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2009 Mississippi Mu Alpha Theta Inter-School Test 1. Let ABC be a triangle inscribed in a circle with center O. If the vertices of the triangle partition the circle into three arcs of lengths 15, 20, 25, find the area of the triangle. 2. A number x is selected uniformly at random between 250 and 300. If [ x ] = 16, find the probability that [ 100 x ] = 160. (Note: [y] is the greatest integer less than or equal to y.) 3. Find the sum of the solutions of tan 2 x – 9tan x + 1 = 0 that are in the interval (0, 2π) (the endpoints of the interval are represented in terms of radians). 4. Find all primes p such that 31p + 4 is a perfect square. 5. Prove that any two consecutive integers are relatively prime, that is, gcd(a, a + 1) = 1 for any integer a. 6. Find the number of proper divisors of 9878400. 7. Find the number of ways in which n boys and n girls can be seated in a row of 2n chairs if boys and girls must alternate. 8. Use mathematical induction to prove that for each natural number n, 7 n 2 n is divisible by 5. 9. Let { Ai }iI denote the set of all sets Ai as i ranges over set I. Let A' denote the complement of set A. Prove that ( Ai )' = Ai ' , where iI iI A i iI is the union of all the sets Ai and Ai is iI the intersection of all sets Ai . 10. Five married couples are standing is a room. If the 10 people are divided into 5 pairs, find the probability p that (a) each pair is married, (b) each pair contains a male and a female (notice that there are 5 males and 5 females). n ) denote the reminder of n when divided by p. Find the p n smallest number n that satisfies r( ) = p – 1, for p = 2, 3, 4, …, 10. p 11. For natural numbers n and p, let r( xz xy yz = a, = b, and = c, where a, b, and c are real numbers different from xz x y yz zero, represent x in terms of a, b, and c. 12. If 13. Let P be a fixed external point to a circle with center O and radius r. Find the locus of the midpoint M of segment PA as A moves around the circle. 14. A projectile, fired straight upward with initial velocity of 500 ft/sec, moves according to s(t) = – 16 t 2 + 500t, where s is the distance above the ground after t seconds after being fired. (a) Find the velocity and acceleration at the time the projectile hits the ground. (b) Find the greatest height reached. 15. Use the theorem that lim x 0 sin x = 1 to prove that x lim x 0 1 cos x = 0. x 2009 Mississippi Mu Alpha Theta Inter-School Test Tie breakers 1. Prove that there does not exist integers m, n, and p, except 0, 0, 0, for which m + n 2 + p 3 = 0. 2. The sequence of natural numbers is partitioned as follows 1, (2, 3), (4, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14, 15), … Find the sum of the natural numbers in the nth group. 3. Let E, F, G, and H be the midpoints of the sides of a quadrilateral ABCD. If EFGH is a square, what type of quadrilateral is ABCD? C F E D G A H B