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is-this-true-and-is-there-a-proof-k-N-k-gt-1-n-1-k-2-1-k-n-n-1-k-2-1-k-n-1-2-

Question Number 149342 by MJS_new last updated on 04/Aug/21 $$\mathrm{is}\:\mathrm{this}\:\mathrm{true}\:\mathrm{and}\:\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{proof}? \\ $$$$\forall{k}\in\mathbb{N}\mid{k}>\mathrm{1} \\ $$$$\frac{\underset{{n}=\mathrm{1}} {\overset{{k}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}+\sqrt{{n}}}}{\underset{{n}=\mathrm{1}} {\overset{{k}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}−\sqrt{{n}}}}=\mathrm{1}+\sqrt{\mathrm{2}} \\ $$ Commented by mr…

How-many-ordered-pairs-a-b-with-b-lt-a-lt-100-a-b-N-such-that-a-b-both-a-1-b-1-are-integers-

Question Number 149323 by naka3546 last updated on 04/Aug/21 $${How}\:\:{many}\:\:{ordered}\:\:{pairs}\:\left({a},{b}\right)\:{with}\:\:{b}\:<\:{a}\:<\:\mathrm{100}\:\:,\:\:{a},{b}\:\in\:\mathbb{N}\:\:{such}\:\:{that}\:\:\frac{{a}}{{b}}\:\:{both}\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\:\:{are}\:\:{integers}\:\:.\: \\ $$ Answered by Olaf_Thorendsen last updated on 05/Aug/21 $$\mathrm{N}\:=\:\underset{{b}=\mathrm{1}} {\overset{\mathrm{98}} {\sum}}\underset{{a}={b}+\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\lfloor\frac{\lfloor\frac{{a}}{{b}}\rfloor}{\frac{{a}}{{b}}}×\frac{\lfloor\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\rfloor}{\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}}\rfloor \\…

lim-x-x-6-5x-3-x-

Question Number 149317 by tabata last updated on 04/Aug/21 $${lim}_{{x}\rightarrow\infty} \sqrt{{x}^{\mathrm{6}} +\mathrm{5}{x}^{\mathrm{3}} }−{x} \\ $$ Answered by EDWIN88 last updated on 04/Aug/21 $$\left.\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{{x}^{\mathrm{6}} \left(\mathrm{1}+\mathrm{5}{x}^{−\mathrm{3}}…

Find-all-real-solutions-of-x-y-such-that-x-3y-4-x-y-5-y-3x-5-x-y-7-

Question Number 83756 by naka3546 last updated on 05/Mar/20 $${Find}\:\:{all}\:\:{real}\:\:{solutions}\:\:{of}\:\:\left({x},\:{y}\right)\:\:{such}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:{x}\:+\:\mathrm{3}{y}\:+\:\frac{\mathrm{4}}{{x}\:+\:{y}}\:\:=\:\:\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:{y}\:+\:\mathrm{3}{x}\:+\:\frac{\mathrm{5}}{{x}\:+\:{y}}\:\:=\:\:\mathrm{7} \\ $$ Commented by Prithwish Sen 1 last updated on 05/Mar/20…

Let-a-b-c-be-positive-real-numbers-with-sum-3-Prove-that-1-a-1-b-1-c-3-2a-2-bc-3-2b-2-ac-3-2c-2-ab-

Question Number 149244 by EDWIN88 last updated on 04/Aug/21 $$\:\:{Let}\:{a},{b},{c}\:{be}\:{positive}\:{real}\:{numbers} \\ $$$${with}\:{sum}\:\mathrm{3}.\:{Prove}\:{that}\: \\ $$$$\:\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}{a}^{\mathrm{2}} +{bc}}+\frac{\mathrm{3}}{\mathrm{2}{b}^{\mathrm{2}} +{ac}}+\frac{\mathrm{3}}{\mathrm{2}{c}^{\mathrm{2}} +{ab}} \\ $$ Terms of Service Privacy Policy Contact:…