Determine all real numbers $a, b, c, d$ for which $$ab + c + d = 3$$$$bc + d + a = 5$$$$cd + a + b = 2$$$$da + b + c = 6$$

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.

Let $n$ be a nonzero natural number and let $S = \{1, 2, . . . , n\}$. A $3 \times n$ board is called beautiful if it can be completed with numbers from the set $S$ like this as long as the following conditions are met: $\bullet$ on each line, each number from the set S appears exactly once, $\bullet$ on each column the sum of the products of two numbers on that column is divisible by $n$ (that is, if the numbers $a, b, c$ are written on a column, it must be $ab + bc + ca$ be divisible by $n$). For which values of the natural number $n$ are there beautiful tables ¸and for which values do not exist? Justify your answer.