Arctangent sum

Evaluate the sum of arctan(1/(2n^2)) from n=1 to infinity.

Line joining tangent circles

A circle of radius 4 is tangent to a circle of radius 9, and a line is tangent to both circles at A and B. Find |AB|.

x^y = y^x

If 0 < x < y and x^y = y^x then what is the greatest possible value (least upper bound) of x?

3D Tic-Tac-Toe

In how many ways can you get three in a row in three-dimensional tic-tac-toe (noughts and crosses)?

Colorado balloon incident

I gave my students an assignment related to the Colorado balloon incident. Any comments or suggestions would be appreciated. Are there other questions that should have been included? The last question seems to be unanswerable with the given information, but I am including it anyway.

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Read the excerpt from the Wikipedia article on the Colorado Balloon Incident, and then answer the following questions.

1. What is the volume of the balloon, in cubic meters?
2. How many cubic feet are in one cubic meter?
3. What is the lift capacity of helium at sea level at 0° C?
4. The article states that the balloon would be sufficient to lift 29 kg at sea level. Show how this would be calculated.
5. How high above Fort Collins did the balloon reach?
6. How much helium would have been needed to carry Falcon Heene to an altitude of 8,000 feet?

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Note: These questions are released to the public domain, and may be used or modified without attribution.

f(f(x)) = x^2

Find a real function f so that f(f(x)) = x^2 for all x.

x^(x^3) = 3

Find a real solution to x^(x^3) = 3.

Equiangular hexagon

Let ABCDEF be an equiangular hexagon. If |AB|=3, |BC|=5, and |DE|=7, find |EF|.

Spider on a ladder

A 4 meter ladder leans against a wall, the base 1 meter from the wall. A spider is sitting at the midpoint of the ladder. If the ladder slides down the wall, how far will the spider travel?

Parabola and Circle

A circle intersects the parabola y=x^2 at the points (1,1), (2,4), and (3,9). Find the fourth point of intersection.

Viewing angle of a painting

The top of a painting is 4 feet above eye level, and the bottom is 1 foot above eye level. How far should one stand from the wall to get the widest view?

Pentagon area

A pentagon is formed by placing a triangle atop a rectangle. Find the maximum possible area as a function of the perimeter P.

Product of divisors

The product of the divisors of N is 10^7200. What is N? [Typo corrected]

Remainders

Find a positive number that when divided by 5 leaves remainder 1, when divided by 7 leaves 3, and when divided by 9 leaves 5.

Lazy student

A student guesses randomly on a 50-question True/False test. Find the probability that the number of correct guesses is divisible by 4.

Sum of k(k+1)(k+2)

Calculate 1*2*3 + 2*3*4 + 3*4*5 + ... + 998*999*1000.

Random subsets

Two random subsets A and B are selected from an n-element set. What is the probability that A is a subset of B?

1/x + 1/y = 1/210

Find the number of integer solutions to 1/x + 1/y = 1/210.

Numbers less than one trillion containing '99'

How many numbers less than one trillion (10^12) have two or more consecutive nines in their digits?

Last two digits of (sqrt(2)+sqrt(3))^(10^100)

Find the last two digits of the integer closest to (sqrt(2)+sqrt(3))^(10^100).

Semicircle

What is the probability that three points, chosen at random from a circle, lie on a common semicircle?

Tricky definite integral

Evaluate the integral from -1 to 1 of cos(x)/(e^x + 1) dx.

(x+y+z)^100

How many terms does (x+y+z)^100 have when expanded? (After combining like terms.)

Winning percentage

A team won 90% of their games during a season, but lost the first game. Was there a point where their winning percentage was exactly 80%? Exactly 70%?

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?