Problems

Solve the equation with natural numbers $1+x+x^2+x^3=2y$.

Prove there are no natural numbers $a$ and $b$, such as $a^2-3b^2=8$.

Prove there are no integer solutions for the equation $x^2+1990=y^2$.

Robinson found a chest with books and instruments after the ship wreck. Not all the books were in readable condition, but some of the books he managed to read. One sentence read “72 chickens cost *619* p”. (The starred digits were not readable). He has not tasted a chicken for quite some time, and it was pleasant to imagine a properly cooked chicken in front of him. He also was able to decipher the cost of one chicken. Can you?

Jack believes that he can place $99$ integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?

Prove the magic trick for the number $1089={33}^2$: if you take any $3$-digit number $\overline{abc}$ with digits coming in strictly descending order and subtract from it the number obtained by reversing the digits of the original number $\overline{abc}-\overline{cba}$ you get another $3$-digit number, call it $\overline{xyz}$. Then, no matter which number you started with, the sum $\overline{xyz}+\overline{zyx}=1089$.
Recall that a number $\overline{abc}$ is divisible by $11$ if and only if $a-b+c$ also is.

Katie and Charlotte had $4$ sheets of paper. They cut some of the sheets into $4$ pieces. They then cut some of the newly obtained papersheets also into $4$ pieces. They did this several more times, cutting a piece of paper into $4$. In the end they counted the number of sheets. Could this number be $2024$?

Let $p$ and $q$ be two prime numbers such that $q=p+2$. Prove that $p^q+q^p$ is divisible by $p+q$.

Is it possible to construct a 485 × 6 table with the integers from 1 to 2910 such that the sum of the 6 numbers in each row is constant, and the sum of the 485 numbers in each column is also constant?

Prove for any natural number $n$ that $(n+1)(n+2)...(2n)$ is divisible by $2^n$.