Practice (4)
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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 $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.
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 $2020$.
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 $\{a_n\}$ 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 $n\mid g(n)[g(n)+1]$.
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 $3 \mid a_n$.
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 $m^2$.
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.
A positive integer is written on each face of a cube. Then for each vertex of the cube, the product of the numbers on the three faces associated with this vertex is calculated. If the sum of these eight products equals 2015, find the sum of all the numbers on the 6 faces.
Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
Compute the sum of all the roots of $(2x+3)(x-4)+(2x+3)(x-6)=0$
According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is
Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
Consider the sequence of numbers: $4,7,1,8,9,7,6,\dots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is:
Let $A_n$ be the average of all the integers between 1 and 101 which are the multiples of $n$ . Which is the largest among $A_2, A_3, A_4, A_5$ and $A_6$?
The sum of 2008 consecutive positive integers is a perfect square. What is the minimum value of the largest of these integers?
Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.