Let $n$ be a positive integer, show that $11^{n+2} + 12^{2n+1}$ is a multiple of 133.
Show that for any positive integer $n$, the following relationship holds: $$2^n+2 > n^2$$
Colour all the points on a plane either white or black randomly. Show that it is always possible to find a triangle whose vertices have the same colour and its side length is either $1$ or $\sqrt{3}$.
Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.
Tom and Jerry are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2,\cdots, 2014$ are to be written. (Each number will be written exactly once). Tom fills in a piece of paper first. How many pieces of paper must Jerry fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Tom's in at least one position?
Show that:
(a) It is possible to divide all positive integers into three groups $A_1$, $A_2$, and $A_3$ such that for every integer $n\ge 15$, it is possible to find two distinct elements whose sum equals $n$ from all of $A_1$, $A_2$, and $A_3$.
(b) Dividing all positive integers into four groups, then regardless of the partition, there must exists an integer $n\ge 15$ such that it is impossible to find two distinct numbers whose sum is $n$ in at least one of these four groups.
Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)
- Find, with proof, the maximum possible value of $n$.
- For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.
Show that the next integer bigger than $(\sqrt{2}+1)^{2n}$ is divisible by $2^{n+1}$.
What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?
If $-1 < a < b < 0$, then which relationship below holds?
$(A)\quad a < a^3 < ab^2 < ab \qquad (B)\quad a < ab^2 < ab < a^3 \qquad (C)\quad a< ab < ab^2 < a^3 \qquad (D) a^3 < ab^2 < a < ab$
Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.
Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.
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 $2^k\mid a_n$.
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 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 $m^2$.
Let $n$ be a positive integer, show that $(3^{3n}-26n-1)$ is divisible by $676$.
If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$, find the coefficient of $x^2$.
$\textbf{Cover the Board}$
Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.
$\textbf{Cover the Board (II)}$
Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?
Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.
There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.