[u]Round 1[/u] [b]p1.[/b] Five girls and three boys are sitting in a room. Suppose that four of the children live in California. Determine the maximum possible number of girls that could live somewhere outside California. [b]p2.[/b] A $4$-meter long stick is rotated $60^o$ about a point on the stick $1$ meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters. [b]p3.[/b] Let $f(x) = 2x(x - 1)^2 + x^3(x - 2)^2 + 10(x - 1)^3(x - 2)$. Compute $f(0) + f(1) + f(2)$. [u]Round 2[/u] [b]p4.[/b] Twenty boxes with weights $10, 20, 30, ... , 200$ pounds are given. One hand is needed to lift a box for every $10$ pounds it weighs. For example, a $40$ pound box needs four hands to be lifted. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time. [b]p5.[/b] Let $ABC$ be a right triangle with a right angle at $A$, and let $D$ be the foot of the perpendicular from vertex$ A$ to side $BC$. If $AB = 5$ and $BC = 7$, compute the length of segment $AD$. [b]p6.[/b] There are two circular ant holes in the coordinate plane. One has center $(0, 0)$ and radius $3$, and the other has center $(20, 21)$ and radius $5$. Albert wants to cover both of them completely with a circular bowl. Determine the minimum possible radius of the circular bowl. [u]Round 3[/u] [b]p7.[/b] A line of slope $-4$ forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle. [b]p8.[/b] Let $ABC$ be a right triangle with a right angle at $B$, $AB = 9$, and $BC = 7$. Suppose that point $P$ lies on segment $AB$ with $AP = 3$ and that point $Q$ lies on ray $BC$ with $BQ = 11$. Let segments $AC$ and $P Q$ intersect at point $X$. Compute the positive difference between the areas of triangles $AP X$ and $CQX$. [b]p9.[/b] Fresh Mann and Sophy Moore are racing each other in a river. Fresh Mann swims downstream, while Sophy Moore swims $\frac12$ mile upstream and then travels downstream in a boat. They start at the same time, and they reach the finish line 1 mile downstream of the starting point simultaneously. If Fresh Mann and Sophy Moore both swim at $1$ mile per hour in still water and the boat travels at 10 miles per hour in still water, find the speed of the current. [u]Round 4[/u] [b]p10.[/b] The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. The first few terms of the Fibonacci sequence are $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$. Every positive integer can be expressed as the sum of nonconsecutive, distinct, positive Fibonacci numbers, for example, $7 = 5 + 2$. Express $121$ as the sum of nonconsecutive, distinct, positive Fibonacci numbers. (It is not permitted to use both a $2$ and a $1$ in the expression.) [b]p11.[/b] There is a rectangular box of surface area $44$ whose space diagonals have length $10$. Find the sum of the lengths of all the edges of the box. [b]p12.[/b] Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes to $BC$ and $CA$, respectively. Suppose that segments $AD$ and $BE$ intersect at point $H$ with $AH = 20$ and $HD = 13$. Compute $BD \cdot CD$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c4h2809420p24782524]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$? [i]Proposed by Oleksiy Masalitin[/i]

The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

In a triangle $ABC$ , we have a point $O$ on $BC$ . Now show that there exists a line $l$ such that $l||AO$ and $l$ divides the triangle $ABC$ into two halves of equal area .

Let $ ABCD$ be a convex quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.

The figure below has a 1 × 1 square, a 2 × 2 square, a 3 × 3 square, a 4 × 4 square, and a 5 × 5 square. Each of the larger squares shares a corner with the 1 × 1 square. Find the area of the region covered by these five squares. [center][img]https://snag.gy/1AfJWt.jpg[/img][/center]

Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$

Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$. [i]Z. Daroczy, Gy. Maksa[/i]

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$. [i]Proposed by Anant Mudgal, India[/i]

Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers. Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ . Let $a_0$ be a real number and let $f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ . Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.

Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient \[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\] is a rational number.

Integer $n$ has $k$ different prime factors. Prove that $\sigma (n) \mid (2n-k)!$

Suppose that $m$ and $n$ are positive integers such that $m<n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m,n)$ satisfy these conditions? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }2007$

Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number.

In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

The number $1$ is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a - 1$ and $a + 1$; if the number $0$ occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show $1$ after $0$ steps; $2$ after $1$ step; $1, 3$ after $2$ steps; $2, 2, 4$ after $3$ steps, and so on. How many numbers will there be on the blackboard after $n$ steps?

If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$

Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$

Let $AD$ be the perpendicular to the hypotenuse $BC$ of the right triangle $ABC$. Let $DE$ be the height of the triangle $ADB$ and $DZ$ be the height of the triangle $ADC$. On the line $AB$ is chosen the point $N$ so that $CN$ is parallel to $EZ$. Let $A'$ be symmetrical of $A$ to $EZ$ and $I, K$ projections of $A'$ on $AB$, respectively, on $AC$. Prove that $<$ $NA'T$ $=$ $<$ $ADT$, where $T$ is the point of intersection of $IK$ and $DE$.

Mike rides a bike for $30$ minutes, traveling $8$ miles. He started riding at $20$ miles per hour, but by the end of his journey he was only traveling at $10$ miles per hour. What was his average speed, in miles per hour? [i]Proposed by Nathan Ramesh

Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]

For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1, d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.