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 $a+b$.

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?

Consider the randomly generated base 10 real number $r = 0.\overline{p_0p_1p_2\ldots}$, where each $p_i$ is a digit from $0$ to $9$, inclusive, generated as follows: $p_0$ is generated uniformly at random from $0$ to $9$, inclusive, and for all $i \geq 0$, $p_{i + 1}$ is generated uniformly at random from $p_i$ to $9$, inclusive. Compute the expected value of $r$.

Let $N=10^{2024}$. $S$ is a square in the Cartesian plane with side length $N$ and the sides parallel to the coordinate axes. Inside there are $N$ points $P_1$, $P_2$, $\dots$, $P_N$ all of which have different $x$ coordinates, and the absolute value of the slope of any connected line between these points is at most $1$. Prove that there exists a line $l$ such that at least $2024$ of these points is at most distance $1$ away from $l$.

Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars. (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents? [i]Proposed by Lewis Chen[/i]

Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$. [i]Proposed by Aaron Lin[/i]

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

Let $R$ be the rectangle in the coordinate plane with corners $(0, 0)$, $(20, 0)$, $(20, 22)$, and $(0, 22)$, and partition $R$ into a $20\times 22$ grid of unit squares. For a given line in the coordinate plane, let its [i]pixelation[/i] be the set of grid squares in $R$ that contain part of the line in their interior. If $P$ is a point chosen uniformly at random in $R$, then compute the expected number of sets of grid squares that are pixelations of some line through $P$.

A gambler has $\$25$ and each turn, if the gambler has a positive amount of money, a fair coin is flipped. If it is heads, the gambler gains a dollar and if it is tails, the gambler loses a dollar. But, if the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expected number of turns for the gambler to double his money?

Each of the $9$ cells in a $3\times 3$ grid is colored either blue or white with equal probability. The expected value of the area of the largest square of blue cells contained within the grid is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ S$ be the set of $25$ points $(x, y)$ with $0\le x, y \le 4$. A triangle whose three vertices are in $S$ is chosen at random. What is the expected value of the square of its area?

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

Charles is playing a variant of Sudoku. To each lattice point $(x, y)$ where $1\le x,y <100$, he assigns an integer between $1$ and $100$ inclusive. These integers satisfy the property that in any row where $y=k$, the $99$ values are distinct and never equal to $k$; similarly for any column where $x=k$. Now, Charles randomly selects one of his lattice points with probability proportional to the integer value he assigned to it. Compute the expected value of $x+y$ for the chosen point $(x, y)$.

Xavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences. [i]Author: Alex Zhu[/i] [hide="Clarification"]An increasing contiguous subsequence is an increasing subsequence all of whose terms are adjacent in the original sequence. For example, 1,3,4,5,2 has two maximal increasing contiguous subsequences: (1,3,4,5) and (2). [/hide]

Calamitous Clod deceives the math beasts by changing a clock at Beast Academy. First, he removes both the minute and hour hands, then places each of them back in a random position, chosen uniformly along the circle. Professor Grok notices that the clock is not displaying a valid time. That is, the hour and minute hands are pointing in an orientation that a real clock would never display. One such example is the hour hand pointed at $6$ and the minute hand pointed at $3$. [center] [asy] import olympiad; size(4cm); defaultpen(fontsize(8pt)); draw(circle(origin, 4)); dot(origin); for(int i = 1; i <= 12; ++i){ label("$"+string(i)+"$", (3.6*sin(i * pi/6), 3.6*cos(i * pi/6))); } draw(origin -- (3.2, 0), EndArrow(5)); draw(origin -- (0, -2.2), EndArrow(5)); [/asy] [/center] The math beasts can fix this, though. They can turn both hands by the same number of degrees clockwise. On average, what is the minimal number of degrees they must turn the hands so that they display a valid time?

Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -. Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of $E$. (Note: Negative numbers are permitted, so 13-22 gives $E = -9$. Any excess operators are parsed as signs, so -2-+3 gives $E=-5$ and -+-31 gives $E = 31$. Trailing operators are discarded, so 2++-+ gives $E=2$. A string consisting only of operators, such as -++-+, gives $E=0$.) [i]Proposed by Lewis Chen[/i]

A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?

Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.

A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i] Proposed by David Altizio [/i]

Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? [i]2016 CCA Math Bonanza Lightning #4.4[/i]

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?