Question Number 19709 by NECC last updated on 14/Aug/17 $${In}\:{the}\:{cyclic}\:{quadrilateral}\:{ABCD} \\ $$$${AB}=\mathrm{7},{BC}=\mathrm{8},{CD}=\mathrm{8},{DA}=\mathrm{15}. \\ $$$${Calculate}\:{the}\:{angle}\:{ADC}\:{and} \\ $$$${the}\:{length}\:{ofAC}. \\ $$ Answered by Tinkutara last updated on 15/Aug/17…
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Question Number 85241 by Power last updated on 20/Mar/20 Answered by jagoll last updated on 20/Mar/20 $$\mathrm{minimum}\:\frac{\mathrm{4s}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}}\:=\:\frac{\mathrm{4s}^{\mathrm{2}} }{\mathrm{5st}}\:+\:\frac{\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}} \\ $$$$=\:\frac{\mathrm{4}}{\mathrm{5}}\left(\frac{\mathrm{s}}{\mathrm{t}}\right)\:+\:\frac{\mathrm{1}}{\mathrm{5}}\left(\frac{\mathrm{t}}{\mathrm{s}}\right). \\ $$$$\mathrm{let}\:\frac{\mathrm{s}}{\mathrm{t}}\:=\:\mathrm{u}\:\Rightarrow\mathrm{f}\left(\mathrm{u}\right)\:=\:\frac{\mathrm{4}}{\mathrm{5}}\mathrm{u}+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{u}^{−\mathrm{1}}…
Question Number 19704 by Tinkutara last updated on 14/Aug/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\left(\mathrm{in}\:\mathrm{base}\:\mathrm{10}\right)\:\mathrm{of}\:\mathrm{all}\:\mathrm{the} \\ $$$$\mathrm{natural}\:\mathrm{numbers}\:\mathrm{less}\:\mathrm{than}\:\mathrm{64}\:\mathrm{which} \\ $$$$\mathrm{have}\:\mathrm{exactly}\:\mathrm{three}\:\mathrm{ones}\:\mathrm{in}\:\mathrm{their}\:\mathrm{base}\:\mathrm{2} \\ $$$$\mathrm{representation}? \\ $$ Answered by mrW1 last updated on 15/Aug/17…
Question Number 85238 by naka3546 last updated on 20/Mar/20 $${f}\left({x}\right)\:\:=\:\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x} \\ $$$${f}\:^{−\mathrm{1}} \left({x}\right)\:\:=\:\:? \\ $$ Answered by mr W last updated on 20/Mar/20 $${y}={x}^{\mathrm{3}}…
Question Number 85236 by jagoll last updated on 20/Mar/20 $$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{if}\: \\ $$$$\mathrm{f}\:'\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)\:=\:\mathrm{2x}+\mathrm{1} \\ $$ Commented by mathmax by abdo last updated on 20/Mar/20 $${its}\:{clear}\:{that}\:{f}\:{is}\:{polynomial}\:{let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{0}}…
Question Number 19700 by Tinkutara last updated on 14/Aug/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$${k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{2013}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:{k}\:\mathrm{consecutive}\:\mathrm{positive}\:\mathrm{integers}? \\ $$ Answered by mrW1 last updated on 15/Aug/17 $$\mathrm{keep}\:\mathrm{in}\:\mathrm{mind}:\:\mathrm{2013}=\mathrm{3}×\mathrm{11}×\mathrm{61} \\…
Question Number 150769 by mathdanisur last updated on 15/Aug/21 $$\frac{\mathrm{1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} +\mathrm{4}^{\mathrm{3}} +…+\boldsymbol{\mathrm{x}}^{\mathrm{3}} }{\mathrm{1}\centerdot\mathrm{4}+\mathrm{2}\centerdot\mathrm{7}+\mathrm{3}\centerdot\mathrm{10}+…+\boldsymbol{\mathrm{x}}\left(\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{1}\right)}\:=\:\mathrm{2021} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$ Answered by nimnim last updated on…
Question Number 19699 by Tinkutara last updated on 14/Aug/17 $$\mathrm{Let}\:{S}\:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:{O}.\:\mathrm{A}\:\mathrm{chord} \\ $$$${AB},\:\mathrm{not}\:\mathrm{a}\:\mathrm{diameter},\:\mathrm{divides}\:{S}\:\mathrm{into}\:\mathrm{two} \\ $$$$\mathrm{regions}\:{R}_{\mathrm{1}} \:\mathrm{and}\:{R}_{\mathrm{2}} \:\mathrm{such}\:\mathrm{that}\:{O}\:\mathrm{belongs} \\ $$$$\mathrm{to}\:{R}_{\mathrm{2}} .\:\mathrm{Let}\:{S}_{\mathrm{1}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:\mathrm{in} \\ $$$${R}_{\mathrm{1}} ,\:\mathrm{touching}\:{AB}\:\mathrm{at}\:{X}\:\mathrm{and}\:{S}\:\mathrm{internally}. \\ $$$$\mathrm{Let}\:{S}_{\mathrm{2}}…
Question Number 19698 by Tinkutara last updated on 14/Aug/17 $$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}\:+\:{b}\:\mathrm{and}\:{g}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:+ \\ $$$${bx}\:−\:\mathrm{3},\:\mathrm{where}\:{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{b}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equations}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{and}\:{g}\left({x}\right) \\ $$$$=\:\mathrm{0}\:\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{root}? \\ $$ Answered by ajfour…