[u]Round 5[/u] [b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p14.[/b] Maisy chooses a random set $(x, y)$ that satisfies $$x^2 + y^2 -26x -10y \le 482.$$ The probability that $y>0$ can be expressed as $\frac{A\pi -B\sqrt{C}}{D \pi}$. Find $A+B +C +D$. [color=#f00]Due to the problem having a typo, all teams who inputted answers received points[/color] [b]p15.[/b] $6$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). [u]Round 6[/u] [b]p16.[/b] Find the number of $3$ by $3$ grids such that each square in the grid is colored white or black and no two black squares share an edge. [b]p17.[/b] Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 25$, and $AC = 15$. Let $D$ be the point on BC such that $CD = 4$. Let $E$ be the foot of the altitude from $A$ to $BC$. Let $F$ be the intersection of $AE$ with the circle of radius $7$ centered at $A$ such that $F$ is outside of triangle $ABC$. $DF$ can be expressed as $\sqrt{m}$, where $m$ is a positive integer. Find $m$. [b]p18.[/b] Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a $9$ year prison sentence. Given that neither of them talk, they both face a $3$ year sentence. If both of them talk, they both will serve a $6$ year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is $\frac{11}{36}$ , find the expected duration of Bill’s sentence in months. [u]Round 7[/u] [b]p19.[/b] Rectangle $ABCD$ has point $E$ on side $\overline{CD}$. Point $F$ is the intersection of $\overline{AC}$ and $\overline{BE}$. Given that the area of $\vartriangle AFB$ is $175$ and the area of $\vartriangle CFE$ is $28$, find the area of $ADEF$. [b]p20.[/b] Real numbers $x, y$, and $z$ satisfy the system of equations $$5x+ 13y -z = 100,$$ $$25x^2 +169y^2 -z2 +130x y= 16000,$$ $$80x +208y-2z = 2020.$$ Find the value of $x yz$. [color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points. [/color] [b]p21.[/b] Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n$, he can move to $n +1$, $n +2$, or $n +4$. In howmany different ways can he move to the number $10$? [u]Round 8[/u] [b]p22.[/b] A sequence $a_1,a_2,a_3, ...$ of positive integers is defined such that $a_1 = 4$, and for each integer $k \ge 2$, $$2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.$$ Given that $a_6 = 488$, find $a_2 +a_3 +a_4 +a_5$. [b]p23.[/b] $\overline{PQ}$ is a diameter of circle $\omega$ with radius $1$ and center $O$. Let $A$ be a point such that $AP$ is tangent to $\omega$. Let $\gamma$ be a circle with diameter $AP$. Let $A'$ be where $AQ$ hits the circle with diameter $AP$ and $A''$ be where $AO$ hits the circle with diameter $OP$. Let $A'A''$ hit $PQ$ at $R$. Given that the value of the length $RA'$ is is always less than $k$ and $k$ is minimized, find the greatest integer less than or equal to $1000k$. [b]p24.[/b] You have cards numbered $1,2,3, ... ,100$ all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make ${100 \choose 2}$ total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. [img]http://i14.tinypic.com/4tk6mnr.png[/img] a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.

Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash. "That seems fair to me," confirms Hannah encouragingly. While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it. There is a monic polynomial of degree $n$, with real coefficients. The first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, but the rest of the polynomial is cut off where Michael's page is ripped. Wendy barely makes out a little of Michael's scribbling, showing that $a_{n-1}=-a_{n-2}$. Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial. Wendy knows neither the value of $n$, nor the value of $a_{n-1}$, but still she finds a [greatest] lower bound for the answer to the problem. Find the absolute value of that lower bound.

Let $f$ be a function from R to R. Suppose we have: (1) $f(0)=0$ (2) For all $x, y \in (-\infty, -1) \cup (1, \infty)$, we have $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$. (3) If $x \in (-1,0)$, then $f(x) > 0$. Prove: $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$ with $n \in N^+$.

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.

Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .

Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$.

$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$? \[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3 \]

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.

Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]

What is the minimum value of the expression $$xy+yz+zx+\frac 1x+\frac 2y+\frac 5z$$ where $x, y, z$ are positive real numbers?

Find all functions $\mathbb R^+\to\mathbb R^+$ such that \[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d), \quad \forall a,b,c,d\in\mathbb R^+; \quad abcd=1\]

For every sequence $\{a_n\}~(n\in\mathbb N)$ we define the sequences $\{\Delta a_n\}$ and $\{\Delta^2a_n\}$ by the following formulas: \begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all $n\in\mathbb N$ for which $\Delta a_n^2\ne0$, define $$a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}.$$ (a) For which sequences $\{a_n\}$ is the sequence $\{\Delta^2a_n\}$ constant? (b) Find all sequences $\{a_n\}$, for which the numbers $a_n'$ are defined for all $n\in\mathbb N$ and for which the sequence $\{a_n'\}$ is constant. (c) Assume that the sequence $\{a_n\}$ converges to $a=0$, and $a_n\ne a$ for all $n\in\mathbb N$ and the sequence $\{\tfrac{a_{n+1}-a}{a_n-a}\}$ converges to $\lambda\ne1$. i. Prove that $\lambda\in[-1,1)$. ii. Prove that there exists $n_0\in\mathbb N$ such that for all integers $n\ge n_0$ we have $\Delta^2a_n\ne0$. iii. Let $\lambda\ne0$. For which $k\in\mathbb Z$ is the sequence $\{\tfrac{a_n'}{a_{n+k}}\}$ not convergent? iv. Let $\lambda=0$. Prove that the sequences $\{a_n'/a_n\}$ and $\{a_n'/a_{n+1}\}$ converge to $0$. Find an example of $\{a_n\}$ for which the sequence $\{a_n'/a_{n+2}\}$ has a non-zero limit. (d) What happens with part (c) if we remove the condition $a=0$?

Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$

Given that $x, y,$ and $z$ are positive real numbers such that $$x^{\log_2(yz)}=2^8\cdot3^4, \quad y^{\log_2(zx)}=2^9\cdot3^6, \quad \text{and}\quad z^{\log_2(xy)}=2^5 \cdot 3^{10},$$ compute the smallest possible value of $xyz.$

Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions: $f(0, 0) = 0$; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$, $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$. Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.