MathDB

p1. The sides of a triangle

ABC

have lengths

AB = 26

cm,

BC = 17

cm, and

CA = 19

cm. The bisectors of the angles

B

and

C

intersect at the point

I

. By

I

a parallel to

BC

is drawn that intersects the sides

AB

and

BC

at points

M

and

N

respectively. Calculate the perimeter of the triangle

AMN

.
p2. Let

n

be a positive integer. Determine the exact value of the following sum

\frac{1}{1+\sqrt3}+\frac{1}{\sqrt3+\sqrt5}++\frac{1}{\sqrt5+\sqrt7}+...+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}

p3. Find the product of all positive integers that are less than

150

and that have exactly

4

positive divisors. Express your answer as a product of prime factors.
p4. There is a board of

n \times n

, where n is a positive integer and the numbers from

1

n

are written in each of its rows in some order, so that the arrangement obtained is symmetric with respect to the diagonal. It is desired that the diagonal of the board does not contain all the numbers from

1

n

. Show that it is always possible when

n

is even and that it is impossible when

n

is odd.
p5. There are six circles of radius

\frac12

, tangent to each other and to the sides of the rectangle, as shown in the figure. Also the centers

A

B

C

and

D

are collinear. Determine the lengths of the sides of the rectangle. https://cdn.artofproblemsolving.com/attachments/e/1/c8c23f796a73aaab25b78d8f0539750dc3f159.png

Problem Statement