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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

Question Number 223349 Answers: 0 Comments: 0

let gcd(n,m)=1. Determine gcd(5^m +7^m ,5^n +7^n )

$${let}\:{gcd}\left({n},{m}\right)=\mathrm{1}.\:{Determine}\:{gcd}\left(\mathrm{5}^{{m}} +\mathrm{7}^{{m}} ,\mathrm{5}^{{n}} +\mathrm{7}^{{n}} \right) \\ $$

Question Number 223348 Answers: 1 Comments: 1

Determine gcd(13a+19b,ab) given that gcd(a,19)=gcd(b,13)=1

$${Determine}\:{gcd}\left(\mathrm{13}{a}+\mathrm{19}{b},{ab}\right)\:{given}\:{that}\:{gcd}\left({a},\mathrm{19}\right)={gcd}\left({b},\mathrm{13}\right)=\mathrm{1} \\ $$

Question Number 223346 Answers: 1 Comments: 0

proof gcd(2^m −1,2^n −1)=2^(gcd(m,n)) −1

$${proof}\:{gcd}\left(\mathrm{2}^{{m}} −\mathrm{1},\mathrm{2}^{{n}} −\mathrm{1}\right)=\mathrm{2}^{{gcd}\left({m},{n}\right)} −\mathrm{1} \\ $$

Question Number 223340 Answers: 3 Comments: 1

Question Number 223317 Answers: 3 Comments: 0

Question Number 223315 Answers: 0 Comments: 4

Question Number 223304 Answers: 2 Comments: 0

lim_(x→0) ((∫_0 ^x^2 sin((√t))dt )/x^3 ) =...?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\boldsymbol{{sin}}\left(\sqrt{\boldsymbol{{t}}}\right)\boldsymbol{{dt}}\:}{\boldsymbol{{x}}^{\mathrm{3}} }\:=...? \\ $$

Question Number 223301 Answers: 2 Comments: 0

Question Number 223286 Answers: 2 Comments: 3

OC=4.5 & EF=2 & DE^⌢ =EC^⌢ & AC=2AF⇒AB=? ⇓⇓⇓

$${OC}=\mathrm{4}.\mathrm{5}\:\:\:\&\:\:{EF}=\mathrm{2}\:\:\&\:\:{D}\overset{\frown} {{E}}={E}\overset{\frown} {{C}}\:\:\&\:\:{AC}=\mathrm{2}{AF}\Rightarrow{AB}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Downarrow\Downarrow\Downarrow \\ $$

Question Number 223270 Answers: 2 Comments: 0

At a given instant, two cars are at distances 600m and 800m from the point of intersection of the straight roads crossing at right angles and approaching O at uniform speeds of 20 m/s and 30 m/s respectively. Find the shortest distance between the cars and the time taken to reach this position.

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