BD-BDMO 2019

Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

Prove that,if $a,b,c$ are positive real numbers, $$\dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}$$

Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$ . $\omega$ intersects $\alpha$ at $A$ and $B$ .Let $P$ any point on the circumference $\omega$ .The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$ .

$A$ is a positive real number. $n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$ .

Prove that for all positive integers $n$ we can find a permutation of { $1,2,...,n$ } such that the average of two numbers doesn't appear in-between them.For example { $1,3,2,4$ }works,but { $1,4,2,3$ } doesn't because $2$ is between $1$ and $3$ .

When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$ .If $f(x)=\dfrac {e^x}{x}$ .Find the value of $$\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}$$

Given three cocentric circles $\omega_1$ , $\omega_2$ , $\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$ .Constrat a line that intersects $\omega_1$ , $\omega_2$ , $\omega_3$ at $A,B,C$ respectively such that $AB=BC$ .

The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$ ,there are infinitely many multiples of $n$ in $S$ .

Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$ . $\angle {APB}=\angle {CPD}$ .Prove that $AB+BD=AC+CD$ .

Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.