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If P(x) and Q(x) have 'reversed' coefficients, for example: P(x) = x5+3x4+9x3+11x2+6x+2 Q(x) = 2x5+6x4+11x3+9x2+3x+1 What can you say about the roots of P(x) and Q(x)?

6 friends are sitting in a circle whom you have been observing playing the game called 'spin the bottle'. They invite you to play as well. Being a good sport, you agree to play. You happen to notice however that it is HIlga's turn and you would rather not kiss Hilga. You have also noticed that the bottle when spun with the left hand has a tendency to stop on a person an even distance away (e.g. the second, fourth, sixth person from her) and when spun with the right hand will land on a multiple of 3 distance away from HIlga. Currently the bottle is pointing directly across from Hilga but if it were to land on herself she would immediately spin it again from that position, switching hands in the process. You have no idea which hand she will use first nor do you know which way rotation she will place on the bottle [clockwise or counter-clockwise]. Where should you sit?

All the numbers below should be assumed to be positive integers. Definition. An abundant number is an integer n whose divisors add up to more than In. Definition. A perfect number is an integer n whose divisors add up to exactly In. Definition. A deficient number is an integer n whose divisors add up to less than In. Example. 12 is an abundant number, because 1 + 2 + 3+ 4 + 6+12 = 28 and 28 > 2x12. However, 14 is a deficient number, because 1 + 2 + 7 + 14 = 24, and 24 < 2 x 14. Your task is to consider the following conjectures and determine, with proofs, whether they are true or false. Conjecture 1. A number is abundant if and only if it is a multiple of 6. Conjecture 2. If n is perfect, then kn is abundant for any k in N. Conjecture 3. If p1 and p2 are primes, then p1/p2 is abundant. Conjecture 4. If n is deficient, then every divisor of n is deficient. Conjecture 5. If n and m are abundant, then n + m is abundant. Conjecture 6. If n and m are abundant, then nm is abundant. Conjecture 7. If n is abundant, then n is not of the form pm for some natural m and prime p.

Think 7 Letter word that means "lying as if asleep." Change 1 if of the letter that follows it alphabetically. Move that letter to the 2nd position to get a word for something you lay on the floor. What are both words?