Given is the positive integer $n > 2$. Real numbers $\mid x_i \mid \leq 1$ ($i = 1, 2, ..., n$) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$. Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$.

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$

Without using a calculator, prove that $$2^{1995} > 5^{856}$$

For a given number $\alpha{}$ let $f_\alpha$ be a function defined as \[f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.\]Let $\alpha>1$ and $\beta=1/\alpha$. Prove that for any natural $n{}$ the relation $f_\beta(f_\alpha(n))=n$ holds. [i]Proposed by I. Dorofeev[/i]

Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$ $a_{n+1} = 7a_n + 12b_n + 6$ $b_{n+1} = 4a_n + 7b_n + 3$ Prove that $a_n^2$ is the difference of two consecutive cubes.

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

Solve the equation $$(x+1)^2-5(x+1) \sqrt{x}+4x=0$$

[b]p1.[/b] (a) Find three positive integers $a, b, c$ whose sum is $407$, and whose product (when written in base $10$) ends in six $0$'s. (b) Prove that there do NOT exist positive integers $a, b, c$ whose sum is $407$ and whose product ends in seven $0$'s. [b]p2.[/b] Three circles, each of radius $r$, are placed on a plane so that the center of each circle lies on a point of intersection of the other two circles. The region $R$ consists of all points inside or on at least one of these three circles. Find the area of $R$. [b]p3.[/b] Let $f_1(x) = a_1x^2+b_1x+c_1$, $f_2(x) = a_2x^2+b_2x+c_2$ and $f_3(x) = a_3x^2+b_3x+c_3$ be the equations of three parabolas such that $a_1 > a_2 > a-3$. Prove that if each pair of parabolas intersects in exactly one point, then all three parabolas intersect in a common point. [b]p4.[/b] Gigafirm is a large corporation with many employees. (a) Show that the number of employees with an odd number of acquaintances is even. (b) Suppose that each employee with an even number of acquaintances sends a letter to each of these acquaintances. Each employee with an odd number of acquaintances sends a letter to each non-acquaintance. So far, Leslie has received $99$ letters. Prove that Leslie will receive at least one more letter. (Notes: "acquaintance" and "non-acquaintance" refer to employees of Gigaform. If $A$ is acquainted with $B$, then $B$ is acquainted with $A$. However, no one is acquainted with himself.) [b]p5.[/b] (a) Prove that for every positive integer $N$, if $A$ is a subset of the numbers $\{1, 2, ...,N\}$ and $A$ has size at least $2N/3 + 1$, then $A$ contains a three-term arithmetic progression (i.e., there are positive integers $a$ and $b$ so that all three of the numbers $a$,$a + b$, and $a + 2b$ are elements of $A$). (b) Show that if $A$ is a subset of $\{1, 2, ..., 3500\}$ and $A$ has size at least $2003$, then $A$ contains a three-term arithmetic progression. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.

Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.

[b]Problem Section #2 b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.

For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$

A railway line is divided into ten sections by stations $E_1, E_2,..., E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $ 12$ km, and a trip through three consecutive stations is at least $17$ Km. Calculate the distance between $E_2$ and $E_7$.

Let $f: \mathbb{R}^{+}\rightarrow \mathbb{R}+$ be an increasing function. For each $u\in\mathbb{R}^{+}$, we denote $g(u)=\inf\{ f(t)+u/t \mid t>0\}$. Prove that: $(a)$ If $x\leq g(xy)$, then $x\leq 2f(2y)$; $(b)$ If $x\leq f(y)$, then $x\leq 2g(xy)$.

Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that: \[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Let $b_1,b_2,... $ be a sequence of positive real numbers such that for each $ n\ge 1$, $$b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}$$ Show that there is a positive integer $M$ such that $$\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}$$

We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$ Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\ (Ivan Novak)

Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality: $$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$