Find the coefficient of the $x$ term after having expanded $$(x^2+3x+2)^5$$
Find the constant term after $\left(\mid x\mid +\frac{1}{\mid x \mid} -2\right)^2$ is expanded.
Let $n$ be a positive integer. Show that $\left(3^{4n+2} + 5^{2n+1}\right)$ is divisible by $14$.
What is the remainder when $2021^{2020}$ is divided by $10^4$?
Show that $$1+4\binom{n}{1} + 7\binom{n}{2}+\cdots+(3n+1)\binom{n}{n}=(3n+2)\cdot 2^{n-1}$$
How many fractions in simplest form are there between $0$ and $1$ such that the products of their denominators and numerators equal $20!$?
A positive integer is written on each face of a cube. Then for each vertex of the cube, the product of the numbers on the three faces associated with this vertex is calculated. If the sum of these eight products equals 2015, find the sum of all the numbers on the 6 faces.
Simplify the expression $$\binom{2020}{0}^2 + \binom{2020}{1}^2 + \cdots + \binom{2020}{2020}^2$$
Show that $$\sum_{k=0}^m \binom{n}{k}\binom{n-k}{m-k}= 2^m\binom{n}{m}$$
How many fraction numbers between $0$ and $1$ are there whose denominator is $1001$ when written in its simplest form?
How many positive integers not exceeding $10^6$ are there which are neither square nor cubic?
The germination rates of two different seeds are measured at $90\%$ and $80\%$, respectively. Find the probability that
- both will germinate
- at least one will germinate
- exactly one will germinate
A bug crawls from $A$ along a grid. It never goes backward, it crawls towards all the other possible directions with equal probability. For example:
- At $A$, it may crawl to either $B$ or $D$ with a 50-50 chance
- At $E$ (coming from $D$), it may crawl to $B$, $F$, or $H$ with a $\frac{1}{3}$ chance each
- At $C$ (coming from $B$), it will crawl to $F$ for sure
The questions are, from $A$:
- What is the probability of it landing at $E$ in 2 steps?
- What is the probability of it landing at $F$ in 3 steps?
- What is the probability of it landing at $G$ in 4 steps?
Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.
Find the least non-negative residue of $70! \pmod{5183}$.
Compute $50^{250} \pmod{83}$ .
What is the last digit of $7^{222}$?
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?
$\textbf{Cover the Board}$
Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.
$\textbf{Cover the Board (II)}$
Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?
Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.
There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.
Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$?