Found problems: 1111
2004 239 Open Mathematical Olympiad, 2
Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color.
[b]
proposed by S. Berlov[/b]
2018 Harvard-MIT Mathematics Tournament, 1
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
2011 AIME Problems, 12
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 AMC 12/AHSME, 20
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
(I forgot answer choices)
Kvant 2019, M2557
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]
1994 All-Russian Olympiad, 4
Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.
2007 Princeton University Math Competition, 5
Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $.0001$ of the correct answer will be accepted.
1973 Polish MO Finals, 2
Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.
2004 AMC 8, 22
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men?
$\textbf{(A)}\ \frac{1}{3}\qquad
\textbf{(B)}\ \frac{3}{8}\qquad
\textbf{(C)}\ \frac{2}{5}\qquad
\textbf{(D)}\ \frac{5}{12}\qquad
\textbf{(E)}\ \frac{3}{5}$
1989 Spain Mathematical Olympiad, 1
An exam at a university consists of one question randomly selected from the$ n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$. What is the largest lower bound of the probabilities $p_n$?
2001 AMC 12/AHSME, 17
A point $ P$ is selected at random from the interior of the pentagon with vertices $ A \equal{} (0,2)$, $B \equal{} (4,0)$, $C \equal{} (2 \pi \plus{} 1, 0)$, $D \equal{} (2 \pi \plus{} 1,4)$, and $ E \equal{} (0,4)$. What is the probability that $ \angle APB$ is obtuse?
[asy]
size(150);
pair A, B, C, D, E;
A = (0,1.5);
B = (3,0);
C = (2 *pi + 1, 0);
D = (2 * pi + 1,4);
E = (0,4);
draw(A--B--C--D--E--cycle);
label("$A$", A, dir(180));
label("$B$", B, dir(270));
label("$C$", C, dir(0));
label("$D$", D, dir(0));
label("$E$", E, dir(180));
[/asy]
$ \displaystyle \textbf{(A)} \ \frac {1}{5} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)} \ \frac {5}{16} \qquad \textbf{(D)} \ \frac {3}{8} \qquad \textbf{(E)} \ \frac {1}{2}$
2010 ELMO Shortlist, 8
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]
2006 AIME Problems, 5
When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2016 AMC 8, 21
A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
$\textbf{(A) }\frac{3}{10}\qquad\textbf{(B) }\frac{2}{5}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{3}{5}\qquad \textbf{(E) }\frac{7}{10}$
2020 BMT Fall, 13
Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.
2022 JHMT HS, 5
A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.
2006 AMC 8, 17
Jeff rotates spinners $ P$, $ Q$ and $ R$ and adds the resulting numbers. What is the probability that his sum is an odd number?
[asy]size(200);
path circle=circle((0,0),2);
path r=(0,0)--(0,2);
draw(circle,linewidth(1));
draw(shift(5,0)*circle,linewidth(1));
draw(shift(10,0)*circle,linewidth(1));
draw(r,linewidth(1));
draw(rotate(120)*r,linewidth(1));
draw(rotate(240)*r,linewidth(1));
draw(shift(5,0)*r,linewidth(1));
draw(shift(5,0)*rotate(90)*r,linewidth(1));
draw(shift(5,0)*rotate(180)*r,linewidth(1));
draw(shift(5,0)*rotate(270)*r,linewidth(1));
draw(shift(10,0)*r,linewidth(1));
draw(shift(10,0)*rotate(60)*r,linewidth(1));
draw(shift(10,0)*rotate(120)*r,linewidth(1));
draw(shift(10,0)*rotate(180)*r,linewidth(1));
draw(shift(10,0)*rotate(240)*r,linewidth(1));
draw(shift(10,0)*rotate(300)*r,linewidth(1));
label("$P$", (-2,2));
label("$Q$", shift(5,0)*(-2,2));
label("$R$", shift(10,0)*(-2,2));
label("$1$", (-1,sqrt(2)/2));
label("$2$", (1,sqrt(2)/2));
label("$3$", (0,-1));
label("$2$", shift(5,0)*(-sqrt(2)/2,sqrt(2)/2));
label("$4$", shift(5,0)*(sqrt(2)/2,sqrt(2)/2));
label("$6$", shift(5,0)*(sqrt(2)/2,-sqrt(2)/2));
label("$8$", shift(5,0)*(-sqrt(2)/2,-sqrt(2)/2));
label("$1$", shift(10,0)*(-0.5,1));
label("$3$", shift(10,0)*(0.5,1));
label("$5$", shift(10,0)*(1,0));
label("$7$", shift(10,0)*(0.5,-1));
label("$9$", shift(10,0)*(-0.5,-1));
label("$11$", shift(10,0)*(-1,0));[/asy]
$ \textbf{(A)}\ \dfrac{1}{4} \qquad
\textbf{(B)}\ \dfrac{1}{3} \qquad
\textbf{(C)}\ \dfrac{1}{2} \qquad
\textbf{(D)}\ \dfrac{2}{3} \qquad
\textbf{(E)}\ \dfrac{3}{4}$
2018 PUMaC Live Round, Estimation 3
Andrew starts with the $2018$-tuple of binary digits $(0,0,\dots,0)$. On each turn, he randomly chooses one index (between $1$ and $2018$) and flips the digit at that index (makes it $1$ if it was a $0$ and vice versa). What is the smallest $k$ such that, after $k$ steps, the expected number of ones in the sequence is greater than $1008?$
You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$
2021 JHMT HS, 6
Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
2012 Stanford Mathematics Tournament, 4
Two different squares are randomly chosen from an $8\times8$ chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.
1988 Polish MO Finals, 2
For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$. What is the expected value of $X(P)$ if each permutation is equally likely?
2005 AMC 10, 18
Team $ A$ and team $ B$ play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team $ B$ wins the second game and team $ A$ wins the series, what is the probability that team $ B$ wins the first game?
$ \textbf{(A)}\ \frac{1}{5}\qquad
\textbf{(B)}\ \frac{1}{4}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{1}{2}\qquad
\textbf{(E)}\ \frac{2}{3}$
KoMaL A Problems 2022/2023, A. 846
Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin.
[i]Proposed by Péter Pál Pach, Budapest[/i]
2005 AIME Problems, 9
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are distinct primes and $a$,$b$, and $c$ are positive integers, find $a+b+c+p+q+r$.
2009 AMC 8, 12
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
[asy]unitsize(30);
draw(unitcircle);
draw((0,0)--(0,-1));
draw((0,0)--(cos(pi/6),sin(pi/6)));
draw((0,0)--(-cos(pi/6),sin(pi/6)));
label("$1$",(0,.5));
label("$3$",((cos(pi/6))/2,(-sin(pi/6))/2));
label("$5$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy]
[asy]unitsize(30);
draw(unitcircle);
draw((0,0)--(0,-1));
draw((0,0)--(cos(pi/6),sin(pi/6)));
draw((0,0)--(-cos(pi/6),sin(pi/6)));
label("$2$",(0,.5));
label("$4$",((cos(pi/6))/2,(-sin(pi/6))/2));
label("$6$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy]
$ \textbf{(A)}\ \frac {1}{2} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {3}{4} \qquad \textbf{(D)}\ \frac {7}{9} \qquad \textbf{(E)}\ \frac {5}{6}$