p1. Show that if $a, b, c$ are odd integers then the equation $ax^2 + bx + c = 0$ has no rational roots. p2. If $k$ is a positive integer, find the greatest power of $3$ that divides $10^k-1$. p3. The figure shows the triangle $ABC$, right at $C$, its circumscribed circle and semicircles built on the two legs. Show that the sum of the areas of the two shaded regions is $\frac12 AC\cdot CB$. p4. Each vertex of a cube is assigned the value $+1$ or $-1$, and each face the product of the values assigned to its vertices. What values can the sum of the $14$ numbers thus obtained, have? p5. Consider the regular pentagon $ABCDE$ in the figure. If $BI$ is $ 1$, how long is $AB$? p6. Let $n\ge 3$ be an integer. A circle is divided into $2n$ arcs by $2n$ points. Each arch measures one of three possible lengths, and no two adjacent arches are the same length. The $2n$ points are alternately colored red and blue. Show that the $n$-gon with blue vertices and the $n$-gon with red vertices have the same perimeter and the same area. PS. Seniors P3,P4 were also posted as Juniors P1, P5.