[b]p10.[/b] Three rectangles of dimension $X \times 2$ and four rectangles of dimension $Y  \times 1$ are the pieces that form a rectangle of area $3XY$ where $X$ and $Y$ are positive, integer values. What is the sum of all possible values of $X$?


[b]p11.[/b] Suppose we have a polynomial $p(x) = x^2 + ax + b$ with real coefficients $a + b = 1000$ and $b > 0$. Find the smallest possible value of $b$ such that $p(x)$ has two integer roots.


[b]p12.[/b] Ten square slips of paper of the same size, numbered $0, 1, 2, ..., 9$, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. We start with the given equation for the area of the rectangle formed by the pieces:
   \[
   3 \cdot (X \times 2) + 4 \cdot (Y \times 1) = 3XY
   \]
   Simplifying, we get:
   \[
   6X + 4Y = 3XY
   \]
2. Rearrange the equation to isolate terms involving \(X\) and \(Y\):
   \[
   3XY - 6X - 4Y = 0
   \]
   Add 8 to both sides to facilitate factoring:
   \[
   3XY - 6X - 4Y + 8 = 8
   \]
3. Factor the left-hand side:
   \[
   (3X - 4)(Y - 2) = 8
   \]
4. Since \(X\) and \(Y\) are positive integers, we need to find pairs \((3X - 4, Y - 2)\) that multiply to 8. The factor pairs of 8 are:
   \[
   (1, 8), (2, 4), (4, 2), (8, 1)
   \]
5. Solve for \(X\) and \(Y\) for each pair:
   - For \((3X - 4, Y - 2) = (1, 8)\):
     \[
     3X - 4 = 1 \implies 3X = 5 \implies X = \frac{5}{3} \quad (\text{not an integer})
     \]
   - For \((3X - 4, Y - 2) = (2, 4)\):
     \[
     3X - 4 = 2 \implies 3X = 6 \implies X = 2
     \]
     \[
     Y - 2 = 4 \implies Y = 6
     \]
   - For \((3X - 4, Y - 2) = (4, 2)\):
     \[
     3X - 4 = 4 \implies 3X = 8 \implies X = \frac{8}{3} \quad (\text{not an integer})
     \]
   - For \((3X - 4, Y - 2) = (8, 1)\):
     \[
     3X - 4 = 8 \implies 3X = 12 \implies X = 4
     \]
     \[
     Y - 2 = 1 \implies Y = 3
     \]
6. The possible integer values of \(X\) are 2 and 4. Summing these values:
   \[
   2 + 4 = 6
   \]

The final answer is \(\boxed{6}\).