An urn contain a number of yellow and green balls. You extract two balls from the urn (without adding them back) and calculate the probability of both balls being green. Can you choose the number of yellow and green balls such that this probability to be $\frac{1}{4}$?

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$? $\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 57 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 59$

Let $X,Y$ be i.i.d $\operatorname{Geom}(p)$. What is the conditional distribution of $X|X+Y=k$? $\textbf{(A)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\}$ $\textbf{(B)}~\operatorname{Uniform}\left\{1,2,\ldots,k\right\}$ $\textbf{(C)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\}$ $\textbf{(D)}~\text{None of the above}$

Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$. [i]S. Csorgp, L. Horvath[/i]

A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is $\textbf{(A) }.05\qquad\textbf{(B) }.065\qquad\textbf{(C) }.08\qquad\textbf{(D) }.09\qquad \textbf{(E) }.1$

(a) Alenka has two jars, each with $6$ marbles labeled with numbers $1$ through $6$. She draws one marble from each jar at random. Denote by $p_n$ the probability that the sum of the labels of the two drawn marbles is $n$. Compute pn for each $n \in N$. (b) Barbara has two jars, each with $6$ marbles which are labeled with unknown numbers. The sets of labels in the two jars may differ and two marbles in the same jar can have the same label. If she draws one marble from each jar at random, the probability that the sum of the labels of the drawn marbles is $n$ equals the probability $p_n$ in Alenka’s case. Determine the labels of the marbles. Find all solution

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$.

The origami club meets once a week at a fixed time, but this week, the club had to reschedule the meeting to a different time during the same day. However, the room that they usually meet has $5$ available time slots, one of which is the original time the origami club meets. If at any given time slot, there is a $30$ percent chance the room is not available, what is the probability the origami club will be able to meet at that day?

Aaron has a coin that is slightly unbalanced. The odds of getting heads are $60\%$. What are the odds that if he flips it endlessly, at some point during his flipping he has a total of three more tails than heads?

The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. [asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label("A", (6,19), SE); label("B", (23,4), NW); label("C", (23,8), NW); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label("N", (28,13.5), N); label("W", (26,11.5), W); label("E", (30,11.5), E); label("S", (28,9.5), S);[/asy] $\textbf {(A) } \frac{11}{32} \qquad \textbf {(B) } \frac 12 \qquad \textbf {(C) } \frac 47 \qquad \textbf {(D) } \frac{21}{32} \qquad \textbf {(E) } \frac 34$

Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$? [i]2015 CCA Math Bonanza Lightning Round #5.3[/i]

Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is $\frac{1}{K}$, $K\in\mathbb{N}$. Find $K$.

Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$

In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

Tamika selects two different numbers at random from the set $ \{ 8,9,10\} $ and adds them. Carlos takes two different numbers at random from the set $ \{ 3,5,6\} $ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? $ \text{(A)}\ \frac{4}{9}\qquad\text{(B)}\ \frac{5}{9}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{1}{3}\qquad\text{(E)}\ \frac{2}{3} $

A random selector can only select one of the nine integers $ 1,2,\ldots,9$, and it makes these selections with equal probability. Determine the probability that after $ n$ selections ($ n>1$), the product of the $ n$ numbers selected will be divisible by 10.

For an even positive integer $n$ Kevin has a tape of length $4n$ with marks at $-2n,-2n+1,\ldots,2n-1,2n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2,\ldots,n-1,n$ and places a stone on each of these points. We call a stone 'stuck' if it is on $2n$ or $-2n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner: [list] [*]He picks an unstuck stone uniformly at random and then flips a fair coin. [*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. [*]He repeats until all the stones are stuck.[/list] Let $p_n$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate \[\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}\] in terms of $n$.

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]

Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select? $\textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{5}{36} \qquad\textbf{(C)}\ \frac{14}{45} \qquad\textbf{(D)}\ \frac{25}{63} \qquad\textbf{(E)}\ \frac{1}{2}$

[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable $\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$ tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]

Select a random real number $m$ from the interval $(\frac{1}{6}, 1)$. A track is in the shape of an equilateral triangle of side length $50$ feet. Ch, Hm, and Mc are all initially standing at one of the vertices of the track. At the time $t = 0$, the three people simultaneously begin walking around the track in clockwise direction. Ch, Hm, and Mc walk at constant rates of $2, 3$, and $4$ feet per second, respectively. Let $T$ be the set of all positive real numbers $t_0$ satisfying the following criterion: [i]If we choose a random number $t_1$ from the interval $[0, t_0]$, the probability that the three people are on the same side of the track at the time $t = t_1$ is precisely $m$.[/i] The probability that $|T| = 17$ (i.e., $T$ has precisely $17$ elements) equals $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Alex, Pei-Hsin, and Edward got together before the contest to send a mailing to all the invited schools. Pei-Hsin usually just stuffs the envelopes, but if Alex leaves the room she has to lick them as well and has a $25\%$ chance of dying from an allergic reaction before he gets back. Licking the glue makes Edward a bit psychotic, so if Alex leaves the room there is a $20\%$ chance that Edward will kill Pei-Hsin before she can start licking envelopes. Alex leaves the room and comes back to find Pei-Hsin dead. What is the probability that Edward was responsible?

Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.

For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.