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Question Number 111721 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{b}>\mathrm{1},\mathrm{x}>\mathrm{0}\:\mathrm{and}\:\left(\mathrm{2x}\right)^{\mathrm{log}_{\mathrm{b}} \mathrm{2}} −\left(\mathrm{3x}\right)^{\mathrm{log}_{\mathrm{b}} \mathrm{3}} =\mathrm{0}, \\ $$$$\mathrm{then}\:\mathrm{x}\:\mathrm{is} \\ $$
Question Number 111671 Answers: 0 Comments: 0
Question Number 111668 Answers: 0 Comments: 6
Question Number 111650 Answers: 1 Comments: 2
Question Number 111644 Answers: 3 Comments: 4
$${y}'+{y}+\mathrm{7}=\mathrm{0} \\ $$
Question Number 111643 Answers: 1 Comments: 0
$$\left(\frac{\mathrm{7}}{\mathrm{3}}\right)!\left({with}\:{out}\:{calculator}\right) \\ $$
Question Number 111641 Answers: 0 Comments: 0
Question Number 111640 Answers: 0 Comments: 0
Question Number 111635 Answers: 0 Comments: 0
$$ \\ $$
Question Number 111624 Answers: 3 Comments: 0
$${if}\:{x}\:{is}\:{a}\:{cube}\:{root}\:{of}\:{a}\:{unity} \\ $$$${prove}\:{that}\: \\ $$$$\left(\mathrm{1}−{x}\right)^{\mathrm{6}} =−\mathrm{27} \\ $$
Question Number 111623 Answers: 1 Comments: 2
$$\left({x}−\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\geqslant\:\mathrm{0} \\ $$
Question Number 111622 Answers: 1 Comments: 0
$${show}\:{that}\:{the}\:{close}\:{form}\:{of}\: \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\left(−\mathrm{1}\right)^{{k}+{i}} }{\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}{i}+\mathrm{2}\right)}\right]\right)=\frac{\mathrm{1}}{\mathrm{8}}\mathrm{ln2} \\ $$
Question Number 111620 Answers: 1 Comments: 0
$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\Gamma\left({x}\right){dx}=\mathrm{ln}\sqrt{\mathrm{2}\pi} \\ $$
Question Number 111619 Answers: 1 Comments: 1
$${solve}\:{for}\:{x}\:{in}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{4}{x}=\sqrt{\mathrm{40}{x}^{\mathrm{2}} +\mathrm{8}{x}−\mathrm{16}} \\ $$
Question Number 111618 Answers: 0 Comments: 0
$${solve}\:{the}\:{fredholm}\:{integral}\:{equation}\:{of} \\ $$$${the}\:{second}\:{kind}\: \\ $$$${y}\left({x}\right)={x}+\lambda\int_{\mathrm{0}} ^{\mathrm{1}} \left({xt}^{\mathrm{2}} +{x}^{\mathrm{2}} {t}\right){y}\left({t}\right){dt} \\ $$
Question Number 111617 Answers: 2 Comments: 0
$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{ln}\left(\mathrm{2cos}^{\mathrm{2}} {x}\right)}{\mathrm{cos2}{x}}{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{16}} \\ $$
Question Number 111616 Answers: 2 Comments: 0
$${solve}\:{the}\:{integral}\:{problem} \\ $$$$\int\frac{\mathrm{cos}{x}}{\mathrm{2}−\mathrm{sin2}{x}}{dx} \\ $$
Question Number 111615 Answers: 2 Comments: 0
$${solve} \\ $$$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\right)\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{3}}} \\ $$
Question Number 111614 Answers: 0 Comments: 0
$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{sin}{x}}{{x}}\right)^{\mathrm{7}} =\frac{\mathrm{5887}\pi}{\mathrm{23040}} \\ $$
Question Number 111611 Answers: 2 Comments: 0
$${solve}\:{for}\:{x}\:{ad}\:{y}\:{in} \\ $$$$\mathrm{5}{x}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)=\mathrm{12}.........\left(\mathrm{1}\right) \\ $$$$\mathrm{5}{y}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)=\mathrm{4}..........\left(\mathrm{2}\right) \\ $$
Question Number 111610 Answers: 1 Comments: 2
Question Number 111601 Answers: 1 Comments: 0
Question Number 111586 Answers: 0 Comments: 3
$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{DTM}}−\mathrm{2020}\:\:\boldsymbol{{savollaridan}}\:\:\boldsymbol{{biri}}. \\ $$$$\:\int\boldsymbol{{x}}^{\boldsymbol{{x}}} =\:? \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{{t}}.\boldsymbol{{me}}/\boldsymbol{{matematik\_olimpiadachilar}} \\ $$
Question Number 111584 Answers: 2 Comments: 0
Question Number 111583 Answers: 1 Comments: 0
$$\mathrm{without}\:\mathrm{using}\:\mathrm{a}\:\mathrm{substitution}\:\mathrm{for}\:\mathrm{3}^{{x}} \:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{9}^{{x}} −\mathrm{3}^{{x}\:+\:\mathrm{2}} −\mathrm{36}\:=\:\mathrm{0} \\ $$
Question Number 111581 Answers: 1 Comments: 10
$${the}\:{age}\:{of}\:{a}\:{mother}\:{and}\:{daughter}\:{arein} \\ $$$${the}\:{ratio}\:{of}\:\mathrm{8}:\mathrm{5}.{if}\:{the}\:{age}\:{now}\:{is} \\ $$$$\mathrm{15},{what}\:{was}\:{the}\:{age}\:{of}\:{the}\:{mother} \\ $$$${six}\:{years}\:{ago}. \\ $$
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