## Problem of the Day #424: Don’t Brute ForceJanuary 1, 2013

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Determine the number of non-isomorphic connected graphs on 6 vertices.

## Problem of the Day #377: Happy $n_b^\text{th}$ Birthday!March 30, 2012

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Sreenath has foolishly forgotten how old Jon is. He knows Jon is $57$, $73$, or $91$, but can’t remember which. In an attempt to figure it out, Sreenath poses a clever math problem for Jon on his birthday (today). “Happy $n_b^{\text{th}}$ Birthday, Jon!” Sreenath says (where $b$, the base of $n$, is an integer greater than 2). Solve for $n$, given that it is three digits long.

## Problem of the Day #376: Log Entry 01March 29, 2012

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Alex has decided to provide you with the following data:

$$\begin{eqnarray} \ln(2) & \approx & 0.693147181 \\ \ln(3) & \approx & 1.09861229 \\ \ln(5) & \approx & 1.60943791 \\ \ln(7) & \approx & 1.94591015 \end{eqnarray}$$

Your mission, should you choose to accept it, is to find the number of digits in the decimal expansion of $35^{12000}$.

## Problem of the Day #375: Log Entry 00March 28, 2012

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Alex has decided to provide you with the following data:

$$\begin{eqnarray} \ln(2) & \approx & 0.693147181 \\ \ln(3) & \approx & 1.09861229 \\ \ln(5) & \approx & 1.60943791 \\ \ln(7) & \approx & 1.94591015 \end{eqnarray}$$

Your mission, should you choose to accept it, is to find the smallest positive integer $x$ such that $2^{10 x}$ has more than $3x+1$ digits.

## Problem of the Day #340: (ir)Rationality of $\sqrt{2}$February 22, 2012

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Thanks to my dad for this problem.

While no positive integers $M$ and $N$ satisfy $M^2 = 2 \cdot N^2$, determine (with proof) whether there are infinitely many pairs of integers $M$ and $N$ such that $M^2 = 2 \cdot N^2 + 1$.
Thus, as $M, N$ increase, $\frac{M}{N}$ becomes a better approximation for $\sqrt{2}$.

## Problem of the Day #335: Zeroes, Ones, and TwosFebruary 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: PhilogsFebruary 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 #326: Post-AMC WoesFebruary 8, 2012

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Yesterday, Arjun also took the 2012 AMC, with similar rules to the old one (see problem #325), except “no answer” is worth $1.5$ points. This time, he answered $12$ questions with $95\%$ certainty for each problem. What is the optimal number of (whole) guesses Arjun should have made?

It turns out, Arjun did make guesses on that AMC – $3$ guesses with $40\%$ certainty of correctness. What is the probability he got over $100$?

## Problem of the Day #325: AMC WoesFebruary 7, 2012

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Arjun is taking an old AMC, a 25-question contest where a correct answer earns $6$ points, no answer earns $2.5$ points, and an incorrect answer earns $0$ points. Given that Arjun is $90\%$ sure of his $10$ answered questions and that there is $1$ minute left, on how many questions should Arjun guess (chance correct per question = $20\%$) to maximize his probability of getting a score of over $100$? What is this probability?

## Problem of the Day #317: FiboscalJanuary 30, 2012

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Express the following in terms of the Fibonacci function, $F(n)$:

$$\sum\limits_{k=0}^{n} \binom{n+k}{n-k}$$