Question Number 178136 by Acem last updated on 13/Oct/22 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Journey}}\:\boldsymbol{{inside}}\:\boldsymbol{{a}}\:\boldsymbol{{regular}}\:\boldsymbol{{hexagon}} \\ $$$$\:{The}\:{operation}\:{is}\:{to}\:{connect}\:{three}\:{dots}\:{of} \\ $$$$\:{a}\:{regular}\:{hexagon}'{s}\:{heads}. \\ $$$$\mathrm{1}\bullet\:{How}\:{many}\:{types}\:{of}\:{geometric}\:{shapes}\:{will}\:{we}\:{get}? \\ $$$$\mathrm{2}\bullet\:{How}\:{many}\:{each}\:{type}? \\ $$$$ \\ $$ Terms of Service…
Question Number 178137 by Acem last updated on 13/Oct/22 $${Let}\:{f}\left({x}\right)=\:\left({ax}+\mathrm{1}\right)^{\mathrm{5}} .\left(\mathrm{1}+{bx}\right)^{\mathrm{4}} \:;\:{a},{b}\:\in\:\mathbb{N} \\ $$$$\:{if}\:{times}\:{of}\:{x}\:{equal}\:\mathrm{62}\:{so}\:{what}\:{are}\:{possible}\:{values} \\ $$$$\:{of}\:{the}\:{sum}\:{a},\:{b}? \\ $$$$ \\ $$ Commented by Rasheed.Sindhi last updated…
Question Number 178134 by Acem last updated on 13/Oct/22 $${Am}\:{not}\:{a}\:{friend}\:{with}\:{this}\:{isssue}: \\ $$$$\mathrm{6}\:{red},\:\mathrm{1}\:{black}\:{and}\:\mathrm{3}\:{white}\:{balls} \\ $$$${We}\:{draw}\:\mathrm{3}\:{balls}\:{in}\:{a}\:{raw},\:{returning}\:{the} \\ $$$$\:{drawn}\:{ball}\:{each}\:{time}. \\ $$$${How}\:{many}\:{different}\:{results}\:{which}\:{include} \\ $$$$\:\boldsymbol{{at}}\:\boldsymbol{{least}}\:{one}\:{black}\:{ball}. \\ $$$$ \\ $$$$\:{Way}_{\mathrm{1}} :\:\mathrm{1000}−\mathrm{9}^{\mathrm{3}}…
Question Number 178113 by Acem last updated on 12/Oct/22 $$\:\mathrm{8}\:{yellow},\:\mathrm{8}\:{red},\:\mathrm{8}\:{green}\:{and}\:\mathrm{8}\:{blue}\:{cards},\: \\ $$$${each}\:{grouo}\:{is}\:{numbered}\:{from}\:\mathrm{1}\:{to}\:\mathrm{8}\:. \\ $$$$\:{We}\:{want}\:{to}\:{know}\:{how}\:{many}\:{pulls}\:{contain} \\ $$$$\:{at}\:{least}\:{one}\:{card}\:{with}\:{number}\:\mathrm{1}\:{through} \\ $$$$\:{pull}\:{processes}\:\mathrm{5}\:{cards}\:{each}\:{time} \\ $$$$ \\ $$$$ \\ $$ Answered…
Question Number 47019 by ajfour last updated on 04/Nov/18 Commented by ajfour last updated on 04/Nov/18 $${If}\:{at}\:{its}\:{roots}\:\:{x}=−\mathrm{3}\:{and}\:{x}=\mathrm{2} \\ $$$${the}\:{biquadratic}\:{also}\:{has}\:{its} \\ $$$${points}\:{of}\:{inflexion},\:{find}\:{its} \\ $$$${other}\:{two}\:{roots}\:\alpha\:{and}\:\beta. \\ $$…
Question Number 178073 by infinityaction last updated on 12/Oct/22 Commented by Rasheed.Sindhi last updated on 12/Oct/22 $$\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{1}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} −\mathrm{1}}+…+\frac{\mathrm{1}}{\mathrm{2}^{{n}} −\mathrm{1}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}−\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{4}−\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{8}−\mathrm{1}}+…+\frac{\mathrm{1}}{\mathrm{2}^{{n}} −\mathrm{1}} \\…
Question Number 112528 by dw last updated on 08/Sep/20 $${If}\:\:\:\:\:\frac{{n}−\mathrm{7}}{{n}−\mathrm{17}}\:\:\:\:\:\:\:{and}\:\:\:\:\frac{{n}−\mathrm{14}}{{n}−\mathrm{24}}\:\:\:{are}\:{integers}\:{numbers} \\ $$$${n}=? \\ $$ Commented by Dwaipayan Shikari last updated on 08/Sep/20 $$\mathrm{1}+\frac{\mathrm{10}}{{n}−\mathrm{17}}\:\:\:\:\:\:\:{n}\left(\mathrm{7},\mathrm{12},\mathrm{18},\mathrm{22},\mathrm{27}\right) \\ $$$$\frac{{n}−\mathrm{10}}{{n}−\mathrm{24}}=\mathrm{1}+\frac{\mathrm{14}}{{n}−\mathrm{24}}\:\:\:\:{n}\left(\mathrm{10},\mathrm{31},\mathrm{38}\right)…
Question Number 178054 by Acem last updated on 12/Oct/22 $${Mr}.\:{w}\:{wants}\:{to}\:{distribute}\:{n}+\mathrm{1}\:{different}\: \\ $$$${prizes}\:\:{to}\:{n}\:{friends}\:{so}\:{that}\:{each}\:{one}\:{gets} \\ $$$$\:{at}\:{least}\:{one}\:{prize},\:{how}\:{many}\:{results} \\ $$$$\:{of}\:{this}\:{process}.\:{I}\:{hope}\:{to}\:{be}\:{that}\:{one}\:{gets}\:\mathrm{2} \\ $$$$ \\ $$ Commented by mr W last…
Question Number 46978 by rahul 19 last updated on 03/Nov/18 Answered by MrW3 last updated on 03/Nov/18 $$\frac{{q}}{{p}}=\mathrm{4}−{p}\Rightarrow{p}=\mathrm{4}−\frac{{q}}{{p}} \\ $$$$\frac{{r}}{{q}}=\mathrm{4}−{q}\Rightarrow{q}=\mathrm{4}−\frac{{r}}{{q}} \\ $$$$\frac{{p}}{{r}}=\mathrm{4}−{r}\Rightarrow{r}=\mathrm{4}−\frac{{p}}{{r}} \\ $$$$\Rightarrow{p}+{q}+{r}=\mathrm{12}−\left(\frac{{q}}{{p}}+\frac{{r}}{{q}}+\frac{{p}}{{r}}\right)\leqslant\mathrm{12}−\mathrm{3}×\:^{\mathrm{3}} \sqrt{\frac{{q}}{{p}}×\frac{{r}}{{q}}×\frac{{p}}{{r}}}=\mathrm{12}−\mathrm{3}=\mathrm{9}…
Question Number 46974 by rahul 19 last updated on 03/Nov/18 $${Number}\:{of}\:{integers}\:{n}\:{for}\:{which}\: \\ $$$$\mathrm{3}{x}^{\mathrm{3}} −\mathrm{25}{x}+{n}=\mathrm{0}\:{has}\:{three}\:{real}\:{roots}\:{is}\:? \\ $$$$ \\ $$ Commented by rahul 19 last updated on…