Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.
Let the integer and decimal part of $(5\sqrt{2}+7)^{2n+1}$ be $I$ and $D$ respectively. Show that $(I+D)D$ is a constant.
Let $n$ be a non-negative integer. Show that $2^{n+1}$ divides the value of $\left\lfloor{(1+\sqrt{3})^{2n+1}}\right\rfloor$ where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding the give real number $x$.
Let $a$, $b$ be two positive real numbers, and $n$ be a positive integer greater than $2$. Show that $$\frac{a^n+a^{n-1}b+\cdots+ab^{-1}+b^n}{n+1}\ge \Big(\frac{a+b}{2}\Big)^n$$
Show that all the terms of the sequence $a_n=\frac{(2+\sqrt{3})^n-(2-\sqrt{3})^n}{2\sqrt{3}}$ are integers, and also find all the $n$ such that $3 \mid a_n$.
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?