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Years ago, U.S. Savings Bonds were sold via payroll deductions. Say a bond worth $25 at maturity was sold for $18.75. You could have a certain amount deducted each pay period and when you built up enough in your bond account to pay the purchase price, you'd get a savings bond, and you might still have an amount left over in you account to go toward you next bond.

Say you had started with 3.75 taken out each pay period. After five such deductions you'd have exactly enough to buy an 18.75 bond. But suppose at some time when you happened to have 7.50 built up in your account you decided you wanted to get bonds more frequently and shifted to having 6.25 taken out each pay period so you'd get a bond every three pay periods. Next pay period you'd have 13.75 in your account, then next time you'd have $20, which is enough to get a bond and have 1.25 left over in your account, followed by a balance of 7.50 the next time, and this cycle would repeat forever, with your balance never going below 1.25. That 1.25 that hangs around in your balance is not helping you get your bond any sooner.

Devise an algorithm or formula so you can determine if such "dead money" is in a person's account, and how large that perpetually low balance is, based on a bond purchase price, the current balance, and the amount deducted each period, so that that lowest balance in the cycle can be refunded to the employee and the cycle will include a zero balance at some point.

Make sure your method allows for the fact that it may take a while before the balance goes to zero. The number given above were chosen to be simple, but allow for deduction amounts that may bear no simple relation to the purchase price.

Consider a perfect square having 3 as the first digit (reading left to right).

Determine the minimum value of N such that N remains a perfect square when the first digit is changed to 5.

Determine the largest area of an isosceles triangle that is enclosed within a unit cube.

What is the answer if the triangle is equilateral?

If x, y, z satisfy:

x + y + z = 12,
1/x + 1/y + 1/z = 2, and
x³ + y³ + z³ = -480,

find x²y + xy² + x²z + xz² + y²z + yz².

If x, y>0 such that:
(x+y)(1/x+1/y)=5
Then, find the minimum value of:
(x³+y³)(1/x³+1/y³)

Caroline correctly evaluates the expression |(x - y)(y - z)(z - x)| by replacing each of the variables x, y, z with a positive integer. How many possible answers between 1 and 100 (inclusive) could she have gotten?

Let {a_n} be a sequence of numbers that satisfy a₀ = 3, and (3 − a_n+1)(6 + a_n) = 18. Find

  n    1
  ∑   ---
 i=0   ai

From A Squared Divisor we know for all natural numbers n>=2 that n^(n-1)-1 is divisible by (n-1)^2.

Prove that n=2 and n=3 are the only natural numbers such that n^(n-1)-1 is divisible by (n-1)^3.

The first digit of the six digit number 1abcde is 1.

If the first digit is moved to the last place, we get a new number abcde1.

Given that, abcde1:1abcde = 7:2, find the value of abcde.

Farmer Brown lives in a land where farms stretch far and wide. One day he wins the lottery and wants to tell his neighbors. He tells his nearest neighbor of his good fortune. What follows is an odd sort of transmission of the news: each person who hears it (as well as farmer Brown himself) tells only his nearest neighbor, no one else.

Whenever that nearest neighbor is the person who told him the news, that branch of the transmission is closed, as there's no point in telling the person who told you, and there's no substitution of the second nearest neighbor.

1. What's the probability that the only person to get the news is the one farmer Brown called himself, due to farmer Brown being his nearest neighbor's nearest neighbor?
2. What's the expected number of people, besides farmer Brown, who will get the news before the transmission dies out altogether?

Consider the land where this happens an infinite plane, with each farmer a randomly placed point with uniform probability density.

For all positive integers n, the function rev(n) reverses the digits of n. For example, rev(205) = 502 and rev(12340) = 04321 = 4321. Compute the least positive integer m such that gcd(m, rev(m)) = 13.

If n is an integer greater than 1, then show that nⁿ⁻¹−1 is divisible by (n−1)².

If:

2a = 3
3b = 4
4c = 5
5d = 6

Then, find the value of a*b*c*d

If

  5
----- = A(√B + √C)
5-√24

then find integer values of A, B, and C.