Practice (130)

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: $$2^n+2 > n^2$$

2631

Show that the next integer bigger than $(\sqrt{2}+1)^{2n}$ is divisible by $2^{n+1}$.

2668

Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.

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 $676$.

2855

Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.

3168

Let $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$

3196

Let $x$ be a real number, show that if the value of $\left(x+\frac{1}{x}\right)$ is an integer, then the value of $\left(x^n+\frac{1}{x^n}\right)$ is an integer too.

3200

Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

3299

Show that $1\cdot 1! + 2\cdot 2! + \cdots + n\cdot n! = (n+1)!-1$

3643

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that the Cauchy equation $$f(x+y)=f(x)+f(y)$$ holds for all $x, q\in\mathbb{Q}$.

3838

Let $n$ be an integer greater than $2$, prove $n^{n+1} > (n+1)^n$.

3839

Let $\{a_n\}$ be a sequence defined as $a_1=1$ and $a_n=\frac{a_{n-1}}{1+a_{n-1}}$ when $n\ge 2$. Find the general formula of $a_n$.

3840

Prove a positive proper fraction $\frac{m}{n}$ must be a sum of some reciprocals of distinct integers.

3854

$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$

3862

For $m=4k+1$ where $k$ is a positive integer. Show that $$\frac{1}{\sqrt{m}}\Big(\Big(\frac{1+\sqrt{m}}{2}\Big)^n-\Big(\frac{1-\sqrt{m}}{2}\Big)^n\Big)$$ must be an integer for any positive integer $n$.

3918

Prove that $\cos 1^\circ$ is irrational.

4243

Let $n$ be a positive integer and $k$ be an odd positive integer, show $k^{2^n}\equiv 1\pmod{2^{n+2}}$.

4251

Show that there exists an infinite number of squares in the form of $(n\cdot 2^k - 7)$ where $n$ and $k$ are both positive integers.

4284

Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$

4315

Let $n$ be a positive integer greater than $1$. Show $$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}\binom{2(k-1)}{k-1}\binom{2(n-k-1)}{n-k-1}=\frac{1}{n}\binom{2(n-1)}{n-1}$$

4316

Show that $$\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty}\binom{2n}{n}x^n$$

4487

In an election for the Peer Pressure High School student council president, there are $2019$ voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice’s boyfriend Bob votes for Alice as well. Then one by one, each of the remaining $2016$ voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate’s votes. For example, the first undecided voter David has a $2/3$ probability of voting for Alice and a $1/3$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

4681

$\textbf{Cheating Husbands}$

A remote town comprises of $100$ married couples. Everyone in the town lives by the following rule: If a husband cheats on his wife, the husband is executed at the night as soon as his wife finds out about it. All the women in the town only gossip about husbands of other women. No woman ever tells another woman if that woman's husband is cheating on her. So every woman in the town knows about all the cheating husbands in the town except her own. It can also be assumed that a husband remains silent about his infidelity. One day, the mayor of the town announces to the whole town that there is at least $1$ cheating husband in the town. What will happen afterwards?

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