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2019 DMM Tiebreaker Round - Duke Math Meet

Source:

October 2, 2023
DMMalgebranumber theory

Problem Statement

p1. Let a(1),a(2),...,a(n),...a(1), a(2), ..., a(n),... be an increasing sequence of positive integers satisfying a(a(n))=3na(a(n)) = 3n for every positive integer nn. Compute a(2019)a(2019).
p2. Consider the function f(12x−7)=18x3−5x+1f(12x - 7) = 18x^3 - 5x + 1. Then, f(x)f(x) can be expressed as f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d, for some real numbers a,b,ca, b, c and dd. Find the value of (a+c)(b+d)(a + c)(b + d).
p3. Let a,ba, b be real numbers such that 5+26=a+b\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}. Find the largest value of the quantity X=1a+1b+1a+...X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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