Maximum distance

Geometry Level 4

A B C D ABCD is a square. M M is a point inside the square such that A M + B M = 8 AM + BM = 8 , find the maximum of ⌊ C M + D M ⌋ \lfloor {CM + DM} \rfloor .

20 24 32 23 19

Ossama Ismail by Ossama Ismail , 63, Egypt

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2 solutions

There are two possibilities.
M i s m i d p o i n t o f A B . T h e n C M + D M = 8 5 . O R M = B . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e n C M + D M = 8 ( 1 + 2 ) . 1 + 2 > 5 . ∴ M a x = 8 ( 1 + 2 ) = 19.3..... A n s 19 M~ is~ midpoint~ of~ AB.~~~ Then~ CM+DM~=8~\sqrt5.~~~~ \\ OR \\ M=B............................ ~~~~~~~ Then~ CM+DM~=8~(1+\sqrt2). \\ 1+\sqrt2 > \sqrt5.\\ \therefore~ Max~=8~(1+\sqrt2)=19.3.....\\ Ans~\large \color{#D61F06}{19}

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Why are these the 2 possibilities? Why can't we have other arrangments?

@Ossama Ismail Are we assuming that M is contained within the square?

Calvin Lin Staff - 4 years, 4 months ago

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M is inside the square. I will post a solution in which used the inequality of Ptolemy's Theorem . and I got

CM + DM ≤ 8 ( 2 + 1 ) \leq 8( \sqrt{2} +1 ) .

Ossama Ismail - 4 years, 4 months ago

Did the same thing, nice

Jason Chrysoprase - 4 years, 4 months ago
Azadali Jivani
Jan 24, 2017

AM + BM = 8 Put AM = 0
So AB = DA = DM = BC = 8 (ABCD is a sq.) By Phyth. Theorem AC = 11.31
11 + 8 = 19 (ans.)

0 Helpful 0 Interesting 0 Brilliant 1 Confused

How do you know that if AM=0, you get the maximum?

Michael Mendrin - 4 years, 4 months ago

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