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Category: Algebra

If-a-b-c-gt-0-and-a-2-b-2-c-2-3-Then-prove-that-a-3-b-2-b-bc-1-

Question Number 154372 by mathdanisur last updated on 17/Sep/21 $$\mathrm{If}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} =\mathrm{3} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\Sigma\:\frac{\mathrm{a}^{\mathrm{3}} }{\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{b}\:+\:\mathrm{bc}}\:\geqslant\:\mathrm{1} \\ $$ Terms of Service Privacy…

x-1-2-y-1-2-27xy-x-2-1-y-2-1-10xy-

Question Number 88811 by jagoll last updated on 13/Apr/20 $$\begin{cases}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{27}{xy}}\\{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({y}^{\mathrm{2}} +\mathrm{1}\right)\:=\mathrm{10}{xy}}\end{cases} \\ $$ Answered by john santu last updated on 13/Apr/20 $${let}\:{me}\:{try}\:…

S-1-1-10-2-1-10-2-4-1-10-4-8-1-10-8-

Question Number 154328 by liberty last updated on 17/Sep/21 $$\:{S}=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{10}}+\frac{\mathrm{2}}{\mathrm{1}+\mathrm{10}^{\mathrm{2}} }+\frac{\mathrm{4}}{\mathrm{1}+\mathrm{10}^{\mathrm{4}} }+\frac{\mathrm{8}}{\mathrm{1}+\mathrm{10}^{\mathrm{8}} }+\ldots\: \\ $$$$ \\ $$ Answered by MJS_new last updated on 17/Sep/21 $$\mathrm{having}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{we}\:\mathrm{could}\:\mathrm{substract}\:\mathrm{the}…

if-a-b-c-gt-0-and-ab-bc-ca-2-then-cyc-a-b-4-c-4-b-3-b-2-c-bc-2-c-3-1-

Question Number 154320 by mathdanisur last updated on 17/Sep/21 $$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=\mathrm{2}\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{a}\left(\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \right)}{\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{2}} \mathrm{c}\:+\:\mathrm{bc}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{3}} }\:\geqslant\:\mathrm{1} \\ $$ Answered by ghimisi last…

If-x-3-1-y-3-y-3-1-z-3-1-Find-xyz-2025-1-

Question Number 154316 by mathdanisur last updated on 17/Sep/21 $$\mathrm{If}\:\:\mathrm{x}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{3}} }\:=\:\mathrm{y}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{3}} }\:=\:\mathrm{1} \\ $$$$\mathrm{Find}\:\:\left(\mathrm{xyz}\right)^{\mathrm{2025}} \:-\:\mathrm{1}\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated on…

If-1-x-n-C-0-C-1-x-C-2-x-2-C-3-x-3-C-n-x-n-then-prove-that-0-i-lt-j-n-i-n-C-i-j-n-C-j-n-2-2-r-0-n-1-n-C-r-

Question Number 23250 by Tinkutara last updated on 28/Oct/17 $${If}\:\left(\mathrm{1}\:+\:{x}\right)^{{n}} \:=\:{C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} {x}\:+\:{C}_{\mathrm{2}} {x}^{\mathrm{2}} \:+\:{C}_{\mathrm{3}} {x}^{\mathrm{3}} \\ $$$$+\:…\:+\:{C}_{{n}} {x}^{{n}} ,\:{then}\:{prove}\:{that} \\ $$$$\underset{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left(\frac{{i}}{\:^{{n}} {C}_{{i}} }\:+\:\frac{{j}}{\:^{{n}}…