list=1] [*] $-\frac{5}{2}$ [*] $-\frac{3}{2}$ [*] $-\frac{7}{4}$ [*] $\frac{6}{7}$ [/list]

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.

[b]p1.[/b] Joe owns stock. On Monday morning on October $20$th, $2008$, his stocks were worth $\$250,000$. The value of his stocks, for each day from Monday to Friday of that week, increased by $10\%$, increased by $5\%$, decreased by $5\%$, decreased by $15\%$, and decreased by $20\%$, though not necessarily in that order. Given this information, let $A$ be the largest possible value of his stocks on that Friday evening, and let $B$ be the smallest possible value of his stocks on that Friday evening. What is $A - B$? [b]p2.[/b] What is the smallest positive integer $k$ such that $2k$ is a perfect square and $3k$ is a perfect cube? [b]p3.[/b] Two competitive ducks decide to have a race in the first quadrant of the $xy$ plane. They both start at the origin, and the race ends when one of the ducks reaches the line $y = \frac12$ . The first duck follows the graph of $y = \frac{x}{3}$ and the second duck follows the graph of $y = \frac{x}{5}$ . If the two ducks move in such a way that their $x$-coordinates are the same at any time during the race, find the ratio of the speed of the first duck to that of the second duck when the race ends. [b]p4.[/b] There were grammatical errors in this problem as stated during the contest. The problem should have said: You play a carnival game as follows: The carnival worker has a circular mat of radius 20 cm, and on top of that is a square mat of side length $10$ cm, placed so that the centers of the two mats coincide. The carnival worker also has three disks, one each of radius $1$ cm, $2$ cm, and $3$ cm. You start by paying the worker a modest fee of one dollar, then choosing two of the disks, then throwing the two disks onto the mats, one at a time, so that the center of each disk lies on the circular mat. You win a cash prize if the center of the large disk is on the square AND the large disk touches the small disk, otherwise you just lost the game and you get no money. How much is the cash prize if choosing the two disks randomly and then throwing the disks randomly (i.e. with uniform distribution) will, on average, result in you breaking even? [b]p5.[/b] Four boys and four girls arrive at the Highball High School Senior Ball without a date. The principal, seeking to rectify the situation, asks each of the boys to rank the four girls in decreasing order of preference as a prom date and asks each girl to do the same for the four boys. None of the boys know any of the girls and vice-versa (otherwise they would have probably found each other before the prom), so all eight teenagers write their rankings randomly. Because the principal lacks the mathematical chops to pair the teenagers together according to their stated preference, he promptly ignores all eight of the lists and randomly pairs each of the boys with a girl. What is the probability that no boy ends up with his third or his fourth choice, and no girl ends up with her third or fourth choice? [b]p6.[/b] In the diagram below, $ABCDEFGH$ is a rectangular prism, $\angle BAF = 30^o$ and $\angle DAH = 60^o$. What is the cosine of $\angle CEG$? [img]https://cdn.artofproblemsolving.com/attachments/a/1/1af1a7d5d523884703b9ff95aaf301bcc18140.png[/img] [b]p7.[/b] Two cows play a game where each has one playing piece, they begin by having the two pieces on opposite vertices of an octahedron, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex occupied by its opponent’s piece. Because cows are not the most intelligent of creatures, they move their pieces randomly. What is the probability that the first cow to move eventually wins? [b]p8.[/b] Find the last two digits of $$\sum^{2008}_{k=1}k {2008 \choose k}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$

[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real. [b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real? [i]proposed by Mohammad Mansouri[/i]

Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left( a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.

Let $p$ and $q$ be real numbers such that the quadratic equation $x^2 + px + q = 0$ has two distinct real solutions $x_1$ and $x_2$. Suppose $|x_1-x_2|=1$, $|p-q|=1$. Prove that $p, q, x_1, x_2$ are all integers.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$. [i]Proposed by Ankan Bhattacharya, USA[/i]

Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

A sequence $\{a_i\}$ is defined by $a_1 = c$ for some $c > 0$ and $a_{n+1} = a_n + \frac{n}{a_n}$. Prove that $\frac{a_n}{n}$ converges and find its limit.

In the increasing sequence of positive integers $a_1$, $a_2$,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence. (a) Prove that all but finitely terms of the sequence are funny. (b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other? $ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$

Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

Three real numbers $a, b$ and $c$ are given . It is known that $a + b + c >0 , bc+ ca + ab > 0$ and $abc > 0$ . Prove that $a > 0 , b > 0$ and $c > 0$ .

Let $c$ denote the largest possible real number such that there exists a nonconstant polynomial $P$ with \[P(z^2)=P(z-c)P(z+c)\] for all $z$. Compute the sum of all values of $P(\tfrac13)$ over all nonconstant polynomials $P$ satisfying the above constraint for this $c$.

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]

The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition \[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\] for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.

Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, ... , x_7) = f(x_1) + f(x_2) +... + f(x_7)$. If $x_1, x_2,..., x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, ... , x_7)$ does $g(x_1, x_2, ... , x_7)$ obtain its maximal value.