Question Number 169768 by mnjuly1970 last updated on 08/May/22 $$\:\:\: \\ $$$$\:\:\:\left[{x}\right]+\left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]+\left[{x}−\frac{\mathrm{1}}{\mathrm{3}}\right]=\:\mathrm{8} \\ $$$$\:\:\:\:\:\left[\mathrm{2}{x}\right]=\mathrm{8}−\left[{x}−\frac{\mathrm{1}}{\mathrm{3}}\right]={k} \\ $$$$\:\:\:\:\:\:\:\:\:\frac{{k}}{\mathrm{2}}\leqslant{x}<\frac{{k}+\mathrm{1}}{\mathrm{2}}\:,\:\:\:\:\:\mathrm{8}−{k}+\frac{\mathrm{1}}{\mathrm{3}}\leqslant{x}\:<\mathrm{9}−{k}+\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\frac{{k}}{\mathrm{2}}\:<\:\frac{\mathrm{28}−\mathrm{3}{k}}{\mathrm{3}}\:\:\Rightarrow\:\mathrm{9}{k}\:<\:\mathrm{56} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{k}\:<\frac{\mathrm{56}}{\mathrm{9}}\:\:\:\:\bigstar \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{25}}{\mathrm{3}}\:−{k}<\frac{{k}+\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:\mathrm{50}−\mathrm{6}{k}<\mathrm{3}{k}+\mathrm{3} \\ $$$$\:\:\:\:\Rightarrow\:\:\:\:\:\frac{\mathrm{47}}{\mathrm{9}}\:<{x}\:\:\bigstar\bigstar \\…
Question Number 104235 by bemath last updated on 20/Jul/20 $${If}\:{you}\:{are}\:{given}\:{a}\:{triangle} \\ $$$${with}\:{side}\:{length}\:\mathrm{15}\:,\:\mathrm{20}\:{and} \\ $$$$\mathrm{25}.\:{what}\:{is}\:{the}\:{triangle}'{s} \\ $$$${shortest}\:{altitude}? \\ $$ Answered by nimnim last updated on 20/Jul/20…
Question Number 38696 by Sr@2004 last updated on 28/Jun/18 Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jun/18 $$\frac{\mathrm{1}}{{b}+{c}}+\frac{\mathrm{1}}{{c}+{a}}=\frac{\mathrm{2}}{{a}+{b}} \\ $$$$\frac{\mathrm{1}}{{b}+{c}}−\frac{\mathrm{1}}{{a}+{b}}=\frac{\mathrm{1}}{{a}+{b}}−\frac{\mathrm{1}}{{c}+{a}} \\ $$$$\frac{{a}+{b}−{b}−{c}}{\left({a}+{b}\right)\left({b}+{c}\right)}=\frac{{c}+{a}−{a}−{b}}{\left({a}+{b}\right)\left({c}+{a}\right)} \\ $$$$\frac{{a}−{c}}{{b}+{c}}=\frac{{c}−{b}}{{c}+{a}} \\ $$$${a}^{\mathrm{2}}…
Question Number 169760 by Shrinava last updated on 07/May/22 $$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}} \\ $$$$\left(\mathrm{5}\:-\:\mathrm{2x}\right)\left(\mathrm{5}\:+\:\mathrm{2}\:\sqrt{\mathrm{4}\:-\:\mathrm{2x}^{\mathrm{2}} }\right)\:=\:\mathrm{25} \\ $$ Answered by MJS_new last updated on 08/May/22 $$\mathrm{no}\:“\mathrm{nice}''\:\mathrm{solution},\:\mathrm{we}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\ $$$$\mathrm{I}\:\mathrm{found}…
Question Number 104201 by I want to learn more last updated on 20/Jul/20 $$\mathrm{If}\:\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\:=\:\:\mathrm{1}\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\mathrm{b}\:\:+\:\:\mathrm{ic}\:\:=\:\:\left(\mathrm{1}\:\:+\:\:\mathrm{a}\right)\mathrm{z}, \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\:\:\:\:\:\frac{\mathrm{a}\:\:+\:\:\mathrm{ib}}{\mathrm{i}\:\:+\:\:\mathrm{c}}\:\:\:=\:\:\:\frac{\mathrm{1}\:\:+\:\:\mathrm{iz}}{\mathrm{1}\:\:−\:\:\mathrm{iz}} \\ $$ Terms of Service Privacy…
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Question Number 104190 by mr W last updated on 19/Jul/20 $${integers}\:{x},{y}\:{satisfy}\:\mathrm{2}{x}+\mathrm{15}{y}=\mathrm{2019}. \\ $$$${find}\:{the}\:{minimum}\:{of}\:\mid{y}−{x}\mid. \\ $$ Commented by mr W last updated on 20/Jul/20 $${thank}\:{you}\:{all}! \\…
Question Number 169713 by mathlove last updated on 06/May/22 Commented by Shrinava last updated on 06/May/22 $$\mathrm{E}\:=\:\mathrm{2018} \\ $$ Commented by mathlove last updated on…
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Question Number 38636 by pieroo last updated on 27/Jun/18 $$\mathrm{Three}\:\mathrm{variables}\:\mathrm{u},\mathrm{v}\:\mathrm{and}\:\mathrm{w}\:\mathrm{are}\:\mathrm{related}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{u}\:\mathrm{varies}\:\mathrm{directly}\:\mathrm{as}\:\mathrm{v}\:\mathrm{and}\:\mathrm{inversely}\:\mathrm{as}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\: \\ $$$$\mathrm{w}.\:\mathrm{If}\:\mathrm{v}\:\mathrm{increases}\:\mathrm{by}\:\mathrm{15\%}\:\mathrm{and}\:\mathrm{w}\:\mathrm{decreased}\:\mathrm{by}\:\mathrm{10\%}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{percentage}\:\mathrm{change}\:\mathrm{in}\:\mathrm{u}. \\ $$ Answered by MrW3 last updated on 28/Jun/18…