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Some Very Interesting Math Challenges
1) f is a function from the nonnegative integers to themselves.
For all nonnegative integers n, f satisfies: f(3n) = 2f(n) f(3n+1) = f(3n) + 1 f(3n+2) = f(3n) + 2 Prove that f(3n) >= f(2n) > f(n) for n>0. And yes, the second inequality is STRICT. 2) For positive, real, a, b, with a != b, prove that: (a+b)/2 > (1/e)*b^(b/(b-a))*a^(a/(a-b)) > (b-a)/(ln b - ln a). 3) S_n = a^n + b^n + c^n. Show that S_n is a rational polynomial in S_1, S_2, S_3, for all integers n>0. |
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