Found problems: 178
2011 Purple Comet Problems, 29
Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.
2022 JHMT HS, 9
Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.
2022 IMC, 8
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be
chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the
convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices
of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$.
2013 Stanford Mathematics Tournament, 5
An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?
2006 Taiwan National Olympiad, 1
There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe is opened, the key inside the safe can be used to open another safe.)
2007 India IMO Training Camp, 3
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$
(Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)
2014 Harvard-MIT Mathematics Tournament, 2
[4] Let $x_1,x_2,\ldots,x_{100}$ be defined so that for each $i$, $x_i$ is a (uniformly) random integer between $1$ and $6$ inclusive. Find the expected number of integers in the set $\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}$ that are multiples of $6$.
2014 Putnam, 4
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
2009 Harvard-MIT Mathematics Tournament, 5
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?
2009 Princeton University Math Competition, 7
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
2019 PUMaC Combinatorics B, 5
Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?
1976 AMC 12/AHSME, 12
A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }24\qquad \textbf{(E) }25$
2014 Math Prize For Girls Problems, 19
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?
ICMC 7, 3
Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]
1974 Putnam, A4
An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?
2013 Stanford Mathematics Tournament, 9
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 $(x, y)$.
2021-2022 OMMC, 8
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$.
[i]Proposed by Isaac Chen[/i]
2012 Hitotsubashi University Entrance Examination, 5
At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7.
(1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition.
(2) Find the expected value of the number of the spot on the top face after the $n$-repetition.
2013 NIMO Problems, 3
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]
2009 USAMTS Problems, 3
I give you a deck of $n$ cards numbered $1$ through $n$. On each turn, you take the top card of the deck and place it anywhere you choose in the deck. You must arrange the cards in numerical order, with card $1$ on top and card $n$ on the bottom. If I place the deck in a random order before giving it to you, and you know the initial order of the cards, what is the expected value of the minimum number of turns you need to arrange the deck in order?
1994 Poland - First Round, 11
Given are natural numbers $n>m>1$. We draw $m$ numbers from the set $\{1,2,...,n\}$ one by one without putting the drawn numbers back. Find the expected value of the difference between the largest and the smallest of the drawn numbers.
2018 Harvard-MIT Mathematics Tournament, 7
Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$
2022 JHMT HS, 7
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
2013 Online Math Open Problems, 29
Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[i]Proposed by Ray Li[/i]
2019 Harvard-MIT Mathematics Tournament, 4
Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is an $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?