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Problem of the Day #422: Divergent Geometric Mean December 31, 2012

Posted by Saketh in : potd , add a comment

Can you find a convergent sequence of reals with a divergent geometric mean? Alternatively, give a proof that no such sequence exists.

Problem of the Day #409: Sohail’s Function May 1, 2012

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Sohail is tinkering around with a function $f$. Suddenly, he realizes that his function satisfies the equation $f(x+y) + f(xy) = f(x)f(y) + 1$ for all pairs of integer values $x$ and $y$. Find all possible values of $f(2012)$.

Problem of the Day #405: Funny Functions April 27, 2012

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Function $f$ is defined such that $f(x^2 – y^2) = (x-y)(f(x)+f(y))$ for all real $x$ and $y$. If $f(1) = 2012$, find $f(2012)$.

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 #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 #387: Floor It April 9, 2012

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Find all real values of $x$ for which $x \lfloor x \lfloor x \rfloor \rfloor = 387$.

Problem of the Day #386: Minimization under Constraints April 8, 2012

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Determine the least possible value of the expression

$$ \frac{(x+y^3)\cdot(y+z^3)\cdot(z+x^3)}{xyz} $$

if $x$, $y$, and $z$ are all greater than or equal to some positive real value $n$. Express your answer in terms of $n$.

Problem of the Day #383: Infinite Square Roots April 5, 2012

Posted by Alex in : potd , add a comment

Find $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}$ to three decimal places of precision.

Problem of the Day #382: Four Prime Factors April 4, 2012

Posted by Alex in : potd , 1 comment so far

Find the sum of the prime factors of $1,015,074,782$, given that there are exactly four of them and one of them is $499$.