If-x-amp-y-are-both-positive-integers-then-then-show-if-it-possiple-that-x-2-y-1-amp-y-2-4x-3-be-both-perfect-squares-simultaneously-

Question Number 192797 by York12 last updated on 27/May/23

I f x & y a r e b o t h p o s i t i v e i n t e g e r s t h e n

t h e n s h o w i f i t p o s s i p l e t h a t x^{2} + y + 1 & y^{2} + 4 x + 3 b e b o t h

p e r f e c t s q u a r e s s i m u l t a n e o u s l y .

Answered by witcher3 last updated on 27/May/23

x = y \Rightarrow

x^{2} + x + 1 = a^{2}

x^{2} < x^{2} + x + 1 = a^{2} < {(x + 1)}^{2} . . impossibl

x < y \Rightarrow\Rightarrow x ⩽ y + 1

y^{2} < y^{2} + 4 x + 3 < y^{2} + 4 y + 4 = {(y + 2)}^{2}

\Rightarrow y^{2} + 4 x + 3 = {(y + 1)}^{2} = y^{2} + 2 y + 1

y = 2 x + 1

x^{2} + y + 1 = {(x + 1)}^{2} + 1 = a^{2} \Rightarrow 1 + {(x + 1)}^{2} = b^{2}

\Rightarrow (b + x + 1) = 1 impossible x ⩾ 1

x > y \Rightarrow x ⩾ y + 1

\Rightarrow x^{2} < x^{2} + y + 1 ⩽ x^{2} + x + 1 < {(x + 1)}^{2}

impossible . .

Commented by witcher3 last updated on 31/May/23

yes in Hight School Im in Team Select

but i didnt participated to many stressed

in High school final i didnt Go withe the reste

im still regret this

Commented by York12 last updated on 27/May/23

I c a n n o t b e l i e v e t h a t t h i s q u e s i o n w a s a s k e d i n C H M O

w h i c h i s s o m e t i m e s c o n s i d e r e d h a r d e r t h a n

I M O

Commented by witcher3 last updated on 28/May/23

IMO is for High school

but sum Quations Can bee tought

like IMO 2013

Commented by York12 last updated on 28/May/23

I a m a h i g h s c h o l a r b y t h e w a y

I h a v e n o t s e e n i t c a u s e i t i s n o t i n v o l v e d

i n t h e c o m p e n e d i u m

s o h a v e y o u p a r t i c i p a t e d

Commented by York12 last updated on 29/May/23

y e a h y o u a r e r i g h t

b u t y o u h a v e t o l o o k a t C H M O p a s t p a p e r s

Commented by York12 last updated on 06/Jun/23

i a m s u r e y o u w o u l d b e a b l e t o g e t a g o l d e n

m e d a l

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