Question Number 192797 by York12 last updated on 27/May/23
$$\boldsymbol{{If}}\:\boldsymbol{{x}}\:\&\:\boldsymbol{{y}}\:\boldsymbol{{are}}\:\boldsymbol{{both}}\:\boldsymbol{{positive}}\:\boldsymbol{{integers}}\:\boldsymbol{{then}} \\ $$$$\boldsymbol{{then}}\:\boldsymbol{{show}}\:\boldsymbol{{if}}\:\boldsymbol{{it}}\:\boldsymbol{{possiple}}\:\boldsymbol{{that}}\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}+\mathrm{1}\:\&\:\boldsymbol{{y}}^{\mathrm{2}} \:+\:\mathrm{4}\boldsymbol{{x}}\:+\:\mathrm{3}\:\boldsymbol{{be}}\:\boldsymbol{{both}}\: \\ $$$$\boldsymbol{{perfect}}\:\boldsymbol{{squares}}\:\boldsymbol{{simultaneously}}. \\ $$
Answered by witcher3 last updated on 27/May/23
$$\mathrm{x}=\mathrm{y}\Rightarrow \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}=\mathrm{a}^{\mathrm{2}} \\ $$$$\mathrm{x}^{\mathrm{2}} <\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}=\mathrm{a}^{\mathrm{2}} <\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} ..\mathrm{impossibl} \\ $$$$\mathrm{x}<\mathrm{y}\Rightarrow\Rightarrow\mathrm{x}\leqslant\mathrm{y}+\mathrm{1} \\ $$$$\mathrm{y}^{\mathrm{2}} <\mathrm{y}^{\mathrm{2}} +\mathrm{4x}+\mathrm{3}<\mathrm{y}^{\mathrm{2}} +\mathrm{4y}+\mathrm{4}=\left(\mathrm{y}+\mathrm{2}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{y}^{\mathrm{2}} +\mathrm{4x}+\mathrm{3}=\left(\mathrm{y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{y}^{\mathrm{2}} +\mathrm{2y}+\mathrm{1} \\ $$$$\mathrm{y}=\mathrm{2x}+\mathrm{1} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}+\mathrm{1}=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}=\mathrm{a}^{\mathrm{2}} \Rightarrow\mathrm{1}+\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{b}^{\mathrm{2}} \\ $$$$\Rightarrow\left(\mathrm{b}+\mathrm{x}+\mathrm{1}\right)=\mathrm{1}\:\mathrm{impossible}\:\mathrm{x}\geqslant\mathrm{1} \\ $$$$\mathrm{x}>\mathrm{y}\Rightarrow\mathrm{x}\geqslant\mathrm{y}+\mathrm{1} \\ $$$$\Rightarrow\mathrm{x}^{\mathrm{2}} <\mathrm{x}^{\mathrm{2}} +\mathrm{y}+\mathrm{1}\leqslant\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}<\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{impossible}.. \\ $$$$ \\ $$
Commented by witcher3 last updated on 31/May/23
$$\mathrm{yes}\:\mathrm{in}\:\mathrm{Hight}\:\mathrm{School}\:\mathrm{Im}\:\mathrm{in}\:\mathrm{Team}\:\mathrm{Select} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{didnt}\:\mathrm{participated}\:\:\mathrm{to}\:\mathrm{many}\:\mathrm{stressed} \\ $$$$\mathrm{in}\:\mathrm{High}\:\mathrm{school}\:\mathrm{final}\:\mathrm{i}\:\mathrm{didnt}\:\mathrm{Go}\:\mathrm{withe}\:\mathrm{the}\:\mathrm{reste} \\ $$$$\mathrm{im}\:\mathrm{still}\:\mathrm{regret}\:\mathrm{this} \\ $$
Commented by York12 last updated on 27/May/23
$${I}\:{can}\:{not}\:{believe}\:{that}\:{this}\:{quesion}\:{was}\:{asked}\:{in}\:{CHMO} \\ $$$${which}\:{is}\:{sometimes}\:{considered}\:{harder}\:{than} \\ $$$${IMO} \\ $$
Commented by witcher3 last updated on 28/May/23
$$\mathrm{IMO}\:\mathrm{is}\:\mathrm{for}\:\mathrm{High}\:\mathrm{school} \\ $$$$\mathrm{but}\:\mathrm{sum}\:\mathrm{Quations}\:\mathrm{Can}\:\mathrm{bee}\:\mathrm{tought} \\ $$$$\mathrm{like}\:\mathrm{IMO}\:\mathrm{2013} \\ $$
Commented by York12 last updated on 28/May/23
$${I}\:{am}\:{a}\:{high}\:{scholar}\:{by}\:{the}\:{way}\: \\ $$$${I}\:{have}\:{not}\:{seen}\:{it}\:{cause}\:{it}\:{is}\:{not}\:{involved} \\ $$$${in}\:{the}\:{compenedium} \\ $$$${so}\:{have}\:{you}\:{participated}\: \\ $$
Commented by York12 last updated on 29/May/23
$${yeah}\:{you}\:{are}\:{right}\: \\ $$$${but}\:{you}\:{have}\:{to}\:{look}\:{at}\:{CHMO}\:{past}\:{papers} \\ $$$$ \\ $$
Commented by York12 last updated on 06/Jun/23
$${i}\:{am}\:{sure}\:{you}\:{would}\:{be}\:{able}\:{to}\:{get}\:{a}\:{golden} \\ $$$${medal} \\ $$