Found problems: 1111
2016 PUMaC Combinatorics A, 4
A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?
2005 AMC 10, 21
Forty slips are placed into a hat, each bearing a number $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, $ 8$, $ 9$, or $ 10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $ p$ be the probability that all four slips bear the same number. Let $ q$ be the probability that two of the slips bear a number $ a$ and the other two bear a number $ b\not\equal{} a$. What is the value of $ q/p$?
$ \textbf{(A)}\ 162\qquad
\textbf{(B)}\ 180\qquad
\textbf{(C)}\ 324\qquad
\textbf{(D)}\ 360\qquad
\textbf{(E)}\ 720$
1963 Miklós Schweitzer, 10
Select $ n$ points on a circle independently with uniform distribution. Let $ P_n$ be the probability that the center of the
circle is in the interior of the convex hull of these $ n$ points. Calculate the probabilities $ P_3$ and $ P_4$. [A. Renyi]
2008 Pre-Preparation Course Examination, 2
Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?
2008 Harvard-MIT Mathematics Tournament, 15
In a game show, Bob is faced with $ 7$ doors, $ 2$ of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize?
2015 AMC 12/AHSME, 9
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$
2014 AMC 8, 11
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$
2001 AIME Problems, 10
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2016 AMC 10, 12
Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?
$\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$
2007 AMC 12/AHSME, 12
Integers $ a,$ $ b,$ $ c,$ and $ d,$ not necessarily distinct, are chosen independantly and at random from $ 0$ to $ 2007,$ inclusive. What is the probability that $ ad \minus{} bc$ is even?
$ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$
2021 Simon Marais Mathematical Competition, B1
Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$. Call the resulting matrix $A$.
Determine the probability that $A^2 = O$, as a function of $n$.
2012 NIMO Problems, 2
A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$.
[i]Proposed by Aaron Lin[/i]
2012 Singapore MO Open, 3
For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.
2011 Pre-Preparation Course Examination, 2
prove that for almost every real number $\alpha \in [0,1]$ there exists natural number $n_{\alpha} \in \mathbb N$ such that the inequality
$|\alpha-\frac{p}{q}|\le \frac{1}{q^n}$
for natural $n\ge n_{\alpha}$ and rational $\frac{p}{q}$ has no answers.
1998 Hungary-Israel Binational, 1
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ flips.
(b) What is the expected number of flips needed to finish the game?
1983 USAMO, 1
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.
2006 AMC 12/AHSME, 25
How many non-empty subsets $ S$ of $ \{1, 2, 3, \ldots, 15\}$ have the following two properties?
(1) No two consecutive integers belong to $ S$.
(2) If $ S$ contains $ k$ elements, then $ S$ contains no number less than $ k$.
$ \textbf{(A) } 277\qquad \textbf{(B) } 311\qquad \textbf{(C) } 376\qquad \textbf{(D) } 377\qquad \textbf{(E) } 405$
2023 AMC 10, 7
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$
2012 Miklós Schweitzer, 7
Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.
1998 Miklós Schweitzer, 10
Let $\xi_1 , \xi_2 , ...$ be a series of independent, zero-expected-value random variables for which $\lim_{n\to\infty} E(\xi_n ^ 2) = 0$, and $S_n = \sum_{j = 1}^n \xi_j$ . Denote by I(A) the indicator function of event A. Prove that
$$\frac{1}{\log n} \sum_{k = 1}^n \frac1k I\bigg(\max_{1\leq j\leq k} |S_j|>\sqrt k\bigg) \to 0$$
with probability 1 if $n\to\infty$ .
2011 AMC 12/AHSME, 10
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{12} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{4} \qquad
\textbf{(E)}\ \frac{5}{18}
$
2018 Brazil Undergrad MO, 8
A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?
2014 Harvard-MIT Mathematics Tournament, 23
Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.
2006 District Olympiad, 3
Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded;
b) Is it possible that the sequence is not convergent?
2018 PUMaC Team Round, 1
Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.