Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which \[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\] is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.

For each positive integer $n$, put $$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$

Let $m,n$ be positive integers, greater than 2.Find the number of polynomials of degree $2n-1$ with distinct coefficients from the set $\left\{ 1,2,\ldots,m\right\}$ which are divisible by $x^{n-1}+x^{n-2}+\ldots+1.$

A list consists of all positive integers from $1$ to $2020$, inclusive, with each integer appearing exactly once. Define a move as the process of choosing four numbers from the current list and replacing them with the numbers $1,2,3,4$. If the expected number of moves before the list contains exactly two $4$'s can be expressed as $\frac{a}{b}$ for relatively prime positive integers, evaluate $a+b$. [i]Proposed by Richard Chen and Taiki Aiba[/i]

Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.

Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.

$N$ numbers - ones and twos - are arranged in a circle. We mean a number formed by several digits arranged in a row (clockwise or counterclockwise). For what is the smallest value of $N$, all four-digit numbers whose writing contains only numbers $1$ and $2$, could they be among those shown?

Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.

Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

Determine the perfect squares $ \overline{aabcd} $ of five digits such that $ \overline{dcbaa} $ is a perfect square of five digits.

Prove that the product of four consecutive integers plus $1$ is a perfect square.

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

$find$ all non negative integers $m$, $n$ such that $mn-1$ divides $n^3-1$

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions: (1) Each block must have at most $50$ states. (2) Every pair of states must be together in at least one block. Find the minimum number of blocks that must be formed.

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

Starting with the sequence $F_1 = (1,2,3,4, \ldots)$ of the natural numbers further sequences are generated as follows: $F_{n+1}$ is created from $F_n$ by the following rule: the order of elements remains unchanged, the elements from $F_n$ which are divisible by $n$ are increased by 1 and the other elements from $F_n$ remain unchanged. Example: $F_2 = (2,3,4,5 \ldots)$ and $F_3 = (3,3,5,5, \ldots)$. Determine all natural numbers $n$ such that exactly the first $n-1$ elements of $F_n$ take the value $n.$

$(4^{-1} - 3^{-1})^{-1} =$ $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ \frac{1}{12}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 12 $

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

Find all triples ( $\alpha, \beta,\theta$) of acute angles such that the following inequalities are fulfilled at the same time $$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$ $$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$ $$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$

If the answer to this problem is $x$, then compute the value of $\tfrac{x^2}{8} +2$. [i]Proposed by Lewis Chen [/i]

Fix a positive integer $n$. Let $a_1, a_2, \ldots$ be a sequence of positive integers such that for all $1 \leq j \leq n$, $a_j=j$, and for all $j>n$, $a_j$ is the largest value of $\min(a_i,a_{j-i})$ among $i=1,2, \ldots j-1$. For example, if $n=3$, we have $a_1=1$, $a_2=2$, $a_3=3$, and $a_4=2$ since $\min(a_1,a_3)=1$, $\min(a_2,a_2)=2$, and $\min(a_3,a_1)=1$. We will determine the values of $a_k$ for sufficiently large $k$. (a) Show that $a_i \in \{1,2,3, \ldots n\}$ for all $i$. (b) Show that if $a_x \geq n-1$ and $a_y \geq n-1$, $a_{x+y} \geq n-1$. (c) Show that for some positive integer $N$, $a_k \in \{n-1,n\}$ for all $k \geq N$. (d) Show that $a_k = n$ if and only if $n \mid k$.

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is [i]patriotic[/i] if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is [i]dazzling[/i] if its endpoints are of different colors. Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets. [i]Ankan Bhattacharya[/i]