Show that all terms of the sequence $a_n=\left(\frac{3+\sqrt{5}}{2}\right)^n+\left(\frac{3-\sqrt{5}}{2}\right)^n -2$ are integers. And when $n$ is even, $a_n$ can be expressed as $5m^2$, when $n$ is odd $a_n$ can be expressed as $m^2$.
If the $5^{th}$, $6^{th}$ and $7^{th}$ coefficients in the expansion of $(x^{-\frac{4}{3}}+x)^n$ form an arithmetic sequence, find the constant term in the expanded form.
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?
If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$, find the coefficient of $x^2$.
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?
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?
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?
Mr Wise come across a group of $4$ people. He finds that some of these $4$ people always tell the truth and some always tell lies. So he asks each of them and gets the following answers:
- $A$: "All of us tell lies."
- $B$: "There is only one among our four who tells lies."
- $C$: "There are two among us who tell lies."
- $D$: "I am telling the truth."
Do you think whether $D$ tells the truth?
There are $7$ pieces of paper on a table. Cut some of them into $7$ each and put them back to the table. Repeat this process as many times as you wish. Is it possible to have $1990$ pieces of paper on the table at some moment?
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?