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Question Number 223512    Answers: 1   Comments: 0

If ((log x)/(y−z))=((log y)/(z−x))=((log z)/(x−y)) prove xyz=1

$${If}\:\frac{\mathrm{log}\:{x}}{{y}−{z}}=\frac{\mathrm{log}\:{y}}{{z}−{x}}=\frac{\mathrm{log}\:{z}}{{x}−{y}} \\ $$$${prove}\:{xyz}=\mathrm{1} \\ $$

Question Number 223508    Answers: 0   Comments: 0

Question Number 223490    Answers: 2   Comments: 2

Question Number 223487    Answers: 1   Comments: 0

Let Φ be the hyperbola xy = b², b ≠ 0, and P be a point on Φ. Let Q be the image of reflection of P about the origin. Construct a circle ω centred at P with radius PQ. ω cuts Φ at the points B, C, D, Q. Prove that ΔBCD is equilateral, no matter what the value of b is.

Let Φ be the hyperbola xy = b², b ≠ 0, and P be a point on Φ. Let Q be the image of reflection of P about the origin. Construct a circle ω centred at P with radius PQ. ω cuts Φ at the points B, C, D, Q. Prove that ΔBCD is equilateral, no matter what the value of b is.

Question Number 223482    Answers: 0   Comments: 0

Question Number 223483    Answers: 1   Comments: 0

(x−1)(x−2)=1 (x−1)^(15) −(1/((x−1)^(15) ))=??

$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)=\mathrm{1} \\ $$$$\left({x}−\mathrm{1}\right)^{\mathrm{15}} −\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{15}} }=?? \\ $$

Question Number 223480    Answers: 0   Comments: 0

Question Number 223474    Answers: 3   Comments: 1

(√(1+(√(1+x))))=(x)^(1/3)

$$\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+{x}}}=\sqrt[{\mathrm{3}}]{{x}} \\ $$

Question Number 223461    Answers: 1   Comments: 0

Question Number 223459    Answers: 4   Comments: 0

solve for x∈R (√(25−10x−x^2 ))+(√(15−x^2 ))=2(√5)

$${solve}\:{for}\:{x}\in{R} \\ $$$$\sqrt{\mathrm{25}−\mathrm{10}{x}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{15}−{x}^{\mathrm{2}} }=\mathrm{2}\sqrt{\mathrm{5}} \\ $$

Question Number 223449    Answers: 2   Comments: 0

Question Number 223429    Answers: 2   Comments: 1

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Question Number 223424    Answers: 1   Comments: 3

Question Number 223386    Answers: 1   Comments: 0

Question Number 223383    Answers: 3   Comments: 3

Question Number 223374    Answers: 3   Comments: 0

Question Number 223412    Answers: 0   Comments: 4

Good day great problem solvers. Please I need links to resources helpful for preparations for Olympiad mathematics especially books and video recommendations. I′ll be grateful to get our responses.

$${Good}\:{day}\:{great}\:{problem}\:{solvers}. \\ $$$${Please}\:{I}\:{need}\:{links}\:{to}\:{resources}\:{helpful} \\ $$$${for}\:{preparations}\:{for}\:{Olympiad}\:{mathematics} \\ $$$${especially}\:{books}\:{and}\:{video}\:{recommendations}. \\ $$$${I}'{ll}\:{be}\:{grateful}\:{to}\:{get}\:{our}\:{responses}. \\ $$

Question Number 223410    Answers: 0   Comments: 0

Question Number 223405    Answers: 2   Comments: 0

Question Number 223403    Answers: 1   Comments: 1

Question Number 223401    Answers: 1   Comments: 0

Question Number 223414    Answers: 1   Comments: 0

is it possible to prove that mn(m+n)(m−n) divisible by 6 always

$${is}\:{it}\:{possible}\:{to}\:{prove}\:{that}\:{mn}\left({m}+{n}\right)\left({m}−{n}\right)\: \\ $$$${divisible}\:{by}\:\mathrm{6}\:{always}\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 223400    Answers: 1   Comments: 0

f(1)=2025 𝚺_1 ^n f(k)=n^2 .f(n) f(2025)=?

$$\boldsymbol{{f}}\left(\mathrm{1}\right)=\mathrm{2025} \\ $$$$\underset{\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{f}}\left(\boldsymbol{{k}}\right)=\boldsymbol{{n}}^{\mathrm{2}} .\boldsymbol{{f}}\left(\boldsymbol{{n}}\right) \\ $$$$\boldsymbol{{f}}\left(\mathrm{2025}\right)=? \\ $$

Question Number 223368    Answers: 1   Comments: 0

∫_0 ^1 ln(((2 cos(x^2 ) + ln^2 (x/2))/(1 + cos (x/2)))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{\mathrm{ln}}\left(\frac{\mathrm{2}\:\boldsymbol{\mathrm{cos}}\left({x}^{\mathrm{2}} \right)\:+\:\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left({x}/\mathrm{2}\right)}{\mathrm{1}\:+\:\boldsymbol{\mathrm{cos}}\:\left({x}/\mathrm{2}\right)}\right)\:\boldsymbol{\mathrm{d}}{x} \\ $$$$ \\ $$

Question Number 223367    Answers: 0   Comments: 1

∫_0 ^1 ln(2 cos(x^2 ) + ln^2 ((x/2)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\mathrm{ln}\left(\mathrm{2}\:\mathrm{cos}\left({x}^{\mathrm{2}} \right)\:+\:\mathrm{ln}^{\mathrm{2}} \:\left(\frac{{x}}{\mathrm{2}}\right)\:\mathrm{d}{x}\right. \\ $$$$ \\ $$

Question Number 223354    Answers: 0   Comments: 6

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