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Problem of the Day #12: LPofXaak Numbers March 31, 2011

Posted by Alex in : potd , 2 comments

A positive integer is LPofXaak if there exists a way to form two positive integers using its digits (for example, the pair $(8, 97)$ can be formed from $789$ and the pair $(01, 2)$ can be formed from $210$) such that the arithmetic mean of these two integers is equal to the geometric mean of the sum of the two integers and the positive difference of the two integers. Leading zeroes of numbers do not count as digits, and the numbers formed cannot contain leading zeroes. Let $Q$ be the sum of all LPofXaak numbers less than $1000$. Find the remainder when $Q$ is divided by $1000$.

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Problem of the Day #11: Difference of Squares March 30, 2011

Posted by Aziz in : potd , 2 comments

If $$z=\sqrt{\sqrt{6\pi^2-5-4i^2}+i}$$
Find $$Re(z)^2 – Im(z)^2$$

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Problem of the Day #10: Rational Logs March 29, 2011

Posted by Arjun in : potd , add a comment

Given that $a$, $b$, $x$, and $y$ are positive integers, and $a+b = 10$, find the number of ordered pairs $(a,b)$ such that: $$ a^{5\log_2x} = b^{\log_{32}y}$$ where the quotient of $\log x$ and $\log y$ is rational.
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Problem of the Day #9: A Simple Number Theory Problem March 28, 2011

Posted by Seungln in : potd , add a comment

Let $V(x)$ be the possible number of positive integers $n$ such that $n^{-1}$ in modulus $x$ exists – that is, there exists an integer r such that $n \equiv \frac{1}{r} \pmod{x}$. Find $p$ such that $V(1998)\equiv p \pmod{330}$

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Problem of the Day #8: A House with a View March 27, 2011

Posted by Saketh in : potd , add a comment

A (very small) house is to be constructed at the center of a circular plot of land with radius $20$ meters. The developer plans to place trees around the house in a rather peculiar manner. Taking the house to be the origin, a sapling will be planted at every lattice point within the circle. The unit distance is to be $1$ meter.

Due to recent developments in genetic engineering, the trees can be modified to stop growing at any desired girth. To reduce construction costs, however, all of the trees will be identical.

The inhabitants of the house wish to be able to see outside of the plot. That is, there must exist some ray we can draw from the origin to the perimeter of the circle which does not intersect any of the trees. Determine the greatest $R$ such that for any tree radius $r < R$, the view is not blocked.

Assume for the purposes of this problem that the house is a single point at the origin.

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Problem of the Day #7: Probability that spider is above Seungln March 26, 2011

Posted by Sreenath in : potd , add a comment

Albert, Billy, Mitchell, Seungln, and Saketh are standing in a line. They rearrange themselves randomly such that Seungln is not next to Albert or Saketh. There is a spider on the ceiling above the first spot in the line. What is the probability that the spider is directly above Seungln?

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A Tricky Exponent March 25, 2011

Posted by Saketh in : calculus , 1 comment so far

Determine the exact value of  $$\int^{\frac{\pi}{2}}_0 \frac{1}{1+(\tan{x})^{\sqrt{2}}} \,dx$$

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An Integral March 25, 2011

Posted by Arjun in : calculus , 3 comments

Find $$\int^{\frac{\pi}{2}}_0 \frac{1}{\sqrt{\tan{x}}} \,dx$$

Problem of the Day #6: Tiling a 2 by 2 by n Box with Rectangular Prisms March 25, 2011

Posted by Billy in : potd , 1 comment so far

In how many distinct ways can $2n$ identical $1\times1\times2$ rectangular prisms be placed inside a $2\times2\times n$ box?

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Problem of the Day #5 March 24, 2011

Posted by Mitchell in : potd , add a comment

Let $a_n$, for $n$ a positive integer, be the $n$th smallest positive integer whose decimal (base-$10$) representation contains no occurrence of the same digit twice in a row. Find $ a_{ 82^6}$.

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