Twofer

Geometry Level pending

A B C D ABCD is the unit square. F F is the midpoint of A E AE . E E is chosen so the two incircles are congruent. If their radius is a root of f ( x ) = a x 3 + b x 2 + c x + d f(x) = ax^3 + bx^2 + cx + d , where a , b , c , d a,b,c,d are coprime integers, find ∣ f ( 4 ) ∣ |f(4)| .

part 2


The answer is 31.

Fletcher Mattox by Fletcher Mattox , 72, USA

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1 solution

Yuriy Kazakov
Mar 17, 2021

Four equations are here (see picture).

( r + h ) 2 + 1 = ( h + q ) 2 (r+h)^2+1=(h+q)^2

r + q = 1 r+q=1

q = h + 2 p q=h+2p

r = 1 4 + 4 p r=\frac{1}{4+4p}

Equation for r r is

f ( r ) = 4 r 3 − 16 r 2 + 8 r − 1 = 0 f(r)=4r^3-16r^2+8r-1=0

f ( 4 ) = 31 f(4)=31

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Nicely done, Yuriy. (I think your 2nd equation has a typo. g g should be q q , yes?)

Fletcher Mattox - 2 months, 3 weeks ago

Ок - it is mistake - I change it. Thanks for attention. Fine problem.

Yuriy Kazakov - 2 months, 3 weeks ago

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