Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.

Determine if the following infinite series is convergent or divergent:

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

Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$

Compute

$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$

where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.

Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$

For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$

Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by

$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$

achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.

For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.

Let $s\in\mathbb{R}$. Prove that

$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$

converges if and only if $s > 1$.

Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$

Note that the point $(2, 1)$ is always on the curve $x^4 + ky^4 = 16+k$ regardless of the value of $k$. If for a particular non-zero value of $k$, $y'(2)=y''(2)$ along this curve. Find this $k$.

Compute $$\int_0^{\frac{\pi}{4}}(\cos{x} - 2\sin{x}\sin(2x))dx$$

Let $f_0(x)=(\sqrt{e})^x$ , and recursively define $f_{n+1}(x) = f'_n(x)$ for integers $n\ge 0$. Compute $$\sum_{k=0}^{\infty}f_k(1)$$

Consider the parabola $y=ax^2 + 2019x + 2019$. There exists exactly one circle which is centered on the $x$-axis and is tangent to the parabola at exactly two points. It turns out that one of these tangent points is $(0, 2019)$. Find $a$.

What is the smallest natural number $n$ for which the following limit exists?

$$\lim_{x\to 0}\frac{\sin^nx}{\cos^2x(1-\cos{x})^3}$$

Turn the graph of $y=\frac{1}{x}$ by $45^{\circ}$ counter-clockwise and consider the bowl-like top part of the curve (the part above $y=0$). We let a $2D$ fluid accumulate in this $2D$ bowl until the maximum depth of the fluid is $\frac{2\sqrt{2}}{3}$. What’s the area of the fluid used?

Compute $$\int_0^4\frac{dx}{\sqrt{|x-2|}}$$

Compute $$\lim_{x\to 0}\frac{(1-\cos{x})^2}{x^2-x^2\cos^2{x}}$$

Compute $$\int_{-2}^{0}\frac{x^3 + 4x^2 + 7x -20}{x^2+4x+8}dx+\int_0^2\frac{2x^3-7x^2+9x-10}{x^2+4}dx$$

Compute $$\lim_{n\to\infty}n^2\int_0^{\frac{1}{n}}x^{2018x+1}dx$$

Compute $$\int_0^{\pi}\frac{2x\sin{x}}{3+\cos^2x}dx$$

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes through the origin. Compute $a$.

Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between $0$ and $1$ (inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given $5000k$ additional dollars. Otherwise, Moor loses $1000k$ dollars. Moor’s happiness is equal to the log of the amount of money that he has after this gambling game. Find the value of k that Moor should select to maximize his expected happiness.