Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Algebra Subject Test Find the smallest value of r such that a>14 a for all nonnegative a 2. Compute an 2000-5in-0203) 3. Find the smallest n such that n! ends in 290 zeroes 4. Simplify:2V15+V2-(1.5+ 5. Several positive integers are given, not necessarily all different. Their sum is 2003 Suppose that ni of the given numbers are equal to 1, n2 of them are equal to 2 n2003 of them are equal to 2003. Find the largest possible value of +2n3+3 6. Let a1=1, and let an=Ln /an-1l 7. Let a, b, c be the three roots of p(r)=x+12-333 -1001. Find a+b+c Find the value of 3+1+4+2 9. For how many integers n, for 1<ns1000, is the number 2(an)even? 10.S P()is a poly such that P(1) P(2x) P(x+1) x+7 for all real a for which both sides are defined. Find P(1)Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Algebra Subject Test 1. Find the smallest value of x such that a ≥ 14√ a − x for all nonnegative a. 2. Compute tan2 (20◦)−sin2 (20◦) tan2(20◦) sin2 (20◦) . 3. Find the smallest n such that n! ends in 290 zeroes. 4. Simplify: 2q 1.5 + √ 2 − (1.5 + √ 2). 5. Several positive integers are given, not necessarily all different. Their sum is 2003. Suppose that n1 of the given numbers are equal to 1, n2 of them are equal to 2, . . ., n2003 of them are equal to 2003. Find the largest possible value of n2 + 2n3 + 3n4 + · · · + 2002n2003. 6. Let a1 = 1, and let an = bn 3/an−1c for n > 1. Determine the value of a999. 7. Let a, b, c be the three roots of p(x) = x 3 + x 2 − 333x − 1001. Find a 3 + b 3 + c 3 . 8. Find the value of 1 3 2+1 + 1 4 2+2 + 1 5 2+3 + · · ·. 9. For how many integers n, for 1 ≤ n ≤ 1000, is the number 1 2 ³ 2n n ´ even? 10. Suppose P(x) is a polynomial such that P(1) = 1 and P(2x) P(x + 1) = 8 − 56 x + 7 for all real x for which both sides are defined. Find P(−1). 1