Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Lily pads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly and independently with probability $\tfrac12$. The probability that the frog visits pad $7$ is $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

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.

Call a permutation $a_1,a_2,\ldots,a_n$ [i]quasi-increasing[/i] if $a_k\le a_{k+1}+2$ for each $1\le k\le n-1$. For example, $54321$ and $14253$ are quasi-increasing permutations of the integers $1,2,3,4,5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1,2,\ldots,7$.

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime? $ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad \textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad \textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad \textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad \textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$

Let $S=\{1,2\dots,n\}$ for some integer $n>1.$ Say a permutation $\pi$ of $S$ has a local maximum at $k\in S$ if \[\begin{array}{ccc}\text{(i)}&\pi(k)>\pi(k+1)&\text{for }k=1\\ \text{(ii)}&\pi(k-1)<\pi(k)\text{ and }\pi(k)>\pi(k+1)&\text{for }1<k<n\\ \text{(iii)}&\pi(k-1)M\pi(k)&\text{for }k=n\end{array}\] (For example, if $n=5$ and $\pi$ takes values at $1,2,3,4,5$ of $2,1,4,5,3,$ then $\pi$ has a local maximum of $2$ as $k=1,$ and a local maximum at $k-4.$) What is the average number of local maxima of a permutation of $S,$ averaging over all permuatations of $S?$

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, find $a+b$

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? $\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64 $

The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? $ \textbf{(A)} \ 8! \qquad \textbf{(B)} \ 2^8 \cdot 8! \qquad \textbf{(C)} \ (8!)^2 \qquad \textbf{(D)} \ \frac {16!}{2^8} \qquad \textbf{(E)} \ 16!$

A salmon in a mountain river must overpass two waterfalls. In every minute, the probability of the salmon to overpass the first waterfall is $p > 0$, and the probability to overpass the second waterfall is $q > 0$. These two events are assumed to be independent. Compute the probability that the salmon did not overpass the first waterfall in $n$ minutes, assuming that it did not overpass both waterfalls in that time.

If two distinct integers from $1$ to $50$ inclusive are chosen at random, what is the expected value of their product? Note: The expectation is defined as the sum of the products of probability and value, i.e., the expected value of a coin flip that gives you $\$10$ if head and $\$5$ if tail is $\tfrac12\times\$10+\tfrac12\times\$5=\$7.5$.

Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?

Each of $n$ boys and $n$ girls chooses a random number from the set $\{ 1, 2, 3, 4, 5 \}$, uniformly and independently. Let $p_n$ be the probability that every boy chooses a different number than every girl. As $n$ approaches infinity, what value does $\sqrt[n]{p_n}$ approach?

Six cities $A, B, C, D, E$, and $F$ are located on the vertices of a regular hexagon in that order. $G$ is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities $B, C, E, F$ and $G$, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is $p$. Determine the probability of the road between cities $A$ and $D$ is undestroyed.

Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $ 4$, and the second number is divided by $ 5$. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(15pt)); draw(Circle(origin,1)); for(int i = 0;i < 6; ++i) { draw(origin--dir(60i+30)); } label("$7$",midpoint(origin--(dir(0))),E); label("$1$",midpoint(origin--(dir(60))),NE); label("$6$",midpoint(origin--(dir(120))),NW); label("$3$",midpoint(origin--(dir(180))),W); label("$9$",midpoint(origin--(dir(240))),SW); label("$2$",midpoint(origin--(dir(300))),SE); draw((2,0)--(3.5,0)--(3.5,1)--(2,1)--cycle); draw((2,0)--(3.5,0)--(3.5,-1)--(2,-1)--cycle); pair[] V = {(2.5,0.5),(2,0),(3,0),(2.5,-0.5),(2,-1),(3,-1)}; for(int i = 0; i <= 5; ++i) { pair A = V[i]; path p = A--(A.x,A.y + 0.5)--(A.x + 0.5,A.y + 0.5)--(A.x + 0.5, A.y)--cycle; fill(p,mediumgray); draw(p); } path pointer = (-2.5,-0.125)--(-2.5,0.125)--(-1.2,0.125)--(-1.05,0)--(-1.2,-0.125)--cycle; fill(pointer,mediumgray); draw(pointer); label("Pointer",(-1.85,0),fontsize(10pt)); label("$4$",(2,0.5),2N + 2W); label("$3$",(2,0),2N + 2W); label("$2$",(2,-0.5),2N + 2W); label("$1$",(2,-1),2N + 2W); label("$1$",(2,-1),2S + 2E); label("$2$",(2.5,-1),2S + 2E); label("$3$",(3,-1),2S + 2E);[/asy]$ \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 {2}{3}$

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 $C ?$

Jorn wants to cheat at the role play: he intends to cheat the sides to re-label its two octahedra, so that each of the numbers from $1$ to $16$ has the same probability as the sum of the dice occurs. So that the game master does not notice this so easily, he only wants to use numbers from $0$ to $8$ , if necessary several times or not at all. Is this possible?

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 $\lfloor 1000a\rfloor$.