2008 Mock ARML 2 Problems
is a convex quadrilateral such that , , , and . Given that , find the area of .
Given that the sum of all positive integers with exactly two proper divisors, each of which is less than , is , find the sum of all positive integers with exactly three proper divisors, each of which is less than (a proper divisor of is a positive integer that divides but is not equal to ).
A variation of Pascal's triangle is constructed by writing the numbers and in the top row and writing each subsequent term as the sum of the two terms above it. Find the fifth term from the left in the thirteenth row.
Equilateral triangle has a side length of . A ball begins at vertex , rolls through the interior of the triangle, bounces off side , and settles at point P. Given that and , find the total distance that the ball travels.
Al is thinking of a function, . He reveals to Bob that the function is a polynomial of the form , where , , , , and are complex number coefficients. Bob wishes to determine the value of . For any complex number that Bob asks about, Al will tell him the value of . At least how many values of must Bob ask about in order to definitively determine the value of ?
John has a pile of blocks. On top of the pile is one block. Below this block are two smaller blocks. Below each of these two blocks are two even smaller blocks. Below each of these blocks are two still smaller blocks, and so on until the last row, which contains blocks. John removes blocks one at a time, removing only blocks that currently have no blocks on top of them. Find the number of ways (order matters) in which John can remove exactly seven blocks.
Let equal the number of zeroes to the right of the rightmost non-zero digit in the decimal form of , and let . Given that can be written as , where and are relatively prime positive integers, is less than , and is less than , find .
Given that and for all non-negative integers , evaluate .