A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the positive direction or $1$ unit in the negative direction. What is his expected distance from the origin after $6$ moves? [i]2017 CCA Math Bonanza Lightning Round #3.4[/i]

Two points are selected in a convex pentagon. Prove that you can choose a quadrilateral with vertices at the vertices of a pentagon so that both selected points fall into it.

Let each sequence of capital Bulgarian letters be a word (Note that the Bulgarian alphabet consists of 30 letters). We say that a word is [i]calm[/i], if there is no sequence of the letter $P$ in it (two letters $P$ next to each other). Find a clear expression for $f(n)$ (closed formula), which represents the number of calm words with length $n$. Example: [i]РНТ[/i] and [i]ТТРВ[/i] are calm, but [i]ЖГРР[/i] and [i]ЖРРРП[/i] aren’t.

Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?

Compute the number of digits is $2015!$. Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?

Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on $(ABC)$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. Prove that $D$ lies on the line $SM$.

$ S \equal{} \{1,2,\dots,n\}$ is divided into two subsets. How the set is divided, if there exist two elements whose sum is a perfect square, then $ n$ is at least ? $\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ 17$

Three circles through the point $O$ and of radius $r$ intersect pairwise in the additional points $A$,$B$,$C$. Prove that the circle through the points $A$, $B$, and $C$ also has radius $r$.

Find the largest positive integer $n$ for which the inequality \[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\] holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.

A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$. What is the last page to be read?

Let $V_0 = \varnothing$ be the empty set and recursively define $V_{n+1}$ to be the set of all $2^{|V_n|}$ subsets of $V_n$ for each $n=0,1,\dots$. For example \[ V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \quad\text{and}\quad V_3 = \left\{ \varnothing, \left\{ \varnothing \right\}, \left\{ \left\{ \varnothing \right\} \right\}, \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \right\}. \] A set $x \in V_5$ is called [i]transitive[/i] if each element of $x$ is a subset of $x$. How many such transitive sets are there? [i]Proposed by Evan Chen[/i]

In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? $\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying \[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \] Prove that \[ 2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2 \] has no real roots.

Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons? by N.Beluhov

Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$? $\text{(A) }60\qquad\text{(B) }66\qquad\text{(C) }68\qquad\text{(D) }70\qquad\text{(E) }72$

Points $A_1, A_2, ..., A_{10}$ are marked on a circle clockwise. It is known that these points can be divided into pairs of points symmetric with respect to the centre of the circle. Initially at each marked point there was a grasshopper. Every minute one of the grasshoppers jumps over its neighbour along the circle so that the resulting distance between them doesn't change. It is not allowed to jump over any other grasshopper and to land at a point already occupied. It occurred that at some moment nine grasshoppers were found at points $A_1, A_2, ... , A_9$ and the tenth grasshopper was on arc $A_9A_{10}A_1$. Is it necessarily true that this grasshopper was exactly at point $A_{10}$?

Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ $\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}$

If $a_k >0$ [ $k=$1, 2, $\cdots$, $n$], then prove the following inequality $$\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5$$

Equilateral triangle $ABC$ has side length $6$. Circles with centers at $A$, $B$, and $C$ are drawn such that their respective radii $r_A$, $r_B$, and $r_C$ form an arithmetic sequence with $r_A<r_B<r_C$. If the shortest distance between circles $A$ and $B$ is $3.5$, and the shortest distance between circles $A$ and $C$ is $3$, then what is the area of the shaded region? Express your answer in terms of pi. [asy] size(8cm); draw((0,0)--(6,0)--6*dir(60)--cycle); draw(circle((0,0),1)); draw(circle(6*dir(60),1.5)); draw(circle((6,0),2)); filldraw((0,0)--arc((0,0),1,0,60)--cycle, grey); filldraw(6*dir(60)--arc(6*dir(60),1.5,240,300)--cycle, grey); filldraw((6,0)--arc((6,0),2,120,180)--cycle, grey); label("$A$",(0,0),SW); label("$B$",6*dir(60),N); label("$C$",(6,0),SE); [/asy]

Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline? $ \textbf{(A) }10\qquad\textbf{(B) }16\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40 $