Problem 1: Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

Problem 2: Prove that, if a, b, c are positive real numbers,

Problem 3: Let α and ꙍ be 2 circles such that ꙍ goes through the center of α. ꙍ intersects α at A and B. Let P be any point on the circumference ꙍ. The lines PA and PB intersects α again at E and F respectively. Prove that AB = EF.

Problem 4: *A *is a positive real number. Find the set of possible values of the infinite x_{0}* ^{n}+ x_{1}^{n}+ x_{2}^{n} +* … … where

*x*are all positive real numbers so that the infinite series

_{0}, x_{1}, x_{2}*x*

_{0}+

*x*

_{1}+

*x*

_{2}+

*…*has sum

*A*.

Problem 5: 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, 2, 3, 4} works, but {1, 2, 3, 4} doesn’t because 2 is between 1 and 3.

Problem 6: When a function *f*(*x*) is differentiated *n *times, the function we get is denoted by *f*^{(n)}(*x*). If *f*(*x*) = *e ^{x}/x*, Find the value of

Problem 6: Given three concentric circles ꙍ_{1}, ꙍ_{2}, ꙍ_{3} with radius r_{1}, r_{2}, r_{3} such that r_{1} + r_{3} ≥ 2r_{2}, construct a line that intersects ꙍ_{1}, ꙍ_{2}, ꙍ_{3} at A, B, C respectively such that AB = BC.

Problem 8: The set of natural numbers N are partitioned into a finite number of subsets. Prove that

there exists a subset *S *so that for any natural number *n*, there are infinitely many multiples of *n *in

*S*.

Problem 9: Let *ABCD *be a convex quadrilateral. The internal angle bisectors of ∠BAC and ∠BDC*
*meets at

*P*∠APB = ∠CPD. Prove that,

*AB*+

*BD*=

*AC*+

*CD*.

Problem 10: In chess, a normal knight goes two steps forward and one step to the side, in some orientation. Thanic thought that he should spice the game up a bit, so he introduced a new kind of piece called a warrior. A warrior can either go three steps forward and one step to the side, or two steps forward and two steps to the side in some orientation.

In a 2020 *× *2020 chessboard, prove that the maximum number of warriors so that none of them attack each other is less than or equal to 2/5 of the number of cells.