180 lines
10 KiB
Markdown
180 lines
10 KiB
Markdown
---
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excalidraw-plugin: parsed
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tags:
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- excalidraw
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excalidraw-open-md: true
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---
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> [!question] Question
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> Find numbers from 0 to 100, but only using 4 times the digit "4" (and any operator)
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>
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> Trouver les nombres de 100, mais seulement en utilisant 4 fois le chiffre 4 (et des opérateurs)
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# Ideas / Idées
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You can always use up a 4 using : "$\sqrt{ \sqrt{ x^{4} } } = x$" (does nothing). This allows formulas with only 3 or 2 times the 4.
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On peut toujours utiliser un 4 avec : "$\sqrt{ \sqrt{ x^{4} } } = x$" (ne fait rien). Cela permet des formules avec seulement 2 ou 3 fois le 4.
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# Solutions
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## $n = 0$
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- $4 - 4 + 4 - 4$
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- $44 - 44$
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## $n=1$
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## $n = 41$
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- $41 = 44 - \sqrt{ 4 } - (\lim\limits_{ n \to \infty } \sqrt[n]{4})$
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## $n = 71$
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- $71 = (\Gamma(4) + \Gamma(\sqrt{ 4 })) \cdot (\Gamma(\sqrt{ 4 }) \cdot \Gamma(\sqrt{ 4 }))$
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- $71 = (4! + (\sqrt{ 4 })!) \cdot ((\sqrt{ 4 })! \cdot (\sqrt{ 4 })!)$
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# General solution / Solution générale
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## 1
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$\displaystyle\int _{-4}^{4} \dfrac{\underbrace{(x \times x)+(x \times x)+\cdots + (x \times x)}_{6n \text{ times } x}}{4^{4}} \, dx = n$
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## 2
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$\displaystyle\sqrt{ 4 } \cdot\frac{\ln \left( \frac{\overbrace{\sqrt{ \sqrt{ \sqrt{ \sqrt{ \cdots 4 } } } }}^{n \text{ times }} }{\ln(4)} \right)}{\ln(4)}$
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> [!definition] Logarithm / Logarithme
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> $\log_{n}(n^{x}) = x$
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> [!definition] Natural logarithm / Logarithme naturel
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> $\ln(x) = \log_{e}(x)$
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> So : / Alors :
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> $\ln(e^{x}) = x$
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## 3
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$0 = 4 - 4$
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$1 = (\sqrt{ 4 })!$
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$2 = \sqrt{ 4 }$
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$3 = \dfrac{4!}{\sqrt{ 4 }}$
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$4 = $
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> [!definition] Factorial / Factorielle
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> $n! = 1 \times 2 \times 3 \times \cdots \times n$
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> [!definition] Double factorial / Double factorielle
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> $n!! = 1 \cancel{\times 2} \times 3 \cancel{\times 4} \times \cdots \times n$ if $n$ is odd / si $n$ est impair
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> $n!! = \cancel{1 \times} 2 \times \cancel{3 \times} 4\times \cdots \times n$ if $n$ is even / si $n$ est pair
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> [!definition] Concatenation
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> New operator / Nouvel opérateur :
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> $4 \cdot 4 = 44$
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> $71 \cdot 92 = 7192$
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## 2
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$\displaystyle\log_{\log_{4}(\sqrt{ 4 })} \left( \log_{4} \left( \underbrace{\sqrt{ \sqrt{ \sqrt{ \sqrt{ \dots \sqrt{ 4 } } } } }}_{n \text{ times}} \right) \right) = n$
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#### Why ? / Pourquoi ?
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$\log_{4}(\sqrt{ 4 }) = \frac{1}{2}$
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$\log_{4}(4) = 1$
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$\log_{4}(4^{2}) = 2$
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$\vdots$
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$\log_{4}(4^{p}) = p$
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$\log_{4}(\sqrt{ 4 }) = \log_{4}\left( 4^{\frac{1}{2}} \right) = \frac{1}{2}$
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$\log_{4}(\sqrt{ \sqrt{ 4 } }) = \log_{4}\left( 4^{\frac{1}{4}} \right) = \frac{1}{4}$
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$\log_{4}(\sqrt{ \sqrt{ \sqrt{ 4 } } }) = \log_{4}\left( 4^{\frac{1}{8}} \right) = \frac{1}{8}$
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$\log_{4}(\sqrt{ \sqrt{ \sqrt{ \sqrt{ 4 } } } }) = \log_{4}\left( 4^{\frac{1}{16}} \right) = \frac{1}{16}$
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$\vdots$
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$\displaystyle\log_{4}\left( \underbrace{\sqrt{ \sqrt{ \sqrt{ \dots \sqrt{ 4 } } } }}_{n \text{ times}} \right) = \log_{4} \left( 4^{\frac{1}{2^{n}}} \right) = \frac{1}{2^{n}}$
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$\displaystyle\log_{\frac{1}{2}} \left( \frac{1}{2 \times n} \right) = n$
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![[kangourou 2024-07-15 problèmes du jour.svg]]
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%%
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# Excalidraw Data
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## Text Elements
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Velocity
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Vitesse ^H0Z9Cp37
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Acceleration ^y4qpuUdL
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Position ^UY2ok27f
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derivative
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dérivée ^dW5ZQscu
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derivative
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dérivée ^S24o7Dxr
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integral
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intégrale ^dpD4ekhS
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integral
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intégrale ^7kTHCP9V
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## Drawing
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```compressed-json
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```
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%% |