Set 7
p19. The polynomial x4+ax3+bx2−32x, wherea and b are real numbers, has roots that form a square in the complex plane. Compute the area of this square.
p20. Tetrahedron ABCD has equilateral triangle base ABC and apex D such that the altitude from D to ABC intersects the midpoint of BC. Let M be the midpoint of AC. If the measure of ∠DBA is 67o, find the measure of ∠MDC in degrees.
p21. Last year’s high school graduates started high school in year n−4=2017, a prime year. They graduated high school and started college in year n=2021, a product of two consecutive primes. They will graduate college in year n+4=2025, a square number. Find the sum of all n<2021 for which these three properties hold. That is, find the sum of those n<2021 such that n−4 is prime, n is a product of two consecutive primes, and n+4 is a square.
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