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Question Number 112534 Answers: 0 Comments: 3

What is the maximum number of points to be distributed within a 3×6 to ensure that there are no two points whose distance apart is less than (√2)?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{points}\:\mathrm{to}\:\mathrm{be}\:\mathrm{distributed}\:\mathrm{within} \\ $$$$\mathrm{a}\:\mathrm{3}×\mathrm{6}\:\mathrm{to}\:\mathrm{ensure}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{two} \\ $$$$\mathrm{points}\:\mathrm{whose}\:\mathrm{distance}\:\mathrm{apart}\:\mathrm{is}\:\mathrm{less} \\ $$$$\mathrm{than}\:\sqrt{\mathrm{2}}? \\ $$

Question Number 111721 Answers: 1 Comments: 0

If b>1,x>0 and (2x)^(log_b 2) −(3x)^(log_b 3) =0, then x is

$$\mathrm{If}\:\mathrm{b}>\mathrm{1},\mathrm{x}>\mathrm{0}\:\mathrm{and}\:\left(\mathrm{2x}\right)^{\mathrm{log}_{\mathrm{b}} \mathrm{2}} −\left(\mathrm{3x}\right)^{\mathrm{log}_{\mathrm{b}} \mathrm{3}} =\mathrm{0}, \\ $$$$\mathrm{then}\:\mathrm{x}\:\mathrm{is} \\ $$

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+7=0

$${y}'+{y}+\mathrm{7}=\mathrm{0} \\ $$

Question Number 111643 Answers: 1 Comments: 0

((7/3))!(with out calculator)

$$\left(\frac{\mathrm{7}}{\mathrm{3}}\right)!\left({with}\:{out}\:{calculator}\right) \\ $$

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 (1−x)^6 =−27

$${if}\:{x}\:{is}\:{a}\:{cube}\:{root}\:{of}\:{a}\:{unity} \\ $$$${prove}\:{that}\: \\ $$$$\left(\mathrm{1}−{x}\right)^{\mathrm{6}} =−\mathrm{27} \\ $$

Question Number 111623 Answers: 1 Comments: 2

(x−2)(x+3)(x−1)^2 ≥ 0

$$\left({x}−\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\geqslant\:\mathrm{0} \\ $$

Question Number 111622 Answers: 1 Comments: 0

show that the close form of Σ_(k=0) ^∞ (Σ_(i=0) ^∞ [(((−1)^(k+i) )/((k+1)(k+2i+2)))])=(1/8)ln2

$${show}\:{that}\:{the}\:{close}\:{form}\:{of}\: \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\left(−\mathrm{1}\right)^{{k}+{i}} }{\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}{i}+\mathrm{2}\right)}\right]\right)=\frac{\mathrm{1}}{\mathrm{8}}\mathrm{ln2} \\ $$

Question Number 111620 Answers: 1 Comments: 0

show that ∫_0 ^1 lnΓ(x)dx=ln(√(2π))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\Gamma\left({x}\right){dx}=\mathrm{ln}\sqrt{\mathrm{2}\pi} \\ $$

Question Number 111619 Answers: 1 Comments: 1

solve for x in x^2 +4x=(√(40x^2 +8x−16))

$${solve}\:{for}\:{x}\:{in}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{4}{x}=\sqrt{\mathrm{40}{x}^{\mathrm{2}} +\mathrm{8}{x}−\mathrm{16}} \\ $$

Question Number 111618 Answers: 0 Comments: 0

solve the fredholm integral equation of the second kind y(x)=x+λ∫_0 ^1 (xt^2 +x^2 t)y(t)dt

$${solve}\:{the}\:{fredholm}\:{integral}\:{equation}\:{of} \\ $$$${the}\:{second}\:{kind}\: \\ $$$${y}\left({x}\right)={x}+\lambda\int_{\mathrm{0}} ^{\mathrm{1}} \left({xt}^{\mathrm{2}} +{x}^{\mathrm{2}} {t}\right){y}\left({t}\right){dt} \\ $$

Question Number 111617 Answers: 2 Comments: 0

show that ∫_0 ^(π/4) ((ln(2cos^2 x))/(cos2x))dx=(π^2 /(16))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{ln}\left(\mathrm{2cos}^{\mathrm{2}} {x}\right)}{\mathrm{cos2}{x}}{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{16}} \\ $$

Question Number 111616 Answers: 2 Comments: 0

solve the integral problem ∫((cosx)/(2−sin2x))dx

$${solve}\:{the}\:{integral}\:{problem} \\ $$$$\int\frac{\mathrm{cos}{x}}{\mathrm{2}−\mathrm{sin2}{x}}{dx} \\ $$

Question Number 111615 Answers: 2 Comments: 0

solve ∫(dx/((x^2 +2x+3)(√(x^2 +x+3))))

$${solve} \\ $$$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\right)\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{3}}} \\ $$

Question Number 111614 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (((sinx)/x))^7 =((5887π)/(23040))

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{sin}{x}}{{x}}\right)^{\mathrm{7}} =\frac{\mathrm{5887}\pi}{\mathrm{23040}} \\ $$

Question Number 111611 Answers: 2 Comments: 0

solve for x ad y in 5x(1+(1/(x^2 +y^2 )))=12.........(1) 5y(1−(1/(x^2 +y^2 )))=4..........(2)

$${solve}\:{for}\:{x}\:{ad}\:{y}\:{in} \\ $$$$\mathrm{5}{x}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)=\mathrm{12}.........\left(\mathrm{1}\right) \\ $$$$\mathrm{5}{y}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)=\mathrm{4}..........\left(\mathrm{2}\right) \\ $$

Question Number 111610 Answers: 1 Comments: 2

Question Number 111601 Answers: 1 Comments: 0

Question Number 111586 Answers: 0 Comments: 3

DTM−2020 savollaridan biri. ∫x^x = ? t.me/matematik_olimpiadachilar

$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{DTM}}−\mathrm{2020}\:\:\boldsymbol{{savollaridan}}\:\:\boldsymbol{{biri}}. \\ $$$$\:\int\boldsymbol{{x}}^{\boldsymbol{{x}}} =\:? \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{{t}}.\boldsymbol{{me}}/\boldsymbol{{matematik\_olimpiadachilar}} \\ $$

Question Number 111584 Answers: 2 Comments: 0

Question Number 111583 Answers: 1 Comments: 0

without using a substitution for 3^x solve the equation 9^x −3^(x + 2) −36 = 0

$$\mathrm{without}\:\mathrm{using}\:\mathrm{a}\:\mathrm{substitution}\:\mathrm{for}\:\mathrm{3}^{{x}} \:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{9}^{{x}} −\mathrm{3}^{{x}\:+\:\mathrm{2}} −\mathrm{36}\:=\:\mathrm{0} \\ $$

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