Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is a $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the
round. What is the expected number of coins he will have at the end of the game?
How many ways are there to insert $+$’s between the digits of $111111111111111$ (fifteen $1$’s) so that the
result will be a multiple of $30$?
Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a
stone from the bucket, each with probability $\frac{1}{2}$ . If she wants to remove a stone from the bucket and
the bucket is currently empty, she merely does nothing for that second (still with probability $\frac{1}{2}$). What
is the probability that after $2017$ seconds her bucket contains exactly $1337$ stones?
Find the smallest square which can cover $n$ congruent equilateral triangles so that these triangles do not overlap.
Show that $$\lim_{x\to 0}\ \frac{x}{\sin{x}}=1$$
Let $0 < x < \frac{\pi}{2}$. Show that $\sin x < x <\tan x$.
Show the following sequence is convergent:
$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$
Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.
Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.
Compute the limit of the power series below as a rational function in $x$:
$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$
Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$
Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:
Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by
$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$
assuming the density is uniform.
Consider the ellipse $x^2+\frac{y^2}{4}=1$. What is the area of the smallest diamond shape with
two vertices on the $x$-axis and two vertices on the $y$-axis that contains this ellipse?
Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$
Find the maximum and minimal values of the function
$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$
over its domain.
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 $(2, 1)$.
Show that $\ln x < \sqrt{x}$ holds for all positive $x$.
Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(2)+f(\frac{1}{2})$.
Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$
Evaluate
$$\int_0^1 x\arcsin{x}d{x}$$
Compute
$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$
where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.
Compute $$\int \ln{x} dx$$