Let $n$ be an odd positive integer, and $$N=6^n + \binom{n}{1}\cdot 6^{n-1}+\cdots + \binom{n}{n-1}\cdot 6-1$$
Find the remainder when $N$ is being divided by $8$.
Assuming that $$(1-2x)^7=a_0 + a_1x+a_2x^2+\cdots+a_7x^7$$
Find the value of
Show that the following relation holds for any positive integers $1 < k \le m < n$: $$\binom{n}{k}m^k > \binom{m}{k}n^k$$
Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$
where $n$ is a positive integer.
Let $n$ be a positive integer and the coefficient of the $x^3$ term in the expansion of $(1+\frac{x}{n})^n$ be $\frac{1}{16}$. Find $n$.
Calculate the value of $$\displaystyle\sum_{k=0}^{n}\frac{1}{2^k}\binom{n}{k}$$
Show that $$\frac{2n+2}{2n+1}\cdot\frac{1}{\binom{2n}{k}}=\frac{1}{\binom{2n+1}{k}}+\frac{1}{\binom{2n+1}{k+1}}$$
Find the value of $$\sum_{k=0}^{n-1}\binom{2n-1}{k}$$
Calculate the value of $$\displaystyle\sum_{k=0}^{n}(-1)^{k}k\binom{n}{k}$$
What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]
What is the hundreds digit of $(20! - 15!)$?
Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$?
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?
Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement.
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
In a high school wit $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
How many distinct permutations of the letters of the word REDDER are there that do not contain a
palindromic substring of length at least two? (A substring is a contiguous block of letters that is part
of the string. A string is palindromic if it is the same when read backwards.)
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1$, $2$, $\cdots$, $20$ on its
sides). He conceals the results but tells you that at least half of the rolls are $20$. Suspicious, you
examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is
truthful, what is the probability that all three remaining concealed dice show $20$?
Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting
with Reimu, they take turns to color one of the white sides either red or green. After all sides are
colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there
are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides
and green sides, then neither of them wins. Given that both of them play optimally to maximize the
probability of winning, what is the probability that Reimu wins?
Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other
five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b - a$? (A number is a
palindrome if it reads the same forwards and backwards in base $10$.)
Compute $$\lim_{x\to 4}\frac{3-\sqrt{x+5}}{x-4}$$
Is the $y=\frac{1}{x}$ a continuous function?
Show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$
Find the value of
$$\lim_{x\to\infty}\frac{\sin{x}}{x}$$