Harvard-MIT Mathematics Tournament March 15. 2003 HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 1.5] Simplify 2y2/1-35.y89+15 2. [5] The graph of r=12y? is a union of n different lines. What is the value of n? 3. 5 If a and b are positive integers that can each be written as a sum of two squares then ab is also a sum of two squares. Find the smallest positive integer c such that c=ab, where a=x3+y and b=23+y each have solutions in integers(, y),but c=r+y does not HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 4.[6]Letz=1-2. Find+是+是+… 5. [6 Compute the surface area of a cube inscribed in a sphere of surface area T 6. [6 Define the Fibonacci numbers by Fo=0, Fi=l, Fn=Fn-1+ Fn-2 for n 22. For how many n,0≤n≤100, is Fn a multiple of13? HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 7. 6 a and b are integers such that a+ vb=v15+v216. Compute a/b 8. [6 How many solutions in nonnegative integers(a, b, c) are there to the equation 9. [6 For z a real number, let f(r)=0 if x I and f(a)=2. c-2 if x >1. How many solutions are there to the equation f(f((f(a)=cHarvard-MIT Mathematics Tournament March 15, 2003 Guts Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 1. [5] Simplify 2003q 2 √ 11 − 3 √ 5 · 4006q 89 + 12√ 55. 2. [5] The graph of x 4 = x 2 y 2 is a union of n different lines. What is the value of n? 3. [5] If a and b are positive integers that can each be written as a sum of two squares, then ab is also a sum of two squares. Find the smallest positive integer c such that c = ab, where a = x 3 + y 3 and b = x 3 + y 3 each have solutions in integers (x, y), but c = x 3 + y 3 does not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 4. [6] Let z = 1 − 2i. Find 1 z + 2 z 2 + 3 z 3 + · · ·. 5. [6] Compute the surface area of a cube inscribed in a sphere of surface area π. 6. [6] Define the Fibonacci numbers by F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2. For how many n, 0 ≤ n ≤ 100, is Fn a multiple of 13? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 7. [6] a and b are integers such that a + √ b = q 15 + √ 216. Compute a/b. 8. [6] How many solutions in nonnegative integers (a, b, c) are there to the equation 2 a + 2b = c! ? 9. [6] For x a real number, let f(x) = 0 if x < 1 and f(x) = 2x − 2 if x ≥ 1. How many solutions are there to the equation f(f(f(f(x)))) = x? 1