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Problems > High School Math > 2007

## A Really, Really, Ridiculously Big Number

1. When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A, B are written in decimal notation.)

2. What is the largest number that can be obtained as the product of positive integers that add up to 100?

Source: unknown

## Buckets and Springs

1. The classic "Three Jugs" problem: Two friends who have an eight-quart jug of water wish to share it evenly. They also have two empty jars, one holding five quarts, the other three. How can they each measure exactly 4 quarts of water?

Source: Cut-the-knot, who attributes the puzzle to Siméon Denis Poisson.

2. 3 people carry 5 pails (each with capacity 8L) to a place where there are 3 springs. One of the springs gives 2L/minute and the other two give 1L/minute. It is not possible to use one spring to fill two pails simultaneously. It takes less than 2 minutes and more than 1 minute to take a pail from one spring to another. What is the shortest time it takes for them to fill all five pails? How?

Source: unknown

1. The zeros of the polynomial

x3 - 33x2 + 354x + k

are in arithmetic progression. What is the value of k?

2. Let f(x) = px5+qx4+rx3+sx2+tx+1 be a polynomial such that f(1)=4, f(2)=11 and all the coefficients p,q,r,s and t are integers. Prove that the equation f(x)=0 has no integer roots.

Source: unknown

## I'm Thinking of a Number...

1. Sally is thinking of a 6-digit number. The sum of the digits is 43. And only two of the following three statements about the number are true: (1) it's a square number. (2) it's a cube number, and (3) the number is under 500000. What number was Sally thinking of?

2. This is a variation on a classic "guess the numbers" puzzle:

I think of 2 single-digit numbers from 1 to 9. I tell Peter Griffin their product and Lois Griffin their sum.

Peter: "I don't know the numbers."

Lois: "I don't know the numbers."

Peter: "I don't know the numbers."

Lois: "I don't know the numbers."

Peter: "I don't know the numbers."

Lois: "I don't know the numbers."

Peter: "I don't know the numbers."

Lois: "I don't know the numbers."

Peter: "Now I know the numbers."

What are the numbers?

Source: unknown

3. Find three three-digit square numbers that together use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once.