Question Number 158444 by HongKing last updated on 04/Nov/21
$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{y}}\:\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{with} \\ $$ $$\frac{\mathrm{2010}}{\mathrm{2011}}\:<\:\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}}\:<\:\frac{\mathrm{2011}}{\mathrm{2012}}\:\:\mathrm{then}\:\mathrm{compute}\:\mathrm{the} \\ $$ $$\mathrm{minimum}\:\mathrm{value}\:\mathrm{for}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\:\:\mathrm{and}\:\mathrm{the} \\ $$ $$\mathrm{values}\:\mathrm{of}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{y}}\:\:\mathrm{which}\:\mathrm{achieves} \\ $$ $$\mathrm{this}\:\mathrm{minimum} \\ $$
Commented bymr W last updated on 04/Nov/21
$$\frac{{x}}{{y}}=\frac{\mathrm{4021}}{\mathrm{4023}} \\ $$
Commented byHongKing last updated on 04/Nov/21
$$\mathrm{how}\:\mathrm{my}\:\mathrm{dear}\:\boldsymbol{\mathrm{S}}\mathrm{er}\:\mathrm{solution}\:\mathrm{please} \\ $$
Commented byRasheed.Sindhi last updated on 04/Nov/21
$${Sir}\:{mr}\:{W}\:,\:{when}\:{you}\:{have}\:{some}\:{time} \\ $$ $${pl}\:{see}\:{my}\:{answer}\:{to}\:{Q}#\mathrm{158124}. \\ $$ $${Actually}\:{my}\:{answer}\:{doesn}'{t}\:{match} \\ $$ $${the}\:{answer}\:{of}\:{the}\:{questioner}... \\ $$ $$\mathcal{T}{hanks}\:{in}\:{advance}\:{sir}! \\ $$
Commented bymr W last updated on 04/Nov/21
$${i}\:{have}\:{checked}.\:{your}\:{answer}\:{is}\:{correct}. \\ $$
Commented byRasheed.Sindhi last updated on 05/Nov/21
$$\mathcal{THANKS}\:\mathcal{A}\:\mathcal{LOT}\:\:\:\mathcal{SIR}! \\ $$
Answered by mr W last updated on 04/Nov/21
$${x}\:{must}\:{fulfill} \\ $$ $$\lceil\frac{\mathrm{2011}{x}}{\mathrm{2010}}\rceil−\lfloor\frac{\mathrm{2012}{x}}{\mathrm{2011}}\rfloor\geqslant\mathrm{2} \\ $$ $${we}\:{get}\:{x}_{{min}} =\mathrm{4021} \\ $$ $${y}_{{min}} =\lfloor\frac{\mathrm{2012}×\mathrm{4021}}{\mathrm{2011}}\rfloor+\mathrm{1}=\mathrm{4023} \\ $$ $$\left({x}+{y}\right)_{{min}} =\mathrm{4021}+\mathrm{4023}=\mathrm{8044} \\ $$ $$ \\ $$ $$\frac{{x}}{{y}}=\frac{\mathrm{4021}}{\mathrm{4023}},\:\frac{\mathrm{6031}}{\mathrm{6034}},\:\frac{\mathrm{6032}}{\mathrm{6035}},\:\frac{\mathrm{8041}}{\mathrm{8045}},\:... \\ $$
Commented byRasheed.Sindhi last updated on 06/Nov/21
$$\frac{\mathrm{2010}}{\mathrm{2011}}\:<\:\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}}\:<\:\frac{\mathrm{2011}}{\mathrm{2012}} \\ $$ $${Sir}\:{in}\:{this}\:{case}\:: \\ $$ $${x}_{{min}} =\mathrm{2010}+\mathrm{2011}=\mathrm{4021} \\ $$ $${y}_{{min}} =\mathrm{2011}+\mathrm{2012}=\mathrm{4023} \\ $$ $${Is}\:{it}\:{a}\:{coincidence}\: \\ $$ $${or}\:{generally}\:{if}\:\frac{{a}}{{b}}<\frac{{x}}{{y}}<\frac{{c}}{{d}} \\ $$ $${then}\:\:\:\:\:{x}_{{min}} ={a}+{b}\:\&\:{y}_{{min}} ={c}+{d} \\ $$ $${and}\:\left({x}+{y}\right)_{{min}} ={a}+{b}+{c}+{d}\:\:? \\ $$
Commented bymr W last updated on 06/Nov/21
$${a}\:{very}\:{nice}\:{thought}\:{sir}! \\ $$ $${but}\:{i}\:{think}\:{here}\:{it}'{s}\:{just}\:{a}\:{coincidence}. \\ $$ $$ \\ $$ $${with}\:{x}={a}+{c}\:{and}\:{y}={b}+{d} \\ $$ $${it}\:{fulfills}\:{indeed} \\ $$ $$\frac{{a}}{{b}}<\frac{{x}}{{y}}<\frac{{c}}{{d}} \\ $$ $${but}\:{x}={a}+{c}\:{mustn}'{t}\:{be}\:{x}_{{min}} \:{and} \\ $$ $${y}={b}+{d}\:{mustn}'{t}\:{be}\:{y}_{{min}} .\:{for}\:{example} \\ $$ $${when}\:{gcd}\left({a}+{c},{b}+{d}\right)\neq\mathrm{1}. \\ $$
Commented bymr W last updated on 06/Nov/21
$${an}\:{example} \\ $$ $$\frac{\mathrm{13}}{\mathrm{75}}<\frac{{x}}{{y}}<\frac{\mathrm{9}}{\mathrm{50}} \\ $$ $${x}_{{min}} =\mathrm{3},\:{y}_{{min}} =\mathrm{17} \\ $$ $$\frac{\mathrm{13}}{\mathrm{75}}<\frac{\mathrm{3}}{\mathrm{17}}<\frac{\mathrm{9}}{\mathrm{50}} \\ $$
Commented byRasheed.Sindhi last updated on 06/Nov/21
ㄒ卄卂几Ҝ丂 爪尺 山 丂丨尺!