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Problem of the Day #398: Sine Sum April 20, 2012

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Suppose that $M$ is the centroid of triangle $ABC$. Suppose further that the circumcircle of triangle $AMC$ is tangent to line $AB$. Determine the maximum possible value of $$\sin\angle CAM + \sin\angle CBM$$

Problem of the Day #397: Functional Fixedness April 19, 2012

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A function has the property that $f(x) = f(x^2 + 2012)$ for all real $x$. Determine the maximum possible value of $f(a)-f(b)$ for real $a$ and $b$.

Problem of the Day #396: Telling Time April 18, 2012

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Albert glances at the clock and notices that the shorter angle between the hour hand and $12$ is twice the shorter angle between the minute hand and $12$. What time(s) could it be? Make your answer(s) exact.

Problem of the Day #395: Dessert April 17, 2012

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Alex just had a sumptuous dinner at his favorite restaurant. He now has $n$ different choices for dessert.

With so many good options, he decides to pick randomly using a coin. However, he wants to ensure that each dessert has an equal probability of being picked.

Devise a strategy for choosing dessert that minimizes the expected number of flips needed, and express the expected number in terms of $n$.

Bonus: Now solve the same problem in the case that Alex’s coin is weighted to land heads more often than tails.

Problem of the Day #394: Permutations with Shared Elements April 16, 2012

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Find the probability that two random permutations of the integers from $1$ to $N$ share exactly one element.

Problem of the Day #393: A Number Game April 15, 2012

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Scott is playing with numbers. To begin, he selects any four-digit number (leading zeros are allowed).

Each turn, he sorts the digits of his current number in both ascending and descending order to produce two other values. He then subtracts the smaller of these from the larger, and keeps the result as his new number.

Eventually, Scott’s routine reaches the fixed point $6174$. We call it a fixed point because $7641 – 1467$ simply produces $6174$ again. How many possible values are there for his starting number?

Problem of the Day #392: Brandon the Bee April 14, 2012

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Brandon the Bee is keeping watch over his hive, which is laid out in a hexagonal grid. Being a very busy bee, he has some very strict guidelines that he follows whenever he takes a tour of the grid:

  1. Brandon never steps outside the boundaries of the hive.
  2. He can only travel between adjacent cells.
  3. He must visit each cell exactly once.
  4. Cells are grouped in clusters, and he must enter and exit each one exactly once.
  5. Some cell borders are blocked off by dried honey. He cannot cross these.
  6. Brandon must start at the specified starting position.

Consider, for example, the hive shown below. Note that there are no blocked borders in this example.

beehive

Devise an efficient algorithm for finding a valid route for Brandon to take.

Problem of the Day #391: Cyclic Triplets April 13, 2012

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Consider the function $f(x) = \frac{100x^2}{1+100x^2}$. Find all ordered triples of real numbers $(a,b,c)$ such that $f(a) = b$, $f(b) = c$, and $f(c) = a$.

Problem of the Day #390: Pawn-Only Chess Game April 12, 2012

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Suppose you are playing a game of chess where only pawn moves are allowed. Find the number of possible positions after 3 moves from each player.

Problem of the Day #389: Blue or Red April 11, 2012

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Billy places $10$ marbles in a jar, $5$ blue and $5$ red. When he isn’t looking, Arjun comes along and takes one of the marbles.

Billy comes back and draws some number of marbles from the jar, with replacement. He observes that the number of times he drew a blue marble is $5$ greater than the number of times he drew a red marble.

What is the probability that the marble Arjun took was blue?