When flipped, a coin has a probability $p$ of landing heads. When flipped twice, it is twice as likely to land on the same side both times as it is to land on each side once. What is the larger possible value of $p$?

a) Can you pose the numbers $0,1,...,9$ on the circumference in such a way, that the difference between every two neighbours would be either $3$ or $4$ or $5$? b) The same question, but about the numbers $0,1,...,13$.

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$

Amir and Bella play a game on a gameboard with $6$ spaces, labeled $0, 1, 2, 3, 4,$ and $5.$ Each turn, each player flips a coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space $0,$ in which case it moves forward one space instead. If Amir and Bella each have a character that starts at space $0,$ the probability that they end turn $5$ on the same space can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$

Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South? \[\begin{tabular}[t]{c|cccc} & NE & MW & South & West \\ \hline White & 42 & 52 & 57 & 35 \\ Black & 5 & 5 & 15 & 2 \\ Asian & 1 & 1 & 1 & 3 \\ Other & 1 & 1 & 2 & 4 \end{tabular}\] $\text{(A)}\ 20\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 56\% \qquad \text{(E)}\ 80\% $

Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations: $$\left \{\begin{matrix} a\cdot b+c+d & = & 6, \\ b\cdot c+d+a & = & 2, \\ c\cdot d+a+b & = & 5, \\ d\cdot a+b+c & = & 3. \end{matrix}\right .$$

For how many $n$ in $\{1, 2, 3, ..., 100 \}$ is the tens digit of $n^2$ odd? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 $

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

[b]p1.[/b] BmMT is in a week, and we don’t have any problems! Let’s write $1$ on the first day, $2$ on the second day, $4$ on the third, $ 8$ on the fourth, $16$ on the fifth, $32$ on the sixth, and $64$ on the seventh. After seven days, how many problems will we have written in total? [b]p2.[/b] $100$ students are taking a ten-point exam. $50$ students scored $8$ points, $30$ students scored $7$ points, and the rest scored $9$ points. What is the average score for the exam? [b]p3.[/b] Rebecca has four pairs of shoes. Rebecca may or may not wear matching shoes. However, she will always use a left-shoe for her left foot and a right-shoe for her right foot. How many ways can Rebecca wear shoes? [b]p4.[/b] A council of $111$ mathematicians voted on whether to hold their conference in Beijing or Shanghai. The outcome of an initial vote was $70$ votes in favor of Beijing, and 41 votes in favor of Shanghai. If the vote were to be held again, what is the minimum number of mathematicians that would have to change their votes in order for Shanghai to win a majority of votes? [b]p5.[/b] What is the area of the triangle bounded by the line $20x + 16y = 160$, the $x$-axis, and the $y$-axis? [b]p6.[/b] Suppose that $3$ runners start running from the start line around a circular $800$-meter track and that their speeds are $100$, $160$, and $200$ meters per minute, respectively. How many minutes will they run before all three are next at the start line at the same time? [b]p7.[/b] Brian’s lawn is in the shape of a circle, with radius $10$ meters. Brian can throw a frisbee up to $50$ meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn? [b]p8.[/b] A seven digit number is called “bad” if exactly four of its digits are $0$ and the rest are odd. How many seven digit numbers are bad? [b]p9.[/b] Suppose you have a $3$-digit number with only even digits. What is the probability that twice that number also has only even digits? [b]p10.[/b] You have a flight on Air China from Beijing to New York. The flight will depart any time between $ 1$ p.m. and $6$ p.m., uniformly at random. Your friend, Henry, is flying American Airlines, also from Beijing to New York. Henry’s flight will depart any time between $3$ p.m. and $5$ p.m., uniformly at random. What is the probability that Henry’s flight departs before your flight? [b]p11.[/b] In the figure below, three semicircles are drawn outside the given right triangle. Given the areas $A_1 = 17$ and $A_2 = 14$, find the area $A_3$. [img]https://cdn.artofproblemsolving.com/attachments/4/4/28393acb3eba83a5a489e14b30a3e84ffa60fb.png[/img] [b]p12.[/b] Consider a circle of radius $ 1$ drawn tangent to the positive $x$ and $y$ axes. Now consider another smaller circle tangent to that circle and also tangent to the positive $x$ and $y$ axes. Find the radius of the smaller circle. [img]https://cdn.artofproblemsolving.com/attachments/7/4/99b613d6d570db7ee0b969f57103d352118112.png[/img] [b]p13.[/b] The following expression is an integer. Find this integer: $\frac{\sqrt{20 + 16\frac{\sqrt{20+ 16\frac{20 + 16...}{2}}}{2}}}{2}$ [b]p14.[/b] Let $2016 = a_1 \times a_2 \times ... \times a_n$ for some positive integers $a_1, a_2, ... , a_n$. Compute the smallest possible value of $a_1 + a_2 + ...+ a_n$. [b]p15.[/b] The tetranacci numbers are defined by the recurrence $T_n = T_{n-1} + T_{n-2} + T_{n-3} + T_{n-4}$ and $T_0 = T_1 = T_2 = 0$ and $T_3 = 1$. Given that $T_9 = 29$ and $T_{14} = 773$, calculate $T_{15}$. [b]p16.[/b] Find the number of zeros at the end of $(2016!)^{2016}$. Your answer should be an integer, not its prime factorization. [b]p17.[/b] A DJ has $7$ songs named $1, 2, 3, 4, 5, 6$, and $7$. He decides that no two even-numbered songs can be played one after the other. In how many different orders can the DJ play the $7$ songs? [b]p18.[/b] Given a cube, how many distinct ways are there (using $6$ colors) to color each face a distinct color? Colorings are distinct if they cannot be transformed into one another by a sequence of rotations. [b]p19. [/b]Suppose you have a triangle with side lengths $3, 4$, and $5$. For each of the triangle’s sides, draw a square on its outside. Connect the adjacent vertices in order, forming $3$ new triangles (as in the diagram). What is the area of this convex region? [img]https://cdn.artofproblemsolving.com/attachments/4/c/ac4dfb91cd055badc07caface93761453049fa.png[/img] [b]p20.[/b] Find $x$ such that $\sqrt{c +\sqrt{c - x}} = x$ when $c = 4$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way. [i]Attention: you have to really prove that a greater number of dominoes is impossible. [/i]

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that $$r^2 \le \frac19 (r_A^2 + r_B^2+r_C^2)$$ and identify, with justification, one case where the equality is attained.

Let $n$ be a positive integer. Is it possible to express $n^2+3n+3$ into the form $ab$ with $a$ and $b$ being positive integers, and such that the difference between $a$ and $b$ is smaller than $2\sqrt{n+1}$?

Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition

Given triangle $ ABC$. Points $ M$, $ N$ are the projections of $ B$ and $ C$ to the bisectors of angles $ C$ and $ B$ respectively. Prove that line $ MN$ intersects sides $ AC$ and $ AB$ in their points of contact with the incircle of $ ABC$.

Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$ for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

a) Prove that if $x,y>0$ then \[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \] b) Prove that if $a,b,c$ are positive real numbers, then \[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]

Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers

$\odot O_1$ and $\odot O_2$ are escribed circles of $\triangle ABC$ ($\odot O_1$ is in $\angle ACB$, $\odot O_2$ is in $\angle ABC$). $\odot O_1$ touches $CB,CA$ at $E,G$; $\odot O_1$ touches $BC,BA$ at $F,H$. $EG\cap FG=P$, prove that $AP\perp BC$.

Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.

An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge -ax(x + y)$ holds for all real numbers $x$ and $y$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.