If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

Fix integers $n\geq 2$ and $1\leq m\leq n-1$. Let $a_0, a_1, \dots, a_n$ be non-negative real numbers satisfying $a_0+a_1+\dots +a_n=1$. Prove that, if $\sum_{k=0}^n a_kx^k < x^m$ for some $0<x<1$, then $$\sum_{k=0}^{m-1}(m-k)a_k < \sum_{k=m+1}^n (k-m)a_k.$$

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

[b]p1.[/b] If Clark wants to divide $100$ pizzas among $25$ people so that each person receives the same number of pizzas, how many pizzas should each person receive? [b]p2.[/b] In a group of $3$ people, every pair of people shakes hands once. How many handshakes occur? [b]p3.[/b] Dylan and Joey have $14$ costumes in total. Dylan gives Joey $4$ costumes, and Joey now has the number of costumes that Dylan had before giving Joey any costumes. How many costumes does Dylan have now? [b]p4.[/b] At Banjo Borger, a burger costs $7$ dollars, a soda costs $2$ dollars, and a cookie costs $3$ dollars. Alex, Connor, and Tony each spent $11$ dollars on their order, but none of them got the same order. If Connor bought the most cookies, how many cookies did Connor buy? [b]p5.[/b] Joey, James, and Austin stand on a large, flat field. If the distance from Joey to James is $30$ and the distance from Austin to James is $18$, what is the minimal possible distance from Joey to Austin? [b]p6.[/b] If the first and third terms of a five-term arithmetic sequence are $3$ and $8$, respectively, what is the sum of all $5$ terms in the sequence? [b]p7.[/b] What is the area of the $S$-shaped figure below, which has constant vertical height $5$ and width $10$? [img]https://cdn.artofproblemsolving.com/attachments/3/c/5bbe638472c8ea8289b63d128cd6b449440244.png[/img] [b]p8.[/b] If the side length of square $A$ is $4$, what is the perimeter of square $B$, formed by connecting the midpoints of the sides of $A$? [b]p9.[/b] The Chan Shun Auditorium at UC Berkeley has room number $2050$. The number of seats in the auditorium is a factor of the room number, and there are between $150$ and $431$ seats, inclusive. What is the sum of all of the possible numbers of seats in Chan Shun Auditorium? [b]p10.[/b] Krishna has a positive integer $x$. He notices that $x^2$ has the same last digit as $x$. If Krishna knows that $x$ is a prime number less than $50$, how many possible values of $x$ are there? [b]p11.[/b] Jing Jing the Kangaroo starts on the number $1$. If she is at a positive integer $n$, she can either jump to $2n$ or to the sum of the digits of $n$. What is the smallest positive integer she cannot reach no matter how she jumps? [b]p12.[/b] Sylvia is $3$ units directly east of Druv and runs twice as fast as Druv. When a whistle blows, Druv runs directly north, and Sylvia runs along a straight line. If they meet at a point a distance $d$ units away from Druv's original location, what is the value of $d$? [b]p13.[/b] If $x$ is a real number such that $\sqrt{x} + \sqrt{10} = \sqrt{x + 20}$, compute $x$. [b]p14.[/b] Compute the number of rearrangments of the letters in $LATEX$ such that the letter $T$ comes before the letter $E$ and the letter $E$ comes before the letter $X$. For example, $TLEAX$ is a valid rearrangment, but $LAETX$ is not. [b]p15.[/b] How many integers $n$ greater than $2$ are there such that the degree measure of each interior angle of a regular $n$-gon is an even integer? [b]p16.[/b] Students are being assigned to faculty mentors in the Berkeley Math Department. If there are $7$ students and $3$ mentors and each student has exactly one mentor, in how many ways can students be assigned to mentors given that each mentor has at least one student? [b]p17.[/b] Karthik has a paper square of side length $2$. He folds the square along a crease that connects the midpoints of two opposite sides (as shown in the left diagram, where the dotted line indicates the fold). He takes the resulting rectangle and folds it such that one of its vertices lands on the vertex that is diagonally opposite. Find the area of Karthik's final figure. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01aa386f6616cafeed5f95ababb27bf24657f6.png[/img] [b]p18.[/b] Sally is inside a pen consisting of points $(a, b)$ such that $0 \le a, b \le 4$. If she is currently on the point $(x, y)$, she can move to either $(x, y + 1)$, $(x, y - 1)$, or $(x + 1, y)$. Given that she cannot revisit any point she has visited before, find the number of ways she can reach $(4, 4)$ from $(0, 0)$. [b]p19.[/b] An ant sits on the circumference of the circular base of a party hat (a cone without a circular base for the ant to walk on) of radius $2$ and height $\sqrt{5}$. If the ant wants to reach a point diametrically opposite of its current location on the hat, what is the minimum possible distance the ant needs to travel? [img]https://cdn.artofproblemsolving.com/attachments/3/4/6a7810b9862fd47106c3c275c96337ef6d23c2.png[/img] [b]p20.[/b] If $$f(x) = \frac{2^{19}x + 2^{20}}{ x^2 + 2^{20}x + 2^{20}}.$$ find the value of $f(1) + f(2) + f(4) + f(8) + ... + f(220)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.

For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? ${ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8 $

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

Find all positive integers $a,b,c$ and prime $p$ satisfying that \[ 2^a p^b=(p+2)^c+1.\]

In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$. Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$. The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$. Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is $ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$

Given an isosceles trapezoid and draw one of its diagonals (so we get $2$ triangles). On each triangle we get, there is an ant walking along the edge. There speed are equal and unchanged. These $2$ ants also didn't change their walking directions and their directions are opposite. Prove that: for all initial locations of the ants, they will meet each other at some time.

Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.

If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$?

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed. [i]Proposed by Nathan Ramesh

Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?