Find pairs of integers such that:

Let be a natural number and be a prime number. How many different pairs satisfy the equation:

Inspired by https://artofproblemsolving.com/community/c4h3560823

Find all ordered pairs of positive integers such that and are both perfect cubes.

A and B play a game on two table:
1.At first one table got different coloured marbles on it and another one is empty
2.At each move player choose set of marbles that hadn't choose either players before and all chosen marbles from same table, and move all the marbles in that set to another table
3.Player who can not move lose
If A starts and they move alternatily who got the winning strategy?

ChGiven an acute triangle ABC inscribed in circle The altitudes , intersect
each other at . The tangents at and of intersect at . Let be the midpoint of . intersects
at , intersects at
a) Prove that the points lie on the same circle.
b) The circle with diameter intersects the circle at the second point The line intersects
at the second point . Prove that are collinear.

Find the smallest positive integer such that is an exact 5th power, is an exact 6th power, and is an exact 7th power.

Let denote the set of natural numbers and let denote ordered pairs in . Prove that there exist distinct elements in the set for whose product is a perfect cube.

Let . Prove that

a, b and c are three non-negative reel numbers such that a+b+c=1.
What are the extremums of
ab+bc+ca-abc
?

p1. Compute the smallest positive integer such that is a perfect cube.


p2. A four digit integer is chosen at random. What is the probability all digits are distinct?


p3. If Solve for .


p4. In , , , and . Let be the point on such that , and let be the midpoint of . If is a point such that is a rectangle, compute the area of .


p5. Square has a sidelength of . Points , , , and are chosen on , , , and respectively, such that , , , and are length . Compute the area of quadrilateral .


p6. A sequence satisfies for all integers , If and , compute .


p7. In a class, every child has either red hair, blond hair, or black hair. All but children have black hair, all but have red hair, and all but have blond hair. How many children are there in the class?


p8. An Akash set is a set of integers that does not contain two integers such that one divides the other. Compute the minimum positive integer such that the set can be partitioned into n Akash subsets.


PS. You should use hide for answers. Collected here.