## Search found 12 matches

- Sat Mar 23, 2024 10:01 am
- Forum: High-school maths
- Topic: Fun problem
- Replies:
**0** - Views:
**9428**

### Fun problem

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...

- Sat Mar 23, 2024 9:57 am
- Forum: High-school maths
- Topic: Fun integral problem
- Replies:
**2** - Views:
**5456**

### Re: Fun integral problem

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...

- Thu Mar 21, 2024 9:07 pm
- Forum: High-school maths
- Topic: Fun integral problem
- Replies:
**2** - Views:
**5456**

### Fun integral problem

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

- Thu Mar 21, 2024 8:41 pm
- Forum: High-school maths
- Topic: Inequality
- Replies:
**0** - Views:
**5392**

### Inequality

Prove that

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

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

- Thu Mar 21, 2024 8:38 pm
- Forum: High-school maths
- Topic: largest value
- Replies:
**0** - Views:
**5408**

### largest value

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

- Thu Mar 21, 2024 8:37 pm
- Forum: High-school maths
- Topic: Find all positive integers $m$ and $n$ such that $\frac{m^3}{m+n}$ and $\frac{n^3}{m+n}$ are both prime numbers.
- Replies:
**0** - Views:
**4967**

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

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

- Thu Mar 21, 2024 8:36 pm
- Forum: High-school maths
- Topic: Find the square root of $1 + a^2 + \sqrt{1 + a^2 + a^4}$
- Replies:
**0** - Views:
**4983**

### Find the square root of $1 + a^2 + \sqrt{1 + a^2 + a^4}$

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

- Sat Mar 16, 2024 2:17 pm
- Forum: Calculus
- Topic: Nice integral
- Replies:
**1** - Views:
**9264**

### Nice integral

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

- Sat Mar 16, 2024 2:14 pm
- Forum: Calculus
- Topic: Calculate the integral
- Replies:
**0** - Views:
**5220**

### Calculate the integral

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

- Sat Mar 16, 2024 2:13 pm
- Forum: Calculus
- Topic: Calculate the sum
- Replies:
**0** - Views:
**5131**

### Calculate the sum

$${\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...