Found problems: 373
Let the sequence $(a_n)$ be constructed in the following way:
$$
a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots.
$$
Prove that $a_{180}>19$.
[i](Folklore)[/i]
Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that
\[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
Consider a regular polygon with $2^n$ sides, for $n \ge 2$, inscribed in a circle of radius $1$. Denote the area of this polygon by $A_n$. Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$
For any positive integer $n$ and real numbers $a_i>0$ ($i=1,2,...,n$), prove that
\[\sum_{k=1}^n \frac{k}{a_1^{-1}+a_2^{-1}+...+a_k^{-1}}\leq 2\sum_{k=1}^n a_k.\]
Discuss if the "$2$" at the right hand side of the inequality can or cannot be replaced by a smaller real number.
Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that
$\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.
Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$.
[i]Proposed by Lewis Chen[/i]
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.
[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles.
[b]p2.[/b] There was a young lady named Chris,
Who, when asked her age, answered this:
"Two thirds of its square
Is a cube, I declare."
Now what was the age of the miss?
(a) Find the smallest possible age for Chris. You must justify your answer.
(Note: ages are positive integers; "cube" means the cube of a positive integer.)
(b) Find the second smallest possible age for Chris. You must justify your answer.
(Ignore the word "young.")
[b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$
[b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$.
(b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram.
(Note: "distance to a side" means the shortest distance to the line obtained by extending the side.)
[b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a_n = \sqrt{1 + (1 - \tfrac{1}{n})^2} + \sqrt{1 + (1 + \tfrac{1}{n})^2}, n \ge 1$. Evaluate $\tfrac{1}{a_1} + \tfrac{1}{a_2} + \ldots + \tfrac{1}{a_{20}}$.
Evaluate the following sum:
$$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$
[i]Proposed by Prudencio Guerrero Fernández[/i]
The sequence $\{a_n\}$ satisfies $a_0=1, a_1=2011,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let
\[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2} \]
What is $\frac{1}{S}$?
[i]Author: Ray Li[/i]
Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2+2n}$.
The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and
\[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\]
for $n \ge 0$. Evaluate the sum
\[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$
[Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that
\[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}).
\]
[i]Proposed by Romania[/i]
Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]
Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?
If $f(n)=\frac{2n+1+\sqrt{n(n+1)}}{\sqrt{n+1}+\sqrt{n}}$ for all positive integers $n$, evaluate
(a) $f(1)$,
(b) the sum $A=f(1)+f(2)+...+f(400)$.