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Problem of the Day #316: Möbius $\mu$ Sums January 29, 2012

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Let $\mu$ be the Möbius function. Find the smallest value of $n \geq 1$ such that:

$$\sum\limits_{i=1}^n \; i \cdot \mu(i) = 0$$

Can you find the next smallest value of $n$?

Problem of the Day #315: Divisibility into $99 \ldots 99900 \ldots 000$ January 28, 2012

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Thanks to my dad for this problem.

Prove (or find a counterexample) that given a positive integer $n$, $\exists \quad i,j: \quad n \mid (10^i – 1)(10^j)$.

Problem of the Day #314: $\pi$ January 27, 2012

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Saketh graphs $f(x) = \frac{1}{1+x^2}$ for $x \in [0,1]$. He cuts out the square $0 \leq x \leq 1$, $0 \leq y \leq 1$, then cuts along the line $f(x)$, producing piece 1 (below $f(x)$) and piece 2 (above $f(x)$). He finds that the ratio of the weights of piece 1 to piece 2 is $R$. Given this, help him find an expression for $\pi$.

Problem of the Day #302: Optimal Path January 15, 2012

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Alex is trying to run to Saketh as fast as he can in Coordinatesville. Alex starts off at $(0 \text{m}, 0 \text{m})$ and runs to Saketh at $(10 \text{m} , 10 \text{m})$. Given that Coordinatesville gets increasingly muddy for increasing $x$, such that Alex’s speed $v$ (in meters per second) is given by $v(x) = e^{x^2} \frac{\text{m}}{\text{s}}$, find the minimum amount of time it takes for Alex to reach Saketh.

Problem of the Day #266: Odd Products in a Line December 10, 2011

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For how many permutations $P(\{1,2,\ldots,n\}) = \{a_1,a_2,\ldots,a_n\}$ is the following expression odd, in terms of $n$?

$$(a_1 – 1)(a_2 – 2) \cdots (a_n – n)$$

Problem of the Day #262: $2011$ is a Cool Number December 6, 2011

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Evaluate the following, for all $305$ primes $p \leq 2011$:

$$\prod\limits_{p} \left(\left((p-1)! \pmod{p} \right) – p \right)$$

Problem of the Day #261: “Exponentorial” December 5, 2011

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Let the exponentorial of a number $n$ be defined as:

$$ɇ(n) = \left( \left( \cdots \left( \left( \left( n \right)^{n-1} \right)^{n-2} \right)^{\cdots} \right)^2 \right)^1$$

The number $ɇ(2011)$ ends in $m$ zeroes followed by a single $1$. Find $m$.

Problem of the Day #260: Cyclic Diamonds December 4, 2011

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A circle of radius $1$ has diameter $\overline{AB}$ and chord $\overline{CD}$ perpendicular to $\overline{AB}$, intersecting $\overline{AB}$ at point $E$. If the maximum possible area of quadrilateral $ABCD$ is $M$, what is the smallest length of $AE$ so that the area of $ABCD = \frac{M}{n}$? Express your answer in terms of $n$.

Also, prove that for integer $n > 1$, length $AE$ is an irrational number.

Problem of the Day #254: Reactive Vials November 28, 2011

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A mischievous joker places $6$ vials of uranium and $8$ vials of heavy water randomly in a line. He knows that if any three vials of uranium are next to each other, they will react dangerously. What is the chance that the system reacts?

Problem of the Day #244: Sequential Shell Building November 18, 2011

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Arjun is constructing a shell pattern for art as seen below, for some monotonically increasing sequence $\{a_k\}$ with $a_0 = 1$.

The right angles for all triangles are fixed at point $P$. The addition of the last triangle (with leg lengths of $a_{n+3}$ and $a_{n+4}$, $n \ge 1$) increases the shape’s area by a factor of:

$$\frac{8 a_n^2 + 10 a_n a_{n-1} + 3 a_{n-1}^2}{3 a_n^2 + 4 a_n a_{n-1} + a_{n-1}^2}$$

Find $a_{15}$.