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Problem of the Day #337: Ants on a Triangle February 19, 2012

Posted by Alex in : potd , add a comment

$\triangle ABC$ is a right triangle with $\angle C$ as the right angle. One ant starts at point $A$, and another ant starts at point $B$. The ants travel the vertices in the order $A \rightarrow B \rightarrow C \rightarrow A$ in an endless cycle. The midpoint of the line segment connecting the two ants forms a closed region $R$. Find the maximum possible value of $\frac{[R]}{[\triangle ABC]}$.

Problem of the Day #336: Boating across the River February 18, 2012

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$999$ people, numbered $2$ to $1000$, are all on one side of a river and wish to reach the other side. There is a single boat with infinite capacity, but a group of people can only ride the boat together if every pair of people on the boat have numbers that are relatively prime. How many back-and-forth trips are needed to transport every person across the river?

You may use a computer program.

Problem of the Day #335: Zeroes, Ones, and Twos February 17, 2012

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For how many sets $\{a_k\}_{k=0}^\infty$ of integers $0$, $1$, and $2$, does:

$$\sum\limits_{k=0}^\infty a_k 2^k$$

equal $25$? $100$? $2^n – 1$ for some positive integer $n$?

Problem of the Day #334: Philogs February 16, 2012

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Express in terms of $n$:

$$\sum\limits_{k=1}^n \log\left(\phi(k)\right) \;\;\;\;\; – \; \sum\limits_{p \, \in \, primes} \log\left( \left(1 – \frac{1}{p} \right)^{\lfloor{\frac{n}{p}}\rfloor} \right)$$

Problem of the Day #333: Squares within Squares February 15, 2012

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Square $ABCD$ has side length $1$. Albert chooses points $E$, $F$, $G$, and $H$ on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DA}$ respectively, such that each point divides its segment into two parts, with lengths given by the ratio $2:3$. Points $E$, $F$, $G$, and $H$ are closer to points $A$, $B$, $C$, and $D$ respectively. Find the area of square $EFGH$.

Problem of the Day #332: Valentine’s Day February 14, 2012

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Albert knows $1,000,000$ girls, numbered $1$ to $1,000,000$. He wishes to give a Valentine’s card to each girl whose number cannot be expressed as $n^p$, where $n$ and $p$ are integers, $1 \le n \le 1,000,000$. and $p > 1$. How many cards does Albert need?

Problem of the Day #331: Albert February 13, 2012

Posted by Sreenath in : potd , 1 comment so far

Albert and Billy are playing a game:

At the starting of each round, both players write down a number. They then play rock-paper-scissors $8$ times.

If a player wins at least as many times as the number he wrote, his score increases by the cube of the number written. Else, the score is cut in half.

Albert is exceptionally good at rock-paper-scissors, and wins $\frac{3}{4}$ of the time.

Assuming both players play optimally, compute the probability that Albert has the higher score after $5$ such rounds.

Problem of the Day #330: Incircle and Circumcircle February 12, 2012

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Find the ratio of the circumradius of a regular dodecahedron to its inradius.

Bonus: Find this ratio for any regular polygon with $n$ sides.

Problem of the Day #329: Guessing the Answer February 11, 2012

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Saketh is working on an algorithms problem: given a list of $10,000$ items and their prices, find the maximum number of unique items that can be bought with $10,000$ dollars. The price of each item is randomly generated as a value between $0$ and $10,000$. Saketh, confounded, decides to write a computer program that guesses the same answer each time. What answer should his program guess to maximize his chances of being correct?

Problem of the Day #328: Movie Seats February 10, 2012

Posted by Alex in : potd , 2 comments

Albert and his 8 twin brothers are going to the movies and all wish to sit in the same row. The row contains 20 vacant seats. Albert and his twins want to seat themselves such that each person occupies exactly one seat and at least one of the seats at the end of the row is occupied. How many ways can they do this? Assume that Albert and his twins are indistinguishable.