Problem 1483

1. Assume the contrary: Suppose it is possible to arrange the numbers from 51 to 150 in a $10 \times 10$ array such that for any two horizontally or vertically adjacent numbers $a$ and $b$, at least one of the equations $x^2 - ax + b = 0$ or $x^2 - bx + a = 0$ has two integral roots.

2. Consider the number 149:
- The cell containing 149 must have at least two adjacent cells. Let $a$ be one of the numbers in these adjacent cells.
- For the equation $x^2 - 149x + a = 0$ to have two integral roots, the discriminant must be a perfect square:
$$149^2 - 4a = d^2$$
where $d$ is an integer.

3. **Solve for $a$**:
- Rearrange the equation:
$$149^2 - d^2 = 4a$$
$$(149 - d)(149 + d) = 4a$$
- Since $a$ must be an integer between 51 and 150, we need to find integer solutions for $d$ such that $4a$ is an integer in this range.

4. **Check possible values of $d$**:
- The range for $d$ is determined by:
$$\sqrt{149^2 - 4 \cdot 150} \leq d \leq \sqrt{149^2 - 4 \cdot 51}$$
$$\sqrt{22001} \leq d \leq \sqrt{21801}$$
$$146 < d < 148$$
- The only integer value for $d$ in this range is $d = 147$.

5. **Find corresponding $a$**:
- Substitute $d = 147$ into the equation:
$$149^2 - 147^2 = 4a$$
$$(149 - 147)(149 + 147) = 4a$$
$$2 \cdot 296 = 4a$$
$$592 = 4a$$
$$a = 148$$

6. Consider the number 148:
- The cell containing 148 must also have at least two adjacent cells. Let $b$ be one of the numbers in these adjacent cells.
- For the equation $x^2 - 148x + b = 0$ to have two integral roots, the discriminant must be a perfect square:
$$148^2 - 4b = d^2$$
where $d$ is an integer.

7. **Solve for $b$**:
- Rearrange the equation:
$$148^2 - d^2 = 4b$$
$$(148 - d)(148 + d) = 4b$$
- Since $b$ must be an integer between 51 and 150, we need to find integer solutions for $d$ such that $4b$ is an integer in this range.

8. **Check possible values of $d$**:
- The range for $d$ is determined by:
$$\sqrt{148^2 - 4 \cdot 150} \leq d \leq \sqrt{148^2 - 4 \cdot 51}$$
$$\sqrt{21704} \leq d \leq \sqrt{21504}$$
$$146 < d < 148$$
- The only integer value for $d$ in this range is $d = 147$.

9. **Find corresponding $b$**:
- Substitute $d = 147$ into the equation:
$$148^2 - 147^2 = 4b$$
$$(148 - 147)(148 + 147) = 4b$$
$$1 \cdot 295 = 4b$$
$$295 = 4b$$
$$b = 73.75$$
- Since $b$ is not an integer, this contradicts our assumption.

10. Conclusion: Since we have reached a contradiction, it is not possible to arrange the numbers from 51 to 150 in a $10 \times 10$ array such that for any two horizontally or vertically adjacent numbers $a$ and $b$, at least one of the equations $x^2 - ax + b = 0$ or $x^2 - bx + a = 0$ has two integral roots.

$\blacksquare$