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Question Number 198147 Answers: 1 Comments: 0

if a,x,y,b is an AP and a,p,q,b is a GP. prove that xy≥pq. (with a, b >0)

$${if}\:{a},{x},{y},{b}\:{is}\:{an}\:{AP}\:{and}\:{a},{p},{q},{b}\:{is}\:{a}\:{GP}. \\ $$$${prove}\:{that}\:{xy}\geqslant{pq}. \\ $$$$\left({with}\:{a},\:{b}\:>\mathrm{0}\right) \\ $$

Question Number 198146 Answers: 0 Comments: 1

Please suggest youtube playlist to prepare one for mathematics olympiad. Thanks in advance.

$${Please}\:{suggest}\:{youtube}\:{playlist}\:{to} \\ $$$${prepare}\:{one}\:{for}\:{mathematics}\:{olympiad}. \\ $$$${Thanks}\:{in}\:{advance}. \\ $$$$ \\ $$

Question Number 198156 Answers: 1 Comments: 0

Prove that ((2t−1)/(lnt−ln(1−t)))=∫^( 1) _( 0) t^x (1−t)^(1−x) dx and ∫^( 1) _( 0) ((2t−1)/(lnt−ln(1−t)))dt = (π/2)∫^( 1) _( 0) ((x(1−x))/(sin(πx)))dx

$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}=\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{t}^{\mathrm{x}} \left(\mathrm{1}−\mathrm{t}\right)^{\mathrm{1}−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{and}\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}\mathrm{dt}\:\:=\:\:\frac{\pi}{\mathrm{2}}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{sin}\left(\pi\mathrm{x}\right)}\mathrm{dx} \\ $$

Question Number 198141 Answers: 1 Comments: 0

∫^( 1) _( 0) ((x(1−x))/(sin(πx)))dx=???

$$\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \:\frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{sin}\left(\pi\mathrm{x}\right)}\mathrm{dx}=??? \\ $$

Question Number 198136 Answers: 1 Comments: 0

Question Number 198132 Answers: 1 Comments: 0

Solve: ((log(x^2 +7x−5))/(log(x+2)))=2

$${Solve}: \\ $$$$\frac{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{7}\boldsymbol{\mathrm{x}}−\mathrm{5}\right)}{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}+\mathrm{2}\right)}=\mathrm{2} \\ $$

Question Number 198131 Answers: 1 Comments: 0

Resoudre log(x−3)+log(x−2)=log(x^2 −4x−21)

$$\mathrm{Resoudre} \\ $$$$\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}−\mathrm{3}\right)+\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)=\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{x}}−\mathrm{21}\right) \\ $$$$ \\ $$

Question Number 198124 Answers: 2 Comments: 0

solve for x log100+log(2+x)=10

$${solve}\:{for}\:{x}\:{log}\mathrm{100}+{log}\left(\mathrm{2}+{x}\right)=\mathrm{10} \\ $$

Question Number 198123 Answers: 3 Comments: 0

Determiner lim_(x→3) ((x−3)/(^3 (√(x+5)) −2))

$$\mathrm{Determiner} \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{3}} \:\frac{\boldsymbol{\mathrm{x}}−\mathrm{3}}{\:^{\mathrm{3}} \sqrt{\boldsymbol{\mathrm{x}}+\mathrm{5}}\:−\mathrm{2}} \\ $$$$ \\ $$

Question Number 198114 Answers: 1 Comments: 1

Question Number 198104 Answers: 1 Comments: 0

Question Number 198103 Answers: 3 Comments: 0

solve for x, y ∈N (√x)+(√y)=(√(2023))

$${solve}\:{for}\:{x},\:{y}\:\in{N} \\ $$$$\sqrt{{x}}+\sqrt{{y}}=\sqrt{\mathrm{2023}} \\ $$

Question Number 198093 Answers: 1 Comments: 0

Question Number 198276 Answers: 2 Comments: 0

Question Number 198084 Answers: 2 Comments: 0

Question Number 198077 Answers: 1 Comments: 0

Question Number 198074 Answers: 4 Comments: 0

Question Number 198067 Answers: 1 Comments: 0

A father reduced the quantity of food bought for the family by 10% when he found that the cost of living had increased by 15%. What is the fractional increase in the family food bill?

$${A}\:{father}\:{reduced}\:{the}\:{quantity}\:{of}\:{food} \\ $$$${bought}\:{for}\:{the}\:{family}\:{by}\:\mathrm{10\%}\:{when}\:{he} \\ $$$${found}\:{that}\:{the}\:{cost}\:{of}\:{living}\:{had} \\ $$$${increased}\:{by}\:\mathrm{15\%}.\:{What}\:{is}\:{the}\:{fractional} \\ $$$${increase}\:{in}\:{the}\:{family}\:{food}\:{bill}? \\ $$

Question Number 198065 Answers: 1 Comments: 0

2^x +9+2^x =40

$$\mathrm{2}^{{x}} +\mathrm{9}+\mathrm{2}^{{x}} =\mathrm{40} \\ $$

Question Number 198064 Answers: 1 Comments: 0

3×5^x +5^(x+1) =8×5^3

$$\mathrm{3}×\mathrm{5}^{{x}} +\mathrm{5}^{{x}+\mathrm{1}} =\mathrm{8}×\mathrm{5}^{\mathrm{3}} \\ $$

Question Number 198063 Answers: 2 Comments: 0

solve for x, y ∈R (√(x^2 +2x+1))+(√(y^2 −6y+9))+(√(x^2 −4x+4))+(√(x^2 +y^2 −2xy))=4

$${solve}\:{for}\:{x},\:{y}\:\in{R} \\ $$$$\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}+\sqrt{{y}^{\mathrm{2}} −\mathrm{6}{y}+\mathrm{9}}+\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}}+\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{xy}}=\mathrm{4} \\ $$

Question Number 198062 Answers: 0 Comments: 1