[url=https://artofproblemsolving.com/community/c6h2371620p19389503]p1.[/url] 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!
[img]https://cdn.artofproblemsolving.com/attachments/6/3/0bd6b71a0c09cd3aec2f49b31ff5b4d141de03.png[/img]
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.
To determine the values of \( k \) and \( d \) such that no pair of real numbers \((x, y)\) satisfies the system of equations:
\[ x^3 + y^3 = 2 \]
\[ y = kx + d \]

we proceed as follows:

1. **Substitute \( y = kx + d \) into \( x^3 + y^3 = 2 \):**
   \[
   x^3 + (kx + d)^3 = 2
   \]

2. **Expand \( (kx + d)^3 \):**
   \[
   (kx + d)^3 = k^3x^3 + 3k^2dx^2 + 3kd^2x + d^3
   \]

3. **Combine the terms:**
   \[
   x^3 + k^3x^3 + 3k^2dx^2 + 3kd^2x + d^3 = 2
   \]

4. **Group the \( x^3 \) terms together:**
   \[
   (1 + k^3)x^3 + 3k^2dx^2 + 3kd^2x + d^3 = 2
   \]

5. **For the equation to have no real solutions, the polynomial must have no real roots. Consider the case \( k = -1 \):**
   \[
   (1 + (-1)^3)x^3 + 3(-1)^2dx^2 + 3(-1)d^2x + d^3 = 2
   \]
   \[
   (1 - 1)x^3 + 3dx^2 - 3d^2x + d^3 = 2
   \]
   \[
   3dx^2 - 3d^2x + d^3 = 2
   \]

6. **Simplify the polynomial:**
   \[
   3dx^2 - 3d^2x + d^3 - 2 = 0
   \]

7. **For the quadratic equation \( 3dx^2 - 3d^2x + d^3 - 2 = 0 \) to have no real solutions, its discriminant must be less than 0:**
   \[
   \Delta = b^2 - 4ac
   \]
   where \( a = 3d \), \( b = -3d^2 \), and \( c = d^3 - 2 \).

8. **Calculate the discriminant:**
   \[
   \Delta = (-3d^2)^2 - 4(3d)(d^3 - 2)
   \]
   \[
   \Delta = 9d^4 - 12d(d^3 - 2)
   \]
   \[
   \Delta = 9d^4 - 12d^4 + 24d
   \]
   \[
   \Delta = -3d^4 + 24d
   \]
   \[
   \Delta = d(-3d^3 + 24)
   \]

9. **For the discriminant to be less than 0:**
   \[
   d(-3d^3 + 24) < 0
   \]

10. **Analyze the inequality:**
    - If \( d > 0 \), then \( -3d^3 + 24 < 0 \) implies \( d^3 > 8 \) or \( d > 2 \).
    - If \( d < 0 \), then \( -3d^3 + 24 > 0 \) is always true.

Therefore, the values of \( k \) and \( d \) such that no pair of real numbers \((x, y)\) satisfies the system of equations are:
\[ k = -1 \]
\[ d < 0 \text{ or } d > 2 \]

The final answer is \( \boxed{ k = -1 } \) and \( d < 0 \text{ or } d > 2 \).