Advanced Math Problems: Calculus, Algebra, and Number Theory

1. Evaluate a Limit

Evaluate the limit as $x$ approaches infinity of:

$\frac{x^3 - 3x^2 + 2x - 1}{x^3 + 2x^2 - x + 1}$

2. Sum of an Infinite Series

Find the sum of the series:

$\sum_{n=1}^{\infty} \frac{n^2}{n^4 + 4}$

3. Rational Root Theorem

Prove that there is no rational root for the equation:

$x^5 - x^4 + x^3 - x^2 + x - 1 = 0$

4. Differential Equation

Determine the general solution of the differential equation:

$y' - 2y = e^{2x}$

5. Area Between Curves

Determine the area enclosed by the curves:

$y = x^2$ and $y = x^3 - x$

6. Solve a Trigonometric Equation

Solve the equation:

$\sin x + \cos x = 1$, $0 \le x \le 2\pi$

7. Sum of Cubes

Prove that:

$\sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2$

for all natural numbers $n$.

8. Tangent Line to a Curve

Find the equation of the tangent line to the curve $y = e^{3x} \ln x$ at the point $(1,0)$.

9. Radius of Convergence

Determine the radius of convergence for the power series:

$\sum_{n=1}^{\infty} \frac{(x-2)^n}{n}$

10. Prove the Binomial Theorem

Prove that:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} b^k a^{n-k}$

11. Volume of Revolution

Determine the volume of the solid generated by revolving the region bounded by the curves $y = x^2$ and $y = x$ about the $x$-axis.

12. Limit of a Sequence

Prove that $\underset{n \to \infty}{\lim} \frac{n!}{n^n} = 0$.

13. Diophantine Equation

Find the smallest positive integer solution to the Diophantine equation:

$7x + 11y = 2023$.

14. Inequality in a Triangle

Prove that for any triangle with sides $a$, $b$, and $c$, the following inequality holds:

$\frac{a^3 + b^3 + c^3}{3} \geq abc$.

15. Evaluate a Definite Integral

Determine the value of the integral:

$\int_0^\infty \frac{x^3}{{(x^2+1)^2}} \, dx$.

16. Extrema of a Function

Find the maximum and minimum values of the function $f(x) = 3x^4 - 8x^3 + 5x^2$ on the interval $[0, 2]$.

17. Prove Irrationality

Show that $\sqrt[3]{7} + \sqrt[3]{49}$ is irrational.

18. Recurrence Relation

Find the general solution of the recurrence relation $a_n = 5a_{n-1} - 6a_{n-2}$, given $a_0 = 1$ and $a_1 = 3$.

19. Evaluate a Double Integral

Evaluate the double integral:

$\int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \frac{1}{{(x^2+y^2)^2}} \, dy \, dx$.

20. Sum of Angles in a Polygon

Prove that the sum of the angles in an $n$-sided polygon is equal to $180^\circ (n-2)$.

21. Sum of an Infinite Geometric Series

Find the sum of the infinite geometric series:

$\frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \ldots$.

22. Properties of a Prime Number

Prove that for any prime number $p$, the number $\frac{p-1}{2}$ is odd if and only if $p \equiv 3 \pmod{4}$.

23. Arc Length of a Curve

Determine the arc length of the curve $y = \frac{1}{3} x^3 - x$ from $x = 0$ to $x = 2$.

24. Distinct Arrangements

Determine the number of distinct ways to arrange the letters of the word "MATHEMATICS" such that no two "M"s are adjacent.

25. Sum of Cubes Formula

Prove that for all positive integers $n$, the following is true:

$1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2$.

26. Area in the Complex Plane

Find the area of a triangle with vertices at the complex numbers $z_1 = 1 + 2i$, $z_2 = 2 + i$, and $z_3 = 1 + i$ in the complex plane.

27. Roots of a Polynomial

Prove that the roots of the polynomial $P(x) = x^n - a_1 x^{n-1} + a_2 x^{n-2} - \ldots + (-1)^n a_n$ are all real if and only if $a_i \geq 0$ for all $1 \leq i \leq n$.

28. Improper Integral

Evaluate the improper integral:

$\int_{-\infty}^{\infty} \frac{1}{{1 + x^2}} \, dx$.

29. Complex Numbers

If $z$ is a complex number such that $z^4 = 1$, prove that $z^2 - z + 1 = 0$ if and only if $z \neq 1$.

30. Infinitely Many Primes

Show that there are infinitely many prime numbers of the form $4k + 3$, where $k$ is an integer.