Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ Lucian Petrescu

Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n. $ Prove that there exists a real number $ a $ such that $ a+a_1,a+a_2,\ldots ,a+a_n $ are all irrational.

Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $

Let $ ABCD $ be a cyclic quadrilateral,$ M $ midpoint of $ AC $ and $ N $ midpoint of $ BD. $ If $ \angle AMB =\angle AMD, $ prove that $ \angle ANB =\angle BNC. $

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$for any real numbers $ x,y. $

We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.

Let $ APD $ be an acute-angled triangle and let $ B,C $ be two points on the segments (excluding their endpoints) $ AP,PD, $ respectively. The diagonals of $ ABCD $ meet at $ Q. $ Denote by $ H_1,H_2 $ the orthocenters of $ APD,BPC, $ respectively. The circumcircles of $ ABQ $ and $ CDQ $ intersect at $ X\neq Q, $ and the circumcircles of $ ADQ,BCQ $ meet at $ Y\neq Q. $ Prove that if the line $ H_1H_2 $ passes through $ X, $ then it also passes through $ Y. $