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.

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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

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Consider the following figure:

If the radius of the blue quarter-circle is 2, which is also the diameter of the larger half-circle (colored pink), then what is the radius of the smaller half-circle (colored yellow)?

Source: unknown

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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

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1. When 4444^4444 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

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