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

Posted by Albert in : potd , add a comment

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 #351: Points in a Circle March 4, 2012

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Albert draws a unit circle centered at the origin, and then draws a number of points in it. Then, he adds up for each point the square of the distance to the nearest adjacent point. What is the greatest possible sum he could find?

Problem of the Day #347: Evaluating Boolean Expressions February 29, 2012

Posted by Alex in : potd , add a comment

Saketh wants to evaluate the Boolean algebra expression $(AB + CD)(AC + BD)$. The values of $A$, $B$, $C$, and $D$ are randomly generated. In order to evaluate the expression, Saketh will uncover variables, revealing their values, one at a time in whatever order he desires. Given that Saketh chooses optimally, what is the expected number of variables whose values must be revealed before Saketh can evaluate the expression?

Note: $AB$ denotes $A \text{ and } B$ and $C+D$ denotes $C \text{ or } D$.

Problem of the Day #346: Fun with Functions February 28, 2012

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Ana is investigating a function $f(n)$ defined for all positive integers $n$. She knows that $f(ab) = f(a)\cdot f(b)$ for all relatively prime positive integers $a$ and $b$, and that $f(p+q) = f(p)+f(q)$ for all primes $p$ and $q$. Given these facts, help her find the value of $f(2012)$.

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

Posted by Albert in : potd , add a comment

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 #339: Arjun Likes Food February 21, 2012

Posted by Alex in : potd , add a comment

Inspired by a problem from the 2012 Spotify Code Quest

Arjun likes food. He starts at coordinate $0$ with $500$ units of food, and for each distance of $1$ he moves, he consumes $1$ unit of food. Saketh is at coordinate $100$ and he wants food. What is the maximum amount of food that Arjun can deliver to Saketh? Arjun is allowed to place food on the ground wherever he wishes (so that he can pick it up later), but every time he places food on the ground, half of the quantity is consumed by squirrels.

Problem of the Day #338: Albert’s Quadrilaterals February 20, 2012

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Given any quadrilateral $ABCD$, Albert draws in the points $P$, $Q$, $R$, and $S$ – the midpoints of sides $BC$, $CD$, $DA$, and $AB$ respectively. He then determines the value of $$\frac{AP^2 + BQ^2 + CR^2 + DS^2}{AB^2 + BC^2 + CD^2 + DA^2}$$ What is the largest possible ratio Albert could find?

Problem of the Day #321: Oscillating Function February 3, 2012

Posted by Alex in : potd , add a comment

Let $x_n = f(x_{n-1})$ for all $n > 1$, where $f(x) = x^2-4x+3$. Suppose $x_n$ for $n \ge 1$ only takes distinct two values. Find all possible values of $x_1$.

Problem of the Day #320: Algebra 2012 February 2, 2012

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Find all integer solutions $(x, y)$: $$x^4-y^4 = 2012(2012-2x^2)$$

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

Posted by Albert in : potd , add a comment

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 $\pi$.