MathDB

p1. Find all the primes

p

such that

p^2 + 2

is a prime number.
[url=https://artofproblemsolving.com/community/c4h1846777p12437991]p2. In the drawing, the five circles are tangent to each other and tangents to the lines

L_1

and

L_2

as shown in the following figure. The smallest of the circles has radius

8

and the largest has radius

18

. Calculate the radius of the circle

C

. https://cdn.artofproblemsolving.com/attachments/5/1/11f1836e94fc18faaf9bc0184d79a4c7dc47de.png
[url=https://artofproblemsolving.com/community/c4h3323176p30741866]p3. Consider a network composed of four regular hexagons as a sample in the figure: https://cdn.artofproblemsolving.com/attachments/d/8/361c80df79f777975ca03161d6af545a75a703.png A bee and a fly play the following game: initially the bee chooses one of the dots and paints it red, then the fly chooses one of the unpainted dots and paints it blue. Then the bee chooses an unpainted spot and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If in the end of the game there is an equilateral triangle with its red vertices, the bee wins, otherwise the fly wins. Determine which of the two insects has a winning strategy.
p4. A box contains

15

red pencils,

13

blue pencils, and

8

green pencils. Someone asks Constanza to take a number of pencils out of the box blindfolded. What is the minimum number of pencils that Constanza should take out in order to make sure she gets at least

1

red,

2

blue, and

3

green?
[url=https://artofproblemsolving.com/community/c1068820h2927818p26188408]p5. Let us call a

12

-sided regular polygon

P

. How many triangles is it possible to form using the vertices of

P

? How many of them are scalene triangles?
[url=https://artofproblemsolving.com/community/c4h1846776p12437980]p6. Consider two lines

L_1, L_2

that are cut at point

O

and

M

is the bisector of the angle they form, as shown in the following figure. Points

A

and

B

are drawn in

M

in such a way that

OA = 8

and

OB = 15

and the angle

\angle L_1OL_2

measures

45^o

. Calculate the shortest possible path length from

A

B

by touching lines

L_1

and

L_2

. https://cdn.artofproblemsolving.com/attachments/0/b/65e827fc3b2c48b32e0f423d83804df8afc352.png

Problem Statement