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Problem of the Day #156: Dr. Kim’s Solar Cells, Part 2 August 22, 2011

Posted by Sreenath in : potd , 1 comment so far

Dr. Kim has recently ordered a shipment of ten very expensive solar cells. As the cells arrive one at a time, Dr. Kim places them in a stack on his desk (with the last one received on the top). Izzy occasionally walks in, trips, and smashes the solar cell on the top of the stack. Given that she has already smashed the fifth solar cell Dr. Kim received, in how many ways can Izzy smash the remaining cells?

Problem of the Day #155: Sorted Numbers August 21, 2011

Posted by Alex in : potd , add a comment

A sorted number is a positive integer whose digits form a non-decreasing sequence when read from left to right. Let $w(x)$, where $x > 0$, be the $x^{\text{th}}$ smallest sorted number. Find $w(1111)$.

Problem of the Day #154: Factorial Analysis August 20, 2011

Posted by Albert in : potd , add a comment

How many subsets of $\{2,3,4,…,1000\}$ have the property that the product of all of its elements, each taken to a non-zero rational power, can be $1$?

Problem of the Day #153: Shapeception August 19, 2011

Posted by Seungln in : potd , add a comment

AIbert has a shape that is a triangle within a square within a pentagon within a hexagon. If each of the shapes within another shape is as big as it can possibly be, if all of the shapes are regular, and if the area of the hexagon is $100000$ square units, what is the area of the triangle? Round to the nearest integer.

Problem of the Day #152: Factorials Are Fun August 18, 2011

Posted by Alex in : potd , add a comment

Mitchell only likes positive integers $x$ such that the product of the exponents in the prime factorization of $x!$ is odd. Find the largest possible number that Mitchell likes.

Problem of the Day #151: Polygons in Polyhedrons August 17, 2011

Posted by Albert in : potd , add a comment

How many distinct (not congruent) polygons can be formed by connecting vertices of a regular icosahedron?

End of Summer Contest August 16, 2011

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Problem #149 was the last contest problem. The contest will end in one week (Tuesday, August 23 at 4PM EST).

Problem of the Day #150: A Geometric Decagon August 16, 2011

Posted by Seungln in : potd , add a comment

Albert is given a strange decagon. The ten angles, all convex, form a geometric sequence of ratio $r$. If $r$ is the maximum value possible and the decagon is not degenerate, find the first $10$ nonzero digits of $r$ in decimal form. The concatenated form of the answer will be the answer to this problem.

End of Fourth Two-Week Period August 16, 2011

Posted by Sreenath in : announcement , add a comment

Please join us in congratulating Brian Shimanuki, the winner of the third two-week period. This period spanned problems $136$ through $149$.

As the top-scoring solver over the past two weeks that hasn’t already won a two-week prize, Brian wins a $\$10$ gift card of his choice. You can view the full rankings for these two weeks after the jump. (more…)

Problem of the Day #149: Seungln’s favorite function August 15, 2011

Posted by Albert in : potd , add a comment

Seungln’s favorite function, $f(n)$, is defined to be the number of subsets (including $\varnothing$) of $\{1,2,3,\ldots,n\}$ for which the product of the elements of the subset is less than or equal to $n$. Seungln can easily bash $f(25) = 86$ (after all, there’s only $33,554,432$ subsets to check), but he’s having trouble with $f(30)$ and higher. Help him find $$\sum\limits_{n=30}^{40}f(n)$$