Category: Algebra

Question Number 84871 by tw000001 last updated on 17/Mar/20 $$\mathrm{If}\mathrm{you}\mathrm{know}$$$$\left(\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}\right)^{\mathrm{2}} +\left(\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ca}}\right)^{\mathrm{2}} +\left(\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}\right)^{\mathrm{2}} =\mathrm{3}, \…

Question Number 19332 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Let}{S}_n=n^2+20n+12,n\mathrm{a}\mathrm{positive}$$$$\mathrm{integer}.\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{all}\mathrm{possible}$$$$\mathrm{values}\mathrm{of}n\mathrm{for}\mathrm{which}{S}_n\mathrm{is}\mathrm{a}\mathrm{perfect}$$$$\mathrm{square}?$$ Commented by mrW1 last…

Question Number 19329 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Solve}\mathrm{the}\mathrm{equation}{y}^3=x^3+8x^2-6x+8$$$$\mathrm{for}\mathrm{positive}\mathrm{integers}x\mathrm{and}y.$$ Commented by Tinkutara last updated on 12/Aug/17 $$\mathrm{RMO}\:\mathrm{2000}…

Question Number 19312 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Product}\mathrm{of}n,n^{\mathrm{th}}\mathrm{roots}\mathrm{of}\mathrm{unity}$$$$=1.\alpha .{\alpha }^2.{\alpha }^3\dots ..{\alpha }^{n-1}={(-1)}^{n-1}$$$$\mathrm{Why}?\mathrm{How}\mathrm{to}\mathrm{get}\mathrm{RHS}?$$ Answered by ajfour last updated…

Question Number 84835 by Power last updated on 16/Mar/20 Commented by abdomathmax last updated on 16/Mar/20 $$S=\sum _{k=1}^{32}\frac{k}{k+1}=\sum _{k=1}^{32}\frac{k+1-1}{k+1}$$$$=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \left(\mathrm{1}\right)−\sum_{{k}=\mathrm{1}}…