If real numbers $a$ and $b$ satisfy $a^2 + b^2=1$, find the minimal value of $a^4 + ab+b^4$.
Distinct real numbers $a$, $b$ and $c$ satisfy $a+\frac{1}{b}=b+\frac{1}{c} = c+\frac{1}{a}=t$. Find the value of $t$.
How many integers $m$ are there for which $5\times 2^m +1$ is a square number?
If real numbers $a$, $b$ and $c$ satisfy $abc=-1$, $a+b+c=4$, $\frac{a}{a^2-3a-1}+\frac{b}{b^2-3b-1}+\frac{c}{c^2-3c-1}=\frac{4}{9}$, what is the value of $a^2+b^2+c^2$?
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.
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: $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2015 \times 2016$
Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.
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 $2^{n+1}$.
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 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 $x$.
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 $3 \mid a_n$.
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 $3: 4: 5$?
Label the first row of the Pascal triangle as row $0$. How many odd numbers are there in the $2019^{th}$ row?
Let $n$ be a positive integer. Show that $\left(3^{4n+2} + 5^{2n+1}\right)$ is divisible by $14$.
Simplify the expression $$\binom{2020}{0}^2 + \binom{2020}{1}^2 + \cdots + \binom{2020}{2020}^2$$
Show that $$\sum_{k=0}^m \binom{n}{k}\binom{n-k}{m-k}= 2^m\binom{n}{m}$$
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?
Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.