back to index  |  new

64
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. Find the arithmetic mean of all of these least elements.

474
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?

867

The expression $(x+y+z)^{2006}+(x-y-z)^{2006}$ can be simplified by expanding it and combining like terms. How many terms are there in the simplified expression?


Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.


Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$

for some natural number $k < n$.


If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.


For all positive integer $n$, show that $$\sum_{k=1}^n\frac{k\cdot k! \cdot\binom{n}{k}}{n^k}=n$$

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: $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2015 \times 2016$

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 $2^{n+1}$.

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 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 $$\frac{a^n+a^{n-1}b+\cdots+ab^{-1}+b^n}{n+1}\ge \Big(\frac{a+b}{2}\Big)^n$$

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.