6 let p be the set of all polynomials p x x 4 2 x 2

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6. Let P be the set of all polynomials p ( x ) = x 4 + 2 x 2 + mx + n , where m and n range over the positive reals. There exists a unique p ( x ) ∈ P such that p ( x ) has a real root, m is minimized, and p (1) = 99. Find n . 7. Let A, B, C, D be points, in order, on a straight line such that AB = BC = CD . Let E be a point closer to B than D such that BE = EC = CD and let F be the midpoint of DE . Let AF intersect EC at G and let BF intersect EC at H . If [ BHC ] + [ GHF ] = 1, then AD 2 = a b c where a, b, and c are positive integers, a and c are relatively prime, and b is not divisible by the square of any prime. Find a + b + c . 8. Mark has six boxes lined up in a straight line. Inside each of the first three boxes are a red ball, a blue ball, and a green ball. He randomly selects a ball from each of the three boxes and puts them into a fourth box. Then, he randomly selects a ball from each of the four boxes and puts them into a fifth box. Next, he randomly selects a ball from each of the five boxes and puts them into a sixth box, arriving at three boxes with 1 , 3 , and 5 balls, respectively. The probability that the box with 3 balls has each type of color is m n , where m and n are relatively prime positive integers. Find m + n . Mock 2018 AIME I 3 9. Angela, Bill, and Charles each independently and randomly choose a subset of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is m n , where m and n are relatively prime positive integers. Find m + n . 10. In circle Ω, let AB = 65 be the diameter and let points C and D lie on the same side of arc ˜ AB such that CD = 16, with C closer to B and D closer to A . Moreover, let AD, BC, AC, and BD all have integer lengths. Two other circles, circles ω 1 and ω 2 , have AC and BD as their diameters, respectively. Let circle ω 1 intersect AB at a point E 6 = A and let circle ω 2 intersect AB at a point F 6 = B . Then EF = m n , for relatively prime integers m and n . Find m + n .
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  • Fall '19
  • Prime number, Divisor

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