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2022 CMWMC Guts Round 2/8 - Carnegie Mellon University Womens' Competition

Source:

August 12, 2023
CMWMCalgebrageometrycombinatoricsnumber theory

Problem Statement

Set 2
p4. △ABC\vartriangle ABC is an isosceles triangle with AB=BCAB = BC. Additionally, there is DD on BCBC with AC=DA=BD=1AC = DA = BD = 1. Find the perimeter of △ABC\vartriangle ABC.
p5. Let rr be the positive solution to the equation 100r2+2r−1=0100r^2 + 2r - 1 = 0. For an appropriate AA, the infinite series Ar+Ar2+Ar3+Ar4+...Ar + Ar^2 + Ar^3 + Ar^4 +... has sum 11. Find AA.
p6. Let N(k)N(k) denote the number of real solutions to the equation x4−x2=kx^4 -x^2 = k. As kk ranges from −∞-\infty to ∞\infty, the value of N(k)N(k) changes only a finite number of times. Write the sequence of values of N(k)N(k) as an ordered tuple (i.e. if N(k)N(k) went from 11 to 33 to 22, you would write (1,3,2)(1, 3, 2)).
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