Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.

When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Bhairav the Bat lives next to a town where $12.5$% of the inhabitants have Type AB blood. When Bhairav the Bat leaves his cave at night to suck of the inhabitants blood, chooses individuals at random until he bites one with type AB blood, after which he stops. What is the expected value of the number of individuals Bhairav the Bat will bite in any given night? [i]2015 CCA Math Bonanza Lightning Round #3.1[/i]

In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Consider an unbiased coin which is tossed infinitely many times. Let $A_n$ be the event that no two successive heads occur in the first $n$ tosses of this experiment. Then which of the following is incorrect : (A) $\lim_{n \to \infty} P(A_n)=0$ (B) $\lim_{n \to \infty}3^n P(A_n)=0$ (C) $2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}$ (D) $\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}$ is lesser than $1.2$

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.

Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? $\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

Yoknapatawpha County has $500,000$ families. Each family is expected to continue to have children until it has a girl, at which point each family stops having children. If the probability of having a boy is $50\%$, and no families have either fertility problems or multiple children per birthing, how many families are expected to have at least $5$ children?

Let $k\ge t\ge 2$ positive integers. For integers $n\ge k$ let $p_n$ be the probability that if we choose $k$ from the first $n$ positive integers randomly, any $t$ of the $k$ chosen integers have greatest common divisor $1$. Let qn be the probability that if we choose $k-t+1$ from the first $n$ positive integers the product is not divisible by a perfect $t^{th}$ power that is greater then $1$. Prove that sequences $p_n$ and $q_n$ converge to the same value.

In a certain family, a husband and wife made the following agreement: If the wife washes the dishes one day, the husband washes the dishes the next day. However, if the husband washes the dishes one day, then who washes the dishes the next day is decided by drawing a coin. Let $ p_n $ denote the probability of the event that the husband washes the dishes on the $ n $-th day of the contract. Prove that there is a limit $ \lim_{n\to \infty} p_n $ and calculate it. We assume $ p_1 = \frac{1}{2} $.

Find the probability that any given row in Pascal’s Triangle contains a perfect square. [i] (.1 point)[/i]

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$? [i]Author: Ray Li[/i]

Teams $\text{A}$ and $\text{B}$ are playing a series of games. If the odds for either to win any game are even and Team $\text{A}$ must win two or Team $\text{B}$ three games to win the series, then the odds favoring Team $\text{A}$ to win the series are $\textbf{(A) }11\text{ to }5\qquad\textbf{(B) }5\text{ to }2\qquad\textbf{(C) }8\text{ to }3\qquad\textbf{(D) }3\text{ to }2\qquad \textbf{(E) }13\text{ to }6$

Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$. [i]Proposed by Lewis Chen[/i]

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

On the trip home from the meeting where this AMC$ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 24$

Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.