[u]Round 5[/u] [b]p13.[/b] Points $A, B, C$, and $D$ lie in a plane with $AB = 6$, $BC = 5$, and $CD = 5$, and $AB$ is perpendicular to $BC$. Point E lies on line $AD$ such that $D \ne E$, $AE = 3$ and $CE = 5$. Find $DE$. [b]p14.[/b] How many ordered pairs of integers $(x,y)$ are solutions to $x^2y = 36 + y$? [b]p15.[/b] Chicken nuggets come in boxes of two sizes, $a$ nuggets per box and $b$ nuggets per box. We know that $899$ nuggets is the largest number of nuggets we cannot obtain with some combination of $a$-sized boxes and $b$-sized boxes. How many different pairs $(a, b)$ are there with $a < b$? [u]Round 6[/u] [b]p16.[/b] You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value $p$ is chosen randomly and uniformly between $0$ and $1$. Alice then makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 -p$. You design your own biased coin to maximize your chance of winning without knowing $p$. What is the probability that you win? [b]p17.[/b] There are $N$ distinct students, numbered from $1$ to $N$. Each student has exactly one hat: $y$ students have yellow hats, $b$ have blue hats, and $r$ have red hats, where $y + b + r = N$ and $y, b, r > 0$. The students stand in a line such that all the $r$ people with red hats stand in front of all the $b$ people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The $y$ people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is $2016$. What is $100y + 10b + r$? [b]p18.[/b] Let P be a point in rectangle $ABCD$ such that $\angle APC = 135^o$ and $\angle BPD = 150^o$. Suppose furthermore that the distance from P to $AC$ is $18$. Find the distance from $P$ to $BD$. [u]Round 7 [/u] [b]p19.[/b] Let triangle $ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D$ and $E$ lie on $AB$ and $AC$, respectively. Suppose $|AD| = |BC| = |EC|$ and triangle $ADE$ is isosceles. Find the sum of all possible values of $\angle BAC$ in radians. Write your answer in the form $2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi$, where $\frac{a}{b}$ and $\frac{c}{d}$ are in lowest terms, $-1 \le \frac{a}{b} \le 1$, and $-1 \le \frac{c}{d} \le 1$. [b]p20.[/b] Kevin is playing a game in which he aims to maximize his score. In the $n^{th}$ round, for $n \ge 1$, a real number between $0$ and $\frac{1}{3^n}$ is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is $a + b\sqrt{c}$ where $a, b$, and $c$ are integers and $c$ is positive and not divisible by any positive perfect square other than $1$. What is $100a + 10b + c$? [b]p21.[/b] Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at $(2016, 2017)$. Define the “area” of a path as the area below the path and above the $x$-axis. The sum of areas over all paths that Lisa can take can be represented as as $a \cdot {{4033} \choose {2016}}$ . What is the remainder when $a$ is divided by $1000$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.

The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that $$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$

The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all real numbers $x$ and $y$\[f(xf(y)-f(x))=2f(x)+xy.\]

Find all real numbers $x$, such that: a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$ b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$

Find all pairs $(a,b)$ of real numbers such that $\lfloor an + b \rfloor$ is a perfect square, for all positive integer $n$.

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

Simplify $\sqrt[4]{17 + 12\sqrt2} - \sqrt[4]{17 - 12\sqrt2}$.

Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied: $\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$ where $a_{n+1} = a_1$.

Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*} [i]Marius Cavachi[/i]

Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also \[\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},\] provided $n$ is an odd positive integer.

Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$.

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

We are given the equation $x^3-cx^2+(c-3)x+1=0$, where $c$ is an arbitrary number. Prove that, if the equation has at least one rational root, then all of its roots are rational.

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

Find all strictly monotone functions $f : \mathbb{R} \to \mathbb{R}$ such that some polynomial $P(x, y)$ satisfies the equality $$f(x + y) = P(f(x), f(y))$$ for all real numbers $x$ and $y$

Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$. Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.