# Thread: Problem Of The Week #403 Feb 2nd, 2020

1. Here is this week's POTW:

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Monic quadratic polynomials $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23,\,-21,\,-17$ and $-15$ and $Q(P(x))$ has zeros at $x=-59,\,-57,\,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$?

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2.

Hi MHB,

I was told by castor28 that this week's POTW (High School) was a duplicate of POTW #363, which is true. I am truly sorry for letting this thing happened. I therefore want to thank him for catching the mistake.

Please let me make it up by presenting to you the following problem:

A geometric sequence $(a_n)$ has $a_1=\sin x,\,a_2=\cos x$ and $a_3=\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?

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Congratulations to the following members for their correct answer!

1. castor28
2. MegaMoh

Solution from castor28:

Alternate solution from MegaMoh: