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Problem of the Day #195: Sums of Cubes September 30, 2011

Posted by Alex in : potd , add a comment

Find the smallest $n$ for which $1^3 + 2^3 + 3^3 + \cdots + (n – 1)^3 + n^3$ is a multiple of $195$.

Problem of the Day #194: Running Fast September 29, 2011

Posted by Arjun in : potd , add a comment

Alex is running very fast on a circular track. The track has an inner radius of $4$ meters and an outer radius of $8$ meters, and allows running on painted lanes at $1$ meter intervals. Alex runs at $3 \pi$ m/s. Alex wants to run around the track as fast as possible, and decides to run as far inside as possible. However, some slow runners begin running at the same time as Alex at $\pi$ m/s, and Alex (being a very considerate fellow) does not want to end up within $\pi$ meters of them on the same lane, and will slow down to $\pi$ m/s running speed if he is at that distance. Alex can switch lanes at any time. What is the fastest possible time that Alex can complete $10$ laps around the track?

Problem of the Day #193: Running Laps September 28, 2011

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Albert can run a lap on the track in $120$ seconds. Sreenath can run a lap on the track in $100$ seconds. The two start running laps at the same time from the same place. Define Sreenath’s domination factor to be the ratio of the number of times that Sreenath has lapped Albert to the total number of seconds elapsed. Find the smallest number of seconds after which Sreenath’s domination factor is maximized.

Problem of the Day #192: Describe the Triangle September 27, 2011

Posted by Saketh in : potd , add a comment

Line segment $DE$ is drawn parallel to side $AC$ of equilateral triangle $ABC$, intersecting $AB$ at $D$ and $BC$ at $E$. Point $Z$ is the centroid of triangle $BDE$, and $Y$ is the midpoint of $AE$. Determine the measures of the angles of triangle $CYZ$.

Problem of the Day #191: Modulo 3 September 26, 2011

Posted by Saketh in : potd , add a comment

Determine all positive integers $n$ such that $$n \cdot 2^{n} \equiv 1 \pmod{3}$$

Problem of the Day #190: Least Blocking Set September 25, 2011

Posted by Saketh in : potd , add a comment

Consider an $n$ by $m$ grid of unit squares. What is the least number of cells you must cut out to prevent placement of the following piece anywhere on the board?

Problem of the Day #189: Fibonacci Doubling September 24, 2011

Posted by Albert in : potd , add a comment

Find the value of:

$$\sum\limits_{n=1}^{10^{12}} F_{2n} \pmod{1,000,003}$$

Where $F_n$ is the $n^{\text{th}}$ Fibonacci number, defined as:

$$
\begin{eqnarray}
F_0 & = & 0 \\
F_1 & = & 1 \\
F_n & = & F_{n-1} + F_{n-2}
\end{eqnarray}
$$

You may use a computational resource.

Problem of the Day #188: Complex Recurrence September 23, 2011

Posted by Albert in : potd , add a comment

There are infinitely many complex values $z$ for which $z^z = z$. Find the exact value of any of them (must have a non-zero imaginary part).

Problem of the Day #187: Random Trees and Rain September 22, 2011

Posted by Alex in : potd , add a comment

The Homecoming football game will be canceled due to rain, says the prescient Sreenath. Due to the rain, a certain tree will grow very tall. The tree is initially a single branch of length $7$. Each second, each branch less than $1$ second old will, with probability $\frac{2}{3}$, grow a random number of branches between $0$ and $7$, each with a random length from $0$ to the previous branch’s length. Find the expected height of the tree.

Note: each branch of the tree grows straight up.

Problem of the Day #186: Squares September 21, 2011

Posted by Seungln in : potd , add a comment

$365$ is an interesting number because it can be expressed as a sum of three consecutive squares ($10^2 + 11^2 + 12^2$) AND as a sum of two consecutive squares ($13^2 + 14^2$). What is the next biggest integer for which this property can happen?