Let’s analyze and solve the questions in this image step by step.

Question 1:

A three-digit natural number $abc$ is defined as an "exponential number" if $a^b = c$.

We are asked to evaluate the following claims:

I. The largest exponential number is $888$.
II. There are 3 odd exponential numbers.
III. $826$ is an exponential number.

Solution:

• For I: To test the largest exponential number, we can systematically check all three-digit numbers $abc$. $a = 8$, $b = 8$, $c = 8$ leads to $8^8$, which far exceeds three digits. This makes $888$ invalid as an exponential number.
  Thus, I is FALSE.

• For II: Odd exponential numbers occur when $c$ (the result of $a^b$) is odd. This happens if $a$ (the base) is odd. Testing all odd bases from $1$ to $9$, we find exactly 3 valid odd cases.
  Thus, II is TRUE.

• For III: $826$ implies $8^2 = 6$, which is incorrect. Therefore, $826$ is not an exponential number.
  Thus, III is FALSE.

Answer: B) II only

Question 2:

The line segment is divided into 20 equal parts, and points $p, n,$ and $m$ are given positions relative to the left and right ends. The task is to compute the result of the expression $n + m - p$.

Solution: From the image:

• $p$ is 12 units from the right, so $p = 20 - 12 = 8$.
• $n$ is 16 units from the left, so $n = 16$.
• $m$ is 11 units from the left, so $m = 11$.

Now, calculate: $n + m - p = 16 + 11 - 8 = 19$

Answer: C) 1.9

Question 3:

The equation uses four variables $A, B, C, D$, with specific relationships: $A - 4 = B$, $C - A = D + B$, and all variables represent distinct digits. We analyze the claims about their sums.

Solution:

• $A - 4 = B$: This implies $A > 4$ (since $B$ is a digit).
• $C - A = D + B$: Rearranging, $C = A + D + B$. Thus, $C > A$.

The following claims:

1. $A + B > 10$: Clearly true if $A > 6$ and $B > 4$.
2. $C + D \geq 10$: This follows from $C = A + D + B$, making the sum of $C$ and $D$ larger than 10.
3. Only pairs $(C, D)$: The conditions lead to valid pairs (verification involves testing digits).

Answer: E) I, II, and III

Question 4:

A broken calculator displays halved values for even numbers and $x - 1$ for odd numbers. A sequence of inputs produces outputs $4, c, d, e$, where the inputs are $3, 8, 6, 5$. Find the sum $a + b + c + d$.

Solution:

• For $3$ (odd): $a = 3 - 1 = 2$.
• For $8$ (even): $b = 8 / 2 = 4$.
• For $6$ (even): $c = 6 / 2 = 3$.
• For $5$ (odd): $d = 5 - 1 = 4$.

Sum: $a + b + c + d = 2 + 4 + 3 + 4 = 13$

Answer: D) 13

Do you want a detailed explanation of any specific question?

Here are 5 follow-up questions related to the topic:

1. How would you define and analyze "exponential numbers" for larger cases?
2. How can systematic testing of variables like in Question 3 be automated?
3. What techniques help solve piecewise function problems like the broken calculator?
4. How does coordinate reasoning help in solving number line problems?
5. What strategies help in solving equations with constraints on distinct digits?

Tip: Always validate numerical patterns with examples to ensure accuracy in logical questions.