Let $X_1,X_2,X_3$ be $\exp(1)$. Find the conditional distribution of $X_1|X_1+X_2+X_3=k$. $\textbf{(A)}~\operatorname{Uniform}(0,k)$ $\textbf{(B)}~\operatorname{Uniform}\left(0,\frac k3\right)$ $\textbf{(C)}~\operatorname{Uniform}\left(0,\frac{2k}3\right)$ $\textbf{(D)}~\text{None of the above}$

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

An experiment consists of performing $n$ independent tests. The $i$-th test is successful with the probability equal to $p_i$. Let $r_k$ be the probability that exactly $k$ tests succeed. Prove that $$\sum_{i=1}^n p_i =\sum_{k=0}^n kr_k.$$

A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$. [i]Proposed by Ahaan Rungta[/i]

Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? $\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}$ [asy] unitsize(36); draw(circle((-3,0),1)); draw(circle((0,0),1)); draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150)); draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707)); fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black); fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black); label("$5$",(-3,.2),N); label("$3$",(-3.2,0),W); label("$4$",(-3,-.2),S); label("$8$",(-2.8,0),E); label("$6$",(0,.2),N); label("$9$",(-.2,.1),SW); label("$7$",(.2,.1),SE); [/asy]

A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? $\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$

Let $C$ be the unit circle $x^{2}+y^{2}=1 .$ A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diagonal $p q .$ What is the probability that no point of $R$ lies outside of $C ?$

Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $ 4$ replaced by $ 3$ and the other die has the $ 3$ replaced by $ 4$. When these dice are rolled, what is the probability that the sum is an odd number? $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{4}{9}\qquad \textbf{(C)}\ \frac{1}{2}\qquad \textbf{(D)}\ \frac{5}{9}\qquad \textbf{(E)}\ \frac{11}{18}$

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$. [i]Linus Hamilton[/i]

Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing $k$ coins and calculating the value of her handful. After a while, she begins to notice that if $k$ is even, she more often gets even sums, and if $k$ is odd, she more often gets odd sums. Help her prove this true! Given $k$ coins chosen uniformly and at random, prove that. the probability that the parity of $k$ is the same as the parity of the $k$ coins' value is greater than the probability that the parities are different.

Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people. P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$. [hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide] P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer $n$. ii. The first term of the sequence $(U_1)$ is $9n$. iii. For $k \geq 2$, $U_k = U_{k-1} - 17$. Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$. As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$. P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$. [url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed. P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.

Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?

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]

Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

John has a standard four-sided die. Each roll, he gains points equal to the value of the roll multiplied by the number of times he has now rolled that number; for example, if his first rolls were $3,3,2,3$, he would have $3+6+2+9=20$ points. Find the expected number of points John will have after rolling the die 25 times.

Teams $\text{A}$ and $\text{B}$ are playing a series of games. If the odds for either to win any game are even and Team $\text{A}$ must win two or Team $\text{B}$ three games to win the series, then the odds favoring Team $\text{A}$ to win the series are $\textbf{(A) }11\text{ to }5\qquad\textbf{(B) }5\text{ to }2\qquad\textbf{(C) }8\text{ to }3\qquad\textbf{(D) }3\text{ to }2\qquad \textbf{(E) }13\text{ to }6$

A base-$ 10$ three-digit number $ n$ is selected at random. Which of the following is closest to the probability that the base-$ 9$ representation and the base-$ 11$ representation of $ n$ are both three-digit numerals? $ \textbf{(A)}\ 0.3 \qquad \textbf{(B)}\ 0.4 \qquad \textbf{(C)}\ 0.5 \qquad \textbf{(D)}\ 0.6 \qquad \textbf{(E)}\ 0.7$

During the regular season, Washington Redskins achieve a record of $10$ wins and $6$ losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record $LLWWWWWLWWLWWWLL$ contains three winning streaks, while $WWWWWWWLLLLLLWWW$ has just two.)

If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? $\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

Investigate the behaviour at $t\to\infty$ of the solution of the problem \[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]