Mathcounts

A sequence begins 1, 2, …, and each term after the second term is the sum of all preceding terms. What is the 15th term of this sequence?

All points with coordinates (x, y) that are equidistant from the points (1, 3) and (7, 11) lie along a single line. When the equation of the line is written in the form y = mx + b, what is the value of b?

A soccer ball is a polyhedron comprised of 12 pentagons and 20 hexagons. How many vertices does a soccer ball have? 

In how many ways can the vertices of an equilateral triangle be colored with four available colors? Two colorings are considered the same if they can be obtained from each other by any combination of rotations and reflections.

A certain computer program will take 2000 years to run using current technology. Every year, advances in technology make it possible to run the program in half the time it would have taken starting in the previous year. However, once the program is started it cannot be interrupted to apply newer technology. Including the years spent waiting to start, what is the least number of years it will take to finish running the program? Express your answer to the nearest whole number.

The product of three consecutive odd integers is between 64,000,000 and 65,000,000. What is the greatest of the three integers? 

A 3-4-5 right triangle made of paper is cut along the altitude from the right angle, resulting in 2 smaller right triangles. These 2 triangles are then cut along the altitudes from their right angles, then the 4 resulting right triangles are cut the same way and finally, 8 triangles are cut the same way, resulting in 16 smaller right triangles. What is the sum of the perimeters of these 16 triangles? Express your answer as a decimal to the nearest tenth.

What common fraction is equivalent to \(0.2\overline{36}\)? 

If 2015 + a = b for positive integers a and b, both of which are palindromes, what is the smallest possible value of a? 

Maggie writes the numbers 1, 2, 3, …, 10 on separate slips of paper and tosses the 10 slips into a hat. She then randomly pulls three slips from the hat at the same time. What is the probability that the arithmetic mean of the three selected values is in fact written on one of those same three slips? Express your answer as a common fraction.