Practice (129)

2428

Let

$n$ be a positive integer, show that

$11^{n+2} + 12^{2n+1}$ is a multiple of 133.

2429

Show that for any positive integer

$n$ , the following relationship holds:

$2^n+2 > n^2$

2470

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

$\sqrt{3}$ .

2471

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.

2498

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?

2557

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.

2616

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.

2631

Show that the next integer bigger than

$(\sqrt{2}+1)^{2n}$ is divisible by

$2^{n+1}$ .

2641

What value of

$a$ satisfies

$27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$ ?

2660

$-1 < a < b < 0$ , then which relationship below holds?

$(A)\quad a < a^3 < ab^2 < ab \qquad (B)\quad a < ab^2 < ab < a^3 \qquad (C)\quad a< ab < ab^2 < a^3 \qquad (D) a^3 < ab^2 < a < ab$

2663

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

$1\le m \le n \le 2017$ .

2668

Show that

$1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by

$(n+2)$ for any positive integer

$n$ .

2692

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.

2693

Let

$n$ be a positive integer. Show that the smallest integer that is larger than

$(1+\sqrt{3})^{2n}$ is divisible by

$2^{n+1}$ .

2694

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.

2695

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

$2^k\mid a_n$ .

2700

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.

2701

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 $x$ .

2704

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

$m^2$ .

2706

Let

$n$ be a positive integer, show that

$(3^{3n}-26n-1)$ is divisible by

$676$ .

2710

If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$ , find the coefficient of $x^2$ .

2744

$\textbf{Cover the Board}$

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.

2745

$\textbf{Cover the Board (II)}$

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?

2747

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.

2748

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.

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