of a semicircle of radius 
.
into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
be 
,
.
?
through 
,
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
Y by
p1. Show that if 
are odd integers then the equation 
has no rational roots.
p2. If
is a positive integer, find the greatest power of 
that divides 
.
p3. The figure shows the triangle
, right at 
, its circumscribed circle and semicircles built on the two legs.
Show that the sum of the areas of the two shaded regions is
.
p4. Each vertex of a cube is assigned the value
or 
, and each face the product of the values assigned to its vertices. What values can the sum of the 
numbers thus obtained, have?
p5. Consider the regular pentagon
in the figure. If 
is 
, how long is 
?
p6. Let
be an integer. A circle is divided into 
arcs by points. Each arch measures one of three possible lengths, and no two adjacent arches are the same length. The points are alternately colored red and blue. Show that the 
-gon with blue vertices and the -gon with red vertices have the same perimeter and the same area.
PS. Seniors P3,P4 were also posted as Juniors P1, P5.
are odd integers then the equation 
has no rational roots.p2. If
is a positive integer, find the greatest power of 
that divides 
.p3. The figure shows the triangle
, right at 
, its circumscribed circle and semicircles built on the two legs.Show that the sum of the areas of the two shaded regions is
.p4. Each vertex of a cube is assigned the value
or 
, and each face the product of the values assigned to its vertices. What values can the sum of the 
numbers thus obtained, have?p5. Consider the regular pentagon
in the figure. If 
is 
, how long is 
?p6. Let
be an integer. A circle is divided into 
arcs by points. Each arch measures one of three possible lengths, and no two adjacent arches are the same length. The points are alternately colored red and blue. Show that the 
-gon with blue vertices and the -gon with red vertices have the same perimeter and the same area.PS. Seniors P3,P4 were also posted as Juniors P1, P5.
This post has been edited 3 times. Last edited by parmenides51, Oct 14, 2021, 4:13 PM
.![[asy]
draw(circle((0,0),5));
draw((-5,0)--(3,4)--(5,0)--cycle);
draw(arc((-1,2),(3,4),(-5,0)));
draw(arc((4,2), (5,0), (3,4)));
label((5.2, 2.6), "B");
label((-2,5.5), "A");
label((-1.5, 3.2),"C");
label((2,2),"D");
label((4.25,2.25), "E");
[/asy]](http://latex.artofproblemsolving.com/0/d/3/0d3d1e6e4afe58579023f9690f53235f6285d012.png)
and 
as shown in the diagram. Let each leg have lengths 
and 
respectively. Then, the hypothenuse will have length 
and thus the radius will be 
.
by LTE, so the answer is 
.
should be a perfect square. In the present case, where a, b and c are all odd, it would be an odd square, which taken mod 8 should give 1, contrary to 
mod 8 and 
mod 8). Contradiction.
