Student Score Analysis
Description
In a class with a total score of 8250:
- The top 3 students scored 86, 83, and 80 respectively
- The lowest score is 30
- Each score can be shared by at most 3 students
Find the minimum number of students who scored above 60.
Input Example
total_score = 8250
top_scores = [86, 83, 80]
min_score = 30
max_students_per_score = 3
threshold = 60
Output Example
At least 75 students scored above 60
Tangent Line of a Logarithmic Equation
Description
Find the equation of the tangent line to the curve defined by
ln(y) + y = -2ln(x) + x
at the point (1, 1).
Output Example
y = -x + 2
Correlation Between Coin Toss Variables
Description
A fair coin is tossed three times.
Define two random variables:
- X = 1 if the first toss is heads and the third is tails, otherwise X = 0
- Y = 1 if the second and third tosses are both tails, otherwise Y = 0
Compute the correlation between X and Y.
Output Example
Correlation = -1/8
Null Space Dimension of a Linear Transformation
Description
Given a linear transformation from R⁵ to R³ defined by:
(x₁, x₂, x₃, x₄, x₅) → (x₁ + x₂, x₂ + x₃ + x₄, x₄ + x₅)
Find the dimension of the null space.
Output Example
Null space dimension = 2
Expected Value of Uniform Phone Storage
Description
Assume k phone numbers are uniformly and independently stored in n locations.
What is the expected number of occupied locations?
Output Example
Expected value = n * (1 - (1 - 1/n)^k)
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