Calculus anyone?

Algebra Level 4

x 2 − 3710 x + 3510194 + y 2 + 1476225 + x 2 + y 2 − 2430 x − 1058 y + 1756066 \sqrt{x^2-3710x+3510194}+\sqrt{y^2+1476225}+\sqrt{x^2+y^2-2430x-1058y+1756066}

If x x and y y are real numbers, find the minimum value of the expression above.

Happy New Year 2017!
Adapted from IMO 2014 Prelim HK


The answer is 2017.

Lolly Lau by Lolly Lau , 31, Hong Kong

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1 solution

Lolly Lau
Dec 31, 2016

Note that the original expression

x 2 − 3710 x + 3510194 + y 2 + 1476225 + x 2 + y 2 − 2430 x − 1058 y + 1756066 \sqrt{x^2-3710x+3510194}+\sqrt{y^2+1476225}+\sqrt{x^2+y^2-2430x-1058y+1756066} = y 2 + 121 5 2 + ( x − 1215 ) 2 + ( y − 529 ) 2 + ( x − 1855 ) 2 + ( 792 − 529 ) 2 =\sqrt{y^2+1215^2}+\sqrt{(x-1215)^2+(y-529)^2}+\sqrt{(x-1855)^2+(792-529)^2}

Hence if we let

A = ( 0 , 0 ) A=(0,0) , B = ( 1215 , y ) B=(1215,y) , C = ( x , 529 ) C=(x,529) , and D = ( 1855 , 792 ) D=(1855,792) ,

the expression is equal to A B + B C + C D AB+BC+CD ,

which is minimum when A A , B B , C C , and D D are collinear.

Therefore the expression has a minimum value of A D = 185 5 2 + 79 2 2 = 2017 AD = \sqrt{1855^2+792^2}=\textbf{2017} .

Wish you all a Happy New Year!

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Wish you a Happy New Year too! Nice solution! :D

Michael Huang - 4 years, 5 months ago

exactly the same way.

Saya Suka - 4 years, 5 months ago

I dint know how to do this problem, but it was written happy new year.So i deduced the ans to be 2017

genis dude - 4 years, 5 months ago

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Of course. I think many guessed it :)

Lolly Lau - 4 years, 5 months ago

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