MathDB

p1. An ant is about to step on food that is

10

steps in front of it. The ant is being punished, he can only step forward a multiple of three steps and the rest must step backwards. Determine how many steps he takes to reach the food, if he has to take no more than twenty steps. (Note: if the ant takes two steps, each one steps backwards, then it is considered to be the same as two steps back.)
p2. Given that

n

is a natural number. Let's say

x=6+2009 \sqrt{n}

. If

\frac{x^{2009}-x}{x^3-x}

is a rational number, show that

n

is the square of a natural number.
[url=https://artofproblemsolving.com/community/c6h2372256p19397380]p3. Given triangle

ABC

and point

D

on side

AC

. Let

r_1

r_2

and

r

represent the radii of the inscribed circles of triangles

ABD

BCD

, and

ABC

, respectively. Prove that

r_1 + r_2 > r

.
p4. It is known that

p

is a prime number so that the equations

7p = 8x^2 -1

and

p^2 = 2y^2 - 1

have solutions

x

and

y

are integers. Find all

p

-values that satisfy.
p5. It is known that the set

H

has five elements from

\{0, 1, 2, 3,..., 9\}

. Prove that there are two subsets of

H

, which are non-empty and mutually exclusive, in which all the elements have the same sum.
[url=https://artofproblemsolving.com/community/c2476066_indonesia_regional]more

Problem Statement