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Category: Algebra

Question-12909

Question Number 12909 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/May/17 Answered by nume1114 last updated on 07/May/17 $$\:\:\:\:{a}^{\mathrm{2}} \left({a}+\mathrm{1}\right)+{b}^{\mathrm{2}} \left({b}+\mathrm{1}\right)+\mathrm{5}{ab} \\ $$$$={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{5}{ab}…

Question-143979

Question Number 143979 by mathdanisur last updated on 20/Jun/21 Answered by mitica last updated on 20/Jun/21 $${p}={x}+{y}+{z};{q}={xy}+{yz}+{xz};{r}={xyz}\Rightarrow{p}^{\mathrm{2}} \geqslant\mathrm{3}{q} \\ $$$$\mathrm{3}\left({p}+\frac{\mathrm{3}{r}}{{q}}\right)^{\mathrm{4}} =\mathrm{3}\left(\frac{{p}}{\mathrm{3}}+\frac{{p}}{\mathrm{3}}+\frac{{p}}{\mathrm{3}}+\frac{\mathrm{3}{r}}{{q}}\right)^{\mathrm{4}} \overset{{am}−{gm}} {\geqslant} \\ $$$$\geqslant\mathrm{3}\left(\mathrm{4}\centerdot\sqrt[{\mathrm{4}}]{\frac{{p}}{\mathrm{3}}\centerdot\frac{{p}}{\mathrm{3}}\centerdot\frac{{p}}{\mathrm{3}}\centerdot\frac{\mathrm{3}{r}}{{q}}}\right)^{\mathrm{4}}…

for-128-x-127-and-127-y-128-where-x-y-Z-Point-P-x-y-is-a-point-on-the-cartesian-plane-From-the-origin-angle-is-made-counter-clockwise-with-the-positive-x-axis-1-How-many-uniqu

Question Number 12883 by FilupS last updated on 05/May/17 $$\mathrm{for}\:\:\:\:\:−\mathrm{128}\leqslant{x}\leqslant\mathrm{127} \\ $$$$\mathrm{and}\:\:\:−\mathrm{127}\leqslant{y}\leqslant\mathrm{128} \\ $$$$\mathrm{where}\:\:\:{x},{y}\in\mathbb{Z} \\ $$$$\: \\ $$$$\mathrm{Point}\:{P}\left({x},{y}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{cartesian}\:\mathrm{plane}. \\ $$$$\: \\ $$$$\mathrm{From}\:\mathrm{the}\:\mathrm{origin},\:\mathrm{angle}\:\theta\:\mathrm{is}\:\mathrm{made}\:\mathrm{counter} \\…

dear-sir-W-Mjs-the-set-1-4-n-have-the-condition-that-if-two-different-elements-are-selected-and-2112-is-added-to-the-result-then-the-result-is-a-perfect-square-if-n-is-a-positif-number-then

Question Number 78399 by john santu last updated on 17/Jan/20 $${dear}\:{sir}\:{W},\:{Mjs}\: \\ $$$${the}\:{set}\:\left\{\mathrm{1},\mathrm{4},{n}\right\}\:{have}\:{the}\:{condition}\:{that}\: \\ $$$${if}\:{two}\:{different}\:{elements}\:{are} \\ $$$${selected}\:{and}\:\mathrm{2112}\:{is}\:{added}\:{to} \\ $$$${the}\:{result}\:,\:{then}\:{the}\:{result}\: \\ $$$${is}\:{a}\:{perfect}\:{square}\:{if}\:{n}\:{is}\:{a}\: \\ $$$${positif}\:{number}\:.\:{then}\:{the}\:{number}\: \\ $$$${of}\:{possible}\:{values}\:{of}\:{n}\:{is}\:…