In the expansion of
Show that any natural number
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.
Each side in the triangle
How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?
Consider a chess board of size
Tickets cost 50 cents, and
Prove that the Catalan numbers satisfy the recurrence relationship
Determine all prime numbers
Prove that for a real positive