SE-SKOLORNAS 1992

Is $\frac{19^{92} - 91^{29}}{90}$ an integer?

The squares in a $9\times 9$ grid are numbered from $11$ to $99$ , where the first digit is the row and the second the column. Each square is colored black or white. Squares $44$ and $49$ are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square $99$ ?

Solve: $$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 2x_2 - 5x_3 + 3x4 \ge 0 ... 2x_{23} - 5x_{24} + 3x_{25} \ge 0 2x_{24} - 5x_{25} + 3x_1 \ge 0 2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$

Find all positive integers $a, b, c$ such that $a < b$ , $a < 4c$ , and $b c^3 \le a c^3 + b$ .

A triangle has sides $a, b, c$ with longest side $c$ , and circumradius $R$ . Show that if $a^2 + b^2 = 2cR$ , then the triangle is right-angled.

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$ . Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$ .