Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.
Find the total number of sequences of length $n$ containing only letters $A$ and $B$ such that no two $A$s are next to each other. For example, for $n = 2$, there are $3$ possible sequences: $AB$, $BA$, and $BB$.
Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$
where $a_0=a_1=1$.
Show the following sequence is convergent:
$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$
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:
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that
$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$
It is well-known that the solution to the Fibonacci sequence is
$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$
Show that
$$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$$
$\textbf{Average Speed}$
Joe travels at an average of $30$ miles per hour from home to visit a friend who lives $60$ miles away. How fast should he drive on his way straight back to home so that his average speed is $60$ miles per hour for this entire trip?
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
What is the median of the following list of $4040$ numbers?
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
There is a unique positive integer $n$ such that $$\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}$$ What is the sum of the digits of $n?$
There are integers $a$, $b$, and $c$, each greater than 1, such that\[\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]for all $N > 1$. What is $b$?
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\]What is $k?$
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
Solve $x^2 - x -1=0$.
Solve $x^4-x^2-1=0$.
Let $r$ and $s$ be integers. Find the condition such that the expression $\frac{6^{r+s}\times 12^{r-s}}{8^r\times 9^{r+2s}}$ is an integer.
Find the number of real number solutions to the equation: $8^x +4=4^x + 2^{x+2}$.
Let $f_n (x) = (2 + (−2)^n ) x^2 + (n + 3) x + n^2$.
Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} + \sqrt{x} - 2x - 1=0$$