Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.

Click for solution Suppose the sequence contains finitely many powers of $ 2$. Let $ 2^t$ the greatest power met in the sequence. Take $ n>t$. Then for some $ m$, $ m\sqrt2<2^n<2^n+1<(m+1)\sqrt2$, implying $ 2^n>m\sqrt2\ge2^n-(\sqrt2-1)>2^n-\dfrac12$. Let $ m\sqrt2=2^n-\epsilon$. Then $ \epsilon<\dfrac12$. Let $ k$ be a positive integer such that $ 2^k\epsilon<1<2^{k+1}\epsilon$. Clearly $ k\ge1$ and $ \dfrac12<2^k\epsilon$. This means $ 2^{n+k}>2^km\sqrt2=2^{n+k}-2^k\epsilon>2^{n+k}-1$. We must also have $ 2^km\sqrt2=2^{n+k}-2^k\epsilon>2^{n+k}-(\sqrt2-1)>2^{n+k}-\dfrac12$, which is $ 2^k\epsilon<\dfrac12$, contradiction.

Let $a_{n}$ be the last nonzero digit in the decimal representation of the number $n!$. Does the sequence $a_{1}$, $a_{2}$, $a_{3}$, $\cdots$ become periodic after a finite number of terms?

Let $\,n>6\,$ be an integer and $\,a_{1},a_{2},\ldots,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.

Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.

Prove that there exist two strictly increasing sequences $a_{n}$ and $b_{n}$ such that $a_{n}(a_{n} +1)$ divides $b_{n}^2 +1$ for every natural $n$.

Let $\{a_{n}\}$ be a strictly increasing positive integers sequence such that $\gcd(a_{i}, a_{j})=1$ and $a_{i+2}-a_{i+1}>a_{i+1}-a_{i}$. Show that the infinite series \[\sum^{\infty}_{i=1}\frac{1}{a_{i}}\] converges.

Let $\{n_{k}\}_{k \ge 1}$ be a sequence of natural numbers such that for $i<j$, the decimal representation of $n_{i}$ does not occur as the leftmost digits of the decimal representation of $n_{j}$. Prove that \[\sum^{\infty}_{k=1}\frac{1}{n_{k}}\le \frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{9}.\]

An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.

Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that a) for any $ m > n$, $ m - n$ is a factor of $ q_{m} - q_{n}$, b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$. Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} = Q(n)$ for all $ n$.

Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).

The infinite sequence of 2's and 3's \[\begin{array}{l}2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ 3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\cdots \end{array}\] has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1+\lfloor rm \rfloor$ for some nonnegative integer $m$.

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite.

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

Let $a$ be the common difference and $an+b$ the sequence. If $ak+b=m^2$ for some $k,m$, then $\infty$ many squares are given by $(na+m)^2=n^2a^2+2nam+m^2=(k+an^2+2mn)\cdot a + b$. This immeadiatly generalizes to any powers.

In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

Does there exist positive integers $a_{1}<a_{2}<\cdots<a_{100}$ such that for $2 \le k \le 100$, the greatest common divisor of $a_{k-1}$ and $a_{k}$ is greater than the greatest common divisor of $a_{k}$ and $a_{k+1}$?

Suppose that $a$ and $b$ are distinct real numbers such that \[a-b, \; a^{2}-b^{2}, \; \cdots, \; a^{k}-b^{k}, \; \cdots\] are all integers. Show that $a$ and $b$ are integers.