Question Number 151101 by mathdanisur last updated on 18/Aug/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left[\frac{\mathrm{x}}{\mathrm{2}}\right]\:+\:\left[\frac{\mathrm{3x}}{\mathrm{5}}\right]\:=\:\left[\frac{\mathrm{x}}{\mathrm{10}}\right]\:+\:\mathrm{x}\:,\:\:\mathrm{where}\:\mathrm{we} \\ $$$$\mathrm{denoting}\:\mathrm{by}\:\left[\boldsymbol{\mathrm{x}}\right]\:\mathrm{the}\:\mathrm{great}\:\mathrm{integer}\:\mathrm{part} \\ $$$$\mathrm{of}\:\boldsymbol{\mathrm{x}}. \\ $$ Answered by dumitrel last updated on 18/Aug/21…
Question Number 151100 by mathdanisur last updated on 18/Aug/21 $$\mathrm{if}\:\:\mathrm{0}\leqslant\mathrm{x};\mathrm{y};\mathrm{z}\leqslant\mathrm{k}\:\:\mathrm{and}\:\:\mathrm{k}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\mathrm{y}\left(\mathrm{x}\:-\:\mathrm{z}\right)\:-\:\mathrm{z}\left(\mathrm{x}\:-\:\mathrm{k}\right)\:\leqslant\:\mathrm{k}^{\mathrm{2}} \\ $$ Answered by dumitrel last updated on 18/Aug/21 $$\Leftrightarrow{y}\left({x}−{z}\right)+{z}\left({k}−{x}\right)\leqslant{k}^{\mathrm{2}} \\ $$$${I}.\:\:{If}\:{x}\leqslant{z}\Rightarrow{y}\left({x}−{z}\right)\leqslant\mathrm{0} \\…
Question Number 85554 by luxlavanish last updated on 22/Mar/20 $$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{7}=\mathrm{0} \\ $$ Commented by naka3546 last updated on 23/Mar/20 $${what}'{s}\:\:{the}\:\:{question}\:? \\ $$ Terms of…
Question Number 151085 by mathdanisur last updated on 18/Aug/21 Answered by talminator2856791 last updated on 18/Aug/21 $$\: \\ $$$$\:\frac{{a}^{\mathrm{2}} −\left({a}+\mathrm{1}\right)^{\mathrm{2}} −\left({a}+\mathrm{2}\right)^{\mathrm{2}} +\left({a}+\mathrm{3}\right)^{\mathrm{2}} }{\left({a}+\mathrm{4}\right)^{\mathrm{2}} −\left({a}+\mathrm{3}\right)^{\mathrm{2}} −\left({a}+\mathrm{2}\right)^{\mathrm{2}}…
Question Number 151081 by malwan last updated on 18/Aug/21 $${if}\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${and}\:{f}\left({x}+\mathrm{3}\right)=\mathrm{7}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\:{find} \\ $$$$\left.\mathrm{1}\left.\right)\:{a}+{b}+{c}\:\:\:\:\:\mathrm{2}\right)\:{a}−{b}+{c} \\ $$ Answered by mr W last updated on…
Question Number 20013 by Tinkutara last updated on 20/Aug/17 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{{x}\:−\:{a}}\:+\:\frac{\mathrm{1}}{{x}\:−\:{b}} \\ $$$$+\:\frac{\mathrm{1}}{{x}\:−\:{c}}\:=\:\mathrm{0}\:\mathrm{can}\:\mathrm{have}\:\mathrm{a}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{equal} \\ $$$$\mathrm{roots}\:\mathrm{if}\:{a}\:=\:{b}\:=\:{c}. \\ $$ Commented by ajfour last updated on 20/Aug/17 $${how}\:{did}\:{u}\:{solve}\:{it}\:,\:{please}\:{let}\:{me} \\…
Question Number 151082 by mathdanisur last updated on 18/Aug/21 Commented by Mokmokhi last updated on 18/Aug/21 $$\mathrm{This}\:\mathrm{can}\:\mathrm{be}\:\mathrm{proven}\:\mathrm{starting}\:\mathrm{from}\:\mathrm{left}. \\ $$$$\mathrm{By}\:\mathrm{substituting}\:{u}=\frac{\pi}{\mathrm{2}}−{x}. \\ $$$$\mathrm{After}\:\mathrm{evaluation}.\:\mathrm{By}\:\mathrm{dummy}\:\mathrm{variables}\:\mathrm{done}. \\ $$ Terms of…
Question Number 151080 by malwan last updated on 18/Aug/21 $${if}\:\left({f}\circ{f}\circ{f}\circ{f}\right)\left({x}\right)=\mathrm{16}{x}+\mathrm{15} \\ $$$${find}\:{f}\left({x}\right) \\ $$ Answered by Mokmokhi last updated on 18/Aug/21 $$\mathrm{It}\:\mathrm{is}\:\mathrm{reasonable}\:\mathrm{to}\:\mathrm{say}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{linear}\:\mathrm{by}\:\mathrm{rejecting}\:\mathrm{other}\:\mathrm{possibilities}. \\ $$$$\mathrm{Then}\:{f}\left({x}\right)={ax}+{b}\:\mathrm{for}\:\mathrm{some}\:\mathrm{unknowns}\:{a}\:\mathrm{and}\:{b}. \\…
Question Number 151078 by mathdanisur last updated on 18/Aug/21 $$\Omega_{\boldsymbol{\mathrm{k}}} =\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\:\centerdot\underset{\boldsymbol{\mathrm{p}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{p}}\end{pmatrix}}{\begin{pmatrix}{\mathrm{n}+\mathrm{k}}\\{\mathrm{n}+\mathrm{p}}\end{pmatrix}}\:\:;\:\:\mathrm{k}\in\mathbb{N}^{\ast} -\mathrm{fixed} \\ $$$$\mathrm{find}\:\:\Omega=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\Omega_{\boldsymbol{\mathrm{n}}-\mathrm{1}} }\:\centerdot\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\sqrt[{\boldsymbol{\mathrm{i}}^{\mathrm{2}} }]{\boldsymbol{\mathrm{i}}!}\: \\ $$ Terms…
Question Number 20001 by Tinkutara last updated on 20/Aug/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic} \\ $$$$\mathrm{equation}\:\mathrm{8sec}^{\mathrm{2}} \theta\:−\:\mathrm{6sec}\theta\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{is} \\ $$ Answered by mrW1 last updated on 20/Aug/17 $$\mathrm{8sec}^{\mathrm{2}} \theta\:−\:\mathrm{6sec}\theta\:+\:\mathrm{1}\:=\:\mathrm{0} \\…