Imagine there is an infinitive grid. Each grid is a square with side length of 1. Find the ratio of the number of points, number of unit squares and the number of sides of these unit squares.
Find the area of shaded area if the side length of the square is 1.
Show that $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.
A total of $2018$ tickets, numbered $1$, $2$, $3$, $\cdots$, $2014$, $2015$ are placed in an empty bag. Alfrid removes ticket $a$ from the bag. Bernice then removes ticket $b$ from the bag. Finally, Charlie removes ticket $c$ from the bag. They notice that $a < b < c$ and $a + b + c = 2018$. In how many ways could this happen?
Show that $$\binom{n}{k} = \frac{n}{k}\binom{n-1}{k-1}$$
Let positive integers $m\le k \le n$. Show that $$\binom{n}{k}\binom{k}{m} =\binom{n}{m}\binom{n-m}{k-m} =\binom{n}{k-m}\binom{n-k+m}{m}$$
Show that $$\binom{n}{0}+\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2}+\cdots+\frac{1}{n+1}\binom{n}{n}=\frac{2^{n+1}-1}{n+1}$$
Using at least two approaches to prove $$\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots +n\binom{n}{n} = n\cdot 2^{n-1}$$
Compute the value of $$\displaystyle\sum_{k=1}^n k^2\binom{n}{k}$$
Simplify: $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2015 \times 2016$
Show that $$\sum_{k=0}^{n-1}(m+k)(m+k+1)=\frac{n}{3}(3m^2 +3mn+n^2-1)$$
Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.
This four digit number $n$ has 14 positive factors and one of its prime factor has last digit equal to 1. What is $n$?
Let $X$ be the integer part of $\left(3+\sqrt{7}\right)^n$ where $n$ is a positive integer. Show that $X$ must be odd.
Let $n$ be a positive integer. Show that the smallest integer that is larger than $(1+\sqrt{3})^{2n}$ is divisible by $2^{n+1}$.
Let $m=4k+1$ where $k$ is a non-negative integer. Show that $$a=\binom{n}{1}+m\binom{n}{3}+m^2\binom{n}{5}+\cdots+m^{\frac{n-1}{2}}\binom{n}{n}$$
is divisible by $2^{n-1}$, where $n$ is an odd number.
Let sequence $\{a_n\}$ satisfy $a_0=0, a_1=1$, and $a_n = 2a_{n-1}+a_{n-2}$. Show that $2^k\mid n$ if and only if $2^k\mid a_n$.
Show that the following inequality holds for any positive integer $n$: $$(2n+1)^n \ge (2n)^n + (2n-1)^n$$
Let $a$ and $b$ be two positive real numbers. Show that if $\frac{1}{a}+\frac{1}{b}=1$. Prove that the following inequality holds for any positive integer $n$: $$(a+b)^n-a^n-b^n\ge 2^{2n}-2^{n+1}$$
Let $\{a_n\}$ be a sequence defined as $a_n=\lfloor{n\sqrt{2}}\rfloor$ where $\lfloor{x}\rfloor$ indicates the largest integer not exceeding $x$. Show that this sequence has infinitely many square numbers.
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 $3 \mid a_n$.