1. **Problem:** In the figure with BD || AE, given \( \angle DBC = 20^\circ \) and \( \angle BCA = 90^\circ \), find \( \angle CAE \). Step 1: Since BD || AE and BC is a transversal, \( \angle DBC = \angle CAE = 20^\circ \) by alternate interior angles. Step 2: Therefore, the answer is \( 20^\circ \). 2. **Problem:** Find the minimum number of numbers to select from \( \{1, 2, 3, ..., 30\} \) to ensure at least one multiple of 5 is chosen. Step 1: The multiples of 5 in the set are \( 5, 10, 15, 20, 25, 30 \), 6 numbers total. Step 2: The worst case to avoid multiples of 5 is to pick all 24 numbers that are not multiples of 5. Step 3: Selecting one more than 24, i.e., 25 numbers, guarantees at least one multiple of 5. 3. **Problem:** If the radius of a circle is increased by 100%, by what percentage does its area increase? Step 1: Original radius = \( r \), new radius = \( r + 100\% \times r = 2r \). Step 2: Original area = \( \pi r^2 \), new area = \( \pi (2r)^2 = 4 \pi r^2 \). Step 3: Area increased by \( 4 \pi r^2 - \pi r^2 = 3 \pi r^2 \), which is a 300% increase. 4. **Problem:** Given \( \overline{abc} = 13 \times (a + b + c) \) and \( \overline{abc} \leq 200 \), find the minimum and maximum possible values of \( \overline{abc} \). Here, \( \overline{abc} \) is the three-digit number with digits a, b, c. Step 1: Let the number be \( 100a + 10b + c \). Step 2: Given \( 100a + 10b + c = 13 (a + b + c) \). Step 3: Rearranged: \( 100a + 10b + c = 13a + 13b + 13c \) or \( 87a - 3b - 12c = 0 \). Step 4: Rewrite as \( 29a = b + 4c \). Step 5: Try values of \( a \) from 1 to 9, since \( \overline{abc} \) is three digits. For \( a = 3 \): \( 29 \times 3 = 87 = b + 4c \). Check integer solutions \( b, c \) with digits 0–9. Try \( c = 9 \): \( b = 87 - 4 \times 9 = 87 - 36 = 51 \) (not a digit). Try \( c = 6 \): \( b = 87 - 24 = 63 \) (not a digit). Try \( c = 3 \): \( b = 87 - 12 = 75 \) no. Try \( c = 0 \): \( b = 87 \) no. For \( a = 4 \): \( 29 \times 4 = 116 = b + 4c \). Try \( c = 9 \): \( b = 116 - 36 = 80 \) no. \( c=4 \): \( b = 116 - 16 = 100 \) no. Continue this method for other \( a \), or focus on valid digit solutions. Step 6: Known valid solutions are \( 117 = 13 \times (1+1+7=9)\) and \( 195 = 13 \times (1+9+5=15) \). Step 7: Both \( 117 \) and \( 195 \) are \( \leq 200 \). Step 8: So, minimum is 117, maximum is 195 (answer choice C). **Final Answers:** 1. \( 20^\circ \) 2. 25 3. 300% 4. Minimum \( 117 \), Maximum \( 195 \)