MathDB

[url=https://artofproblemsolving.com/community/c6h2371620p19389503]p1. Given triangle

ABC

. Suppose

P

and

P_1

are points on

BC, Q

lies on

CA, R

lies on

AB

, such that

\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}

Let

G

be the centroid of triangle

ABC

and

K = AP_1 \cap RQ

. Prove that points

P,G

, and

K

are collinear.
p2. It is known that

k

is the largest positive integer, such that we can find the integer

n

positive prime numbers (not necessarily different)

q_1, q_2, q_3,... , q_k

, and different prime numbers

p_1, p_2,p_3, ..., p_k

that satisfy

\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_k}=\frac{7+nq_1g_2\cdot\cdot\cdot q_k}{2010}

Determine the number of

n

that satisfy.
p3. Determine the values of

k

and

d

so that no pair of real numbers

(x, y)

satisfies the system of equations:

x^3 + y^3 = 2

y = kx + d

p4. It is known that n is a natural multiple of

2010

. Show that the equation

x + 2y + 3z = 2n

has exactly

1+ \frac{n}{2}+ \frac{n^2}{12}

solution of triples

(x, y, z)

where

x

y

, and

z

are not negative integers .
p5. Given a chessboard as shown in the picture. Can a horse chess piece depart from one tile passes through every other tile only once and returns to its original place ? Explain your answer! https://cdn.artofproblemsolving.com/attachments/6/3/0bd6b71a0c09cd3aec2f49b31ff5b4d141de03.png Explanation: The horse chess move is

L

-shaped, that is, from the original box: (a)

2

squares to the right/left and

1

box to the front/back; or (b)

2

squares to the front/back and

1

box to the right/left.

Problem Statement