Problems

Which term in the expansion $(1+\sqrt{3}{)}^{100}$ will be the largest by the Newton binomial formula?

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.

Find the sums of the following series:

a) $\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\frac{1}{4\times 5}+\dots$;

b) $\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots$;

c) $\frac{0!}{r!}+\frac{1!}{(r-1)!}+\frac{2!}{(r-2)!}+\frac{3!}{(r-3)!}+\dots$ for $r\ge 2$.

Each side in the triangle $ABC$ is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division (the points $A$, $B$, $C$ cannot be the vertices of triangles) in which neither side is parallel to either side of the triangle $ABC$?

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 $n\times n$. It is required to move a rook from the bottom left corner to the upper right corner. You can move only up and to the right, without going into the cells of the main diagonal and the one below it. (The rook is on the main diagonal only initially and in the final moment in time.) How many possible routes does the rook have?

Tickets cost 50 cents, and $2n$ buyers stand in line at a cash register. Half of them have one dollar, the rest – 50 cents. The cashier starts selling tickets without having any money. How many different orders of people can there be in the queue, such that the cashier can always give change?

Prove that the Catalan numbers satisfy the recurrence relationship $C_n=C_0{C}_{n-1}+C_1{C}_{n-2}+\cdots +C_{n-1}{C}_0$. The definition of the Catalan numbers $C_n$ is given in the handbook.

Determine all prime numbers $p$ and $q$ such that $p^2-2q^2=1$ holds.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.