Product of fractions

Simplify (3/4)*(8/9)*(15/16)*(24/25)*...*(9999/10000).

Polynomial remainder

Find the remainder when x^1000 is divided by x^4 - x^2 + 1.

Minimize the area between a curve and a horizontal line

Let f be continuous increasing function on the interval [a,b]. Find c so that the area of the region bounded by x=a, x=b, y=f(x), and y=c is a minimum.

A Fibonacci sum

Evaluate the sum of Fib(n)/10^n where Fib(n) denotes the nth Fibonacci number.

That is, calculate 1/10 + 1/10^2 + 2/10^3 + 3/10^4 + 5/10^5 + 8/10^6 + 13/10^7 + ...

cos(20) * cos(40) * cos(80) = ?

Simplify cos(20°) * cos(40°) * cos(80°) without using a calculator.

Writing 20 as a sum of odd numbers

There are three ways to write 10 as the sum of four odd positive integers, assuming that order does not matter. They are 1+1+1+7, 1+1+3+5, and 1+3+3+3. In how many ways can 20 be written as the sum of eight odd positive integers?

Source: The Moscow Puzzles, by Boris Kordemsky.

Positive integer solutions to 4x + 5y < 1001

Find the number of solutions to the inequality 4x + 5y < 1001, where x and y are positive integers.

Houses on a circular road

Nine houses are built on a circular road. Each house is to be painted either red, white, or blue. Adjacent houses must have different colors. How many color combinations are possible?

Double tangent line

Find the equation of the line that is tangent to the curve
y = x⁴ − 14x³ + 69x² at two points.

Sum of consecutive numbers in seven ways

Find an integer that can be expressed as the sum of two or more consecutive positive integers in exactly seven ways. (Bonus: find the smallest such integer.)

Triple 6

You throw three dice n times. How big does n have to be for you to have greater than a 50% chance of throwing three 6's?

Limit with nth roots

Evaluate the limit of (a^(1/n)+b^(1/n)-1)^n as n approaches infinity, where a, b > 1.

Equilateral triangle in a circle

Let ABC be an equilateral triangle inscribed in a circle, and P a point on the arc AB. Find |CP| given that |AP|=3 and |BP|=4.

Road Trip

Sue drives from San Francisco to New York City. At the halfway point her average speed is 40 miles per hour. How fast must she drive to average 60 miles per hour for whole trip?