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

$$\mathit{I}\mathit{f}\mathit{x}\mathrm{\&}\mathit{y}\mathit{a}\mathit{r}\mathit{e}\mathit{b}\mathit{o}\mathit{t}\mathit{h}\mathit{p}\mathit{o}\mathit{s}\mathit{i}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{e}\mathit{r}\mathit{s}\mathit{t}\mathit{h}\mathit{e}\mathit{n}$$$$\mathit{t}\mathit{h}\mathit{e}\mathit{n}\mathit{s}\mathit{h}\mathit{o}\mathit{w}\mathit{i}\mathit{f}\mathit{i}\mathit{t}\mathit{p}\mathit{o}\mathit{s}\mathit{s}\mathit{i}\mathit{p}\mathit{l}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}{\mathit{x}}^2+\mathit{y}+1\mathrm{\&}{\mathit{y}}^2+4\mathit{x}+3\mathit{b}\mathit{e}\mathit{b}\mathit{o}\mathit{t}\mathit{h}$$$$\mathit{p}\mathit{e}\mathit{r}\mathit{f}\mathit{e}\mathit{c}\mathit{t}\mathit{s}\mathit{q}\mathit{u}\mathit{a}\mathit{r}\mathit{e}\mathit{s}\mathit{s}\mathit{i}\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{a}\mathit{n}\mathit{e}\mathit{o}\mathit{u}\mathit{s}\mathit{l}\mathit{y}.$$

Answered by witcher3 last updated on 27/May/23

$$\mathrm{x}=\mathrm{y}\Rightarrow$$$${\mathrm{x}}^2+\mathrm{x}+1={\mathrm{a}}^2$$$${\mathrm{x}}^2<{\mathrm{x}}^2+\mathrm{x}+1={\mathrm{a}}^2<{(\mathrm{x}+1)}^2..\mathrm{impossibl}$$$$\mathrm{x}<\mathrm{y}\Rightarrow \Rightarrow \mathrm{x}⩽\mathrm{y}+1$$$${\mathrm{y}}^2<{\mathrm{y}}^2+4\mathrm{x}+3<{\mathrm{y}}^2+4\mathrm{y}+4={(\mathrm{y}+2)}^2$$$$\Rightarrow {\mathrm{y}}^2+4\mathrm{x}+3={(\mathrm{y}+1)}^2={\mathrm{y}}^2+2\mathrm{y}+1$$$$\mathrm{y}=2\mathrm{x}+1$$$${\mathrm{x}}^2+\mathrm{y}+1={(\mathrm{x}+1)}^2+1={\mathrm{a}}^2\Rightarrow 1+{(\mathrm{x}+1)}^2={\mathrm{b}}^2$$$$\Rightarrow (\mathrm{b}+\mathrm{x}+1)=1\mathrm{impossible}\mathrm{x}⩾1$$$$\mathrm{x}>\mathrm{y}\Rightarrow \mathrm{x}⩾\mathrm{y}+1$$$$\Rightarrow {\mathrm{x}}^2<{\mathrm{x}}^2+\mathrm{y}+1⩽{\mathrm{x}}^2+\mathrm{x}+1<{(\mathrm{x}+1)}^2$$$$\mathrm{impossible}..$$$$$$

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

$$IcannotbelievethatthisquesionwasaskedinCHMO$$$$whichissometimesconsideredharderthan$$$$IMO$$

Commented by witcher3 last updated on 28/May/23

$$\mathrm{IMO}\mathrm{is}\mathrm{for}\mathrm{High}\mathrm{school}$$$$\mathrm{but}\mathrm{sum}\mathrm{Quations}\mathrm{Can}\mathrm{bee}\mathrm{tought}$$$$\mathrm{like}\mathrm{IMO}2013$$

Commented by York12 last updated on 28/May/23

$$Iamahighscholarbytheway$$$$Ihavenotseenitcauseitisnotinvolved$$$$inthecompenedium$$$$sohaveyouparticipated$$

Commented by York12 last updated on 29/May/23

$$yeahyouareright$$$$butyouhavetolookatCHMOpastpapers$$$$$$

Commented by York12 last updated on 06/Jun/23

$$iamsureyouwouldbeabletogetagolden$$$$medal$$

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