MathDB

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

. https://cdn.artofproblemsolving.com/attachments/b/1/bb5d58b24d2dc68efcd6aa218489fd379f709c.png
p4. Each vertex of a cube is assigned the value

+1

-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

1

, how long is

AB

? https://cdn.artofproblemsolving.com/attachments/d/0/ed3ed73bda0d3fc4a1fb62c3cef5c829ccc937.jpg
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 [url=https://artofproblemsolving.com/community/c4h2690796p23355830]Juniors P1, P5.

Problem Statement