Let $n$ be a positive integer, show that $(3^{3n}-26n-1)$ is divisible by $676$.
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
Label the first row of the Pascal triangle as row $0$. How many odd numbers are there in the $2019^{th}$ row?
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 pairs of two unit square in a $n\times n$ grid share at least one grid point?
How many fractions in simplest form are there between $0$ and $1$ such that the products of their denominators and numerators equal $20!$?
How many positive integers greater than $9$ are there such that every digit is less that the digit on its right?
How many $n$-digit numbers can be formed using just 1 and 2, and no two 1s are next to each other?
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?
Joe marks a stick in three different ways. The first is to mark the stick in 10 equal intervals. The second is to mark it in 12 equal intervals. And finally, he marks it in 15 equal intervals. If Joe cuts the stick at all those marks, how many segments will he get?
How many positive integers not exceeding $10^6$ are there which are neither square nor cubic?
After having taken the same exam, Joe found he answered 1/3 of total problems incorrectly. Mary answered 6 incorrectly. The problems both didn't get right accounts for 1/5 of the total. Can you find how many problems did they both get right?
Mary plans to ask Joe to water the flowers during her summer vacation. Joe has a $10\%$ chance of forgetting this chore. If the flowers have an $85\%$ survival rate when watered but only a $20\%$ survival rate when not watered, what is the probability that the flowers will die upon Mary's return?
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?
The probability that Alice can solve a given problem is $1/2$. Beth has $1/3$ chance to solve the same problem. Carol's chance to solve it is $1/4$. If all them work on this problem independently, what is the probability that one and only one of them solves it?
Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.
Joe shoots the same target four times. If there is an $\frac{80}{81}$ chance he can hit the target at least once, what is his probability of hit the target in a single shoot?