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Problem of the Day #134: Pasture Picking Parade July 31, 2011

Posted by Saketh in : potd , add a comment

Farmer John has $10$ cows on his farm, conveniently labeled from $1$ to $10$. After being milked each morning, the cows are led single-file in order along a path that passes each of $10$ pastures exactly once.

Before leaving the barn, each cow decides which pasture she wishes to graze in. When she reaches that pasture, she will try to occupy it for the day. If she finds that her desired pasture is already occupied, she will take the next available pasture she encounters (if there is one).

Determine the number of distinct mappings of cows to desired pastures for which all of the cows will find a place to graze.

Problem of the Day #133: The Three Sophomores July 30, 2011

Posted by Seungln in : potd , 3 comments

Victoria, Veronica and Ivy are playing a number game.

Victoria is thinking of a number that is congruent to $307 \pmod{821}$. Veronica is thinking of a number that is congruent to $283 \pmod{797}$. Ivy is thinking of a number that is congruent to $383 \pmod{877}$. They tell each other what their number is, and, to their surprise, they find out that they all thought of the same number.

If the number they thought of is a positive integer less than $100000000000$, what is the greatest number for which this could happen?

Problem of the Day #132: Albert’s Lasers July 29, 2011

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Albert has just finished meticulously setting up a row of $100$ lasers for an advanced optics project, all of which are initially pointed upwards. While Albert is taking a break from his many hours of work, Sreenath stumbles upon the lasers and, not knowing why they are there, decides to play with them. For each laser, Sreenath randomly chooses to turn it $60^\circ$ to the right, $30^\circ$ to the right, $30^\circ$ to the left, $60^\circ$ to the left, or not to turn it at all. This causes the beams of some of the lasers to intersect. For example, if Sreenath turns one laser to the right $30^\circ$, it will intersect all other lasers to the right except those already pointing $30^\circ$ or $60^\circ$ to the right. Albert’s Rage Factor when he returns from the break is equal to the number of pairs of lasers that intersect. What is the expected value of Albert’s Rage Factor?

Problem of the Day #131: A Rather Large Sum July 28, 2011

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Evaluate

$$ \Large{ \displaystyle\sum\limits_{x=1}^{2011^2} \;\;\; \displaystyle\sum\limits_{a=1}^{x} \frac{2x}{{x}^{\frac{2a}{x+1}}+x} } $$

Problem of the Day #130: Minimizing acute angles July 27, 2011

Posted by Mitchell in : potd , add a comment

If $A_1, A_2, \cdots, A_{100}$ are points in the plane, no three collinear, what is the minimum possible number of triples $(i, j, k)$ of distinct integers such that $1 \leq i, j, k \leq 100$, $i < k$, and $\angle A_i A_j A_k$ is acute?

Problem of the Day #129: Troubles with AutoCorrect July 26, 2011

Posted by Alex in : potd , 3 comments

Sreenath rewrote Mitchell’s AutoCorrect function! He chose a permutation $\{a_i\}$ of $\{1, 2, 3, 4, 5\}$ such that each $5$-letter word $s_1 s_2 s_3 s_4 s_5$ is replaced with $s_{a_1} s_{a_2} s_{a_3} s_{a_4} s_{a_5}$. Mitchell is saddened because his vocabulary only contains $5$-letter words possible from a $26$-letter alphabet. He then realizes that some words remain unchanged by AutoCorrect. Given that Sreenath chose a permutation $\{a_i\}$ randomly, find the expected value of the total number of words in Mitchell’s vocabulary that are unaffected by AutoCorrect.

You may use a four-function calculator, but one is not necessary.

Problem of the Day #128: qwetyuiosdfhjklzxvbn (Albert’s Revenge) July 25, 2011

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Albert is mad at Sreenath for the number of spam messages that Sreenath sent him. As an act of revenge, Albert decides to get back at him with his mashes. He will, however, use a different set of rules in his spamming.

The top row, again, is defined as the row from ‘$q$’ to ‘]’. The home row is from ‘$a$’ to ‘;’, and the bottom row is from ‘$z$’ to ‘.’ (just like problem #121.) Albert will, however, choose four distinct keys in each row instead of two. Albert will type from the top row first, then go to the home row, then go to the bottom row. For each row, Albert will start from the first key to the left and type the keys until he types the second key, skip to the third key and type the keys until he types the fourth key. (That is, if, in the top row, he chooses ‘$q$’, ‘$e$’, ‘$t$’ and ‘$o$’, he will type ‘$qwetyuio$’.)

How many different mashes can Albert spam Sreenath with, if each mash has to have more than $15$ characters?

Problem of the Day #127: A Silly Number Game July 24, 2011

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Al and Bert are playing a number game in which they take turns modifying a number $n$. Whenever it is Al’s turn, he replaces $n$ with $n – a$, where $n$ and $a$ are relatively prime and $1 < a < n$. Whenever it is Bert's turn, he replaces $n$ with $n - b$, where $n$ and $b$ are not relatively prime and $1 < b < n$. If a player cannot make a move on a given turn, then that player wins. An initial value for $n$ is albert if whoever goes first is guaranteed to win, assuming both players play optimally. How many initial values of $n$ from $1$ to $1,000,000$ are albert?

Problem of the Day #126: Annoying Sums July 23, 2011

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Arjun is locked in jail with paper and a pencil and is told he will be freed if and only if he correctly evaluates the following problem: $$\sum\limits_{k=8}^{\infty} \frac{1}{k(k^2-6k+8)}$$
You are allowed to give him one hint, so once you’ve realized the result is obviously a fraction, you choose to tell him what the sum of the numerator and denominator is…

Problem of the Day #125: Dr. Kim’s Solar Cells July 22, 2011

Posted by Sreenath in : potd , 7 comments

Izzy is standing at one corner of a 20 ft by 20 ft classroom. She begins walking towards the midpoint of one of the opposite walls. When she reaches a wall, she will bounce off in the opposite direction at the same angle at which she hit the wall. If she is ever within 3 feet of one or more solar cells, she will trip towards it, destroying the solar cell(s). Dr. Kim would like to place his solar cells so they will not be destroyed during Izzy’s rampage. The area of the region in which Dr. Kim can safely deploy his solar cells can be expressed in the form $\frac{a-b\sqrt{c}}{d}$, where $a$, $b$, and $d$ are relatively prime positive integers and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$.