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Problem 4. In front of you are $N$boxes numbered $1$ through $N$,and each contains one object, also numbered 1 through N, but objects and box num- bers don't necessarily match. You know where all the objects are, but your friend knows nothing other than the general rules. You need to make a sequence...

First of all we note that $\mathrm{P}(x) + \mathrm{P} \left ( 12 - x \right ) = 12$ since \begin{align*} \mathrm{P}(x) - 12 &= \left ( x -4 \right ) \left ( x - 5 \right ) \left ( x -9 \right ) \\ &= -\left ( 4 - x \right ) \left ( 5- x \right ) \left ( 9 - x \right ) \\ &= - \left [ \l...

Let $P(x) = (x - 3)(x - 7)(x - 8)$. Determine the value of $$\int_{0}^{12} \underbrace{P(P(\ldots P(x) \ldots))}_{69 \text{ times}} \, dx.$$

Prove that
\[\sum_{n=1}^{x}\frac{1}{x+n}<\frac{3}{4}\qquad\forall x\ge 1\]

Let $f(x)=1+\frac{90}{x}$, the largest value of $x$ that satisfies $\underbrace{f(f(...(f(x))...))}_{\text{2019 times}}=x$ is

Find all positive integers $m$ and $n$ such that $\dfrac{m^3}{m+n}$ and $\dfrac{n^3}{m+n}$ are both prime numbers.

Find the number of which the square is the number $1 + a^2 + \sqrt{1 + a^2 + a^4}$

$$\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$$

$$\int_{0}^{1} \frac{\arctan(x)}{1 + x^2} \cdot \operatorname{Li}_{2}\left(\frac{1 - 4x + 6x^2 - 4x^3 + x^4}{1 + 4x + 6x^2 + 4x^3 + x^4}\right) \, dx$$

$${\displaystyle {\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}\\[6pt]={}&{\frac {2\Gamma ^{4}{\bigl (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamm...