MathDB

Set 7
p19. The polynomial

x^4 + ax^3 + bx^2 - 32x

, where

a

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

ABC

intersects the midpoint of

\overline{BC}

. Let

M

be the midpoint of

\overline{AC}

. If the measure of

\angle DBA

67^o

, find the measure of

\angle 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.
PS. You should use hide for answers.

Problem Statement