A Pre-RMO question! -4

Algebra Level 1

If 2 s = a + b + c 2s=a+b+c and ( s − a ) 2 + ( s − b ) 2 + ( s − c ) 2 + s 2 = a x + b x + c x (s-a)^2+(s-b)^2+(s-c)^2+s^2=a^x+b^x+c^x for a real x x , find x 2 x^2 .


The answer is 4.

Zakir Husain by Zakir Husain , 41, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Chew-Seong Cheong
May 31, 2020

X = ( s − a ) 2 + ( s − b ) 2 + ( s − c ) 2 + s 2 = s 2 − 2 a s + a 2 + s 2 − 2 b s + b 2 + s 2 − 2 c s + c 2 + s 2 = 4 s 2 − 2 s ( a + b + c ) + a 2 + b 2 + c 2 = 4 s 2 − 4 s 2 + a 2 + b 2 + c 2 = a 2 + b 2 + c 2 \begin{aligned} X & = (s-a)^2 + (s-b)^2 + (s-c)^2 + s^2 \\ & = s^2 -2as + a^2 + s^2 -2bs + b^2 + s^2 - 2cs + c^2 + s^2 \\ & = 4s^2 - 2s(a+b+c) + a^2 + b^2 + c^2 \\ & = 4s^2 - 4s^2 + a^2+b^2+c^2 \\ & = a^2+b^2+c^2 \end{aligned}

Therefore, x 2 = 4 x^2 = \boxed 4 .

1 Helpful 0 Interesting 1 Brilliant 0 Confused

Bro he askefx^2

vivek teja Sapavath - 8 months, 1 week ago

Log in to reply

Thanks. I have changed it. I don't why people likes to play game.

Chew-Seong Cheong - 8 months, 1 week ago
Zakir Husain
May 31, 2020

( s − a ) 2 + ( s − b ) 2 + ( s − c ) 2 + s 2 = s 2 − 2 s a + a 2 + s 2 − 2 s b + b 2 + s 2 − 2 s c + c 2 + s 2 (s-a)^2+(s-b)^2+(s-c)^2+s^2=\blue{s^2}\green{-2sa}\red{+a^2}\blue{+s^2}\green{-2sb}\red{+b^2}\blue{+s^2}\green{-2sc}\red{+c^2}\blue{+s^2} 4 s 2 + a 2 + b 2 + c 2 − 2 s ( a + b + c ) = ( 2 s ) 2 + a 2 + b 2 + c 2 − 2 s ( 2 s ) = a 2 + b 2 + c 2 = a x + b x + c x 4s^2+a^2+b^2+c^2-2s(a+b+c)=\cancel{(2s)^2}+a^2+b^2+c^2\cancel{-2s(2s)}=a^2+b^2+c^2=a^x+b^x+c^x x = 2 x=2 x 2 = 4 \boxed{x^2=4}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I was however deceived to see 2 s = a + b + c 2s = a + b + c and assume pure geometric solution about semi-perimeter and triangle, because the expression is also close to Heron's formula. Good problem, good solution though

Mahdi Raza - 1 year ago

It is clear that the degree of expression is 2 2 hence x x cannot be any other number except 2 2 .

Vilakshan Gupta - 1 year ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...