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The journey of every round
Every round-trip in the corpus is public. Open one to see the prompt, the zero-point index, the baseline response, the enhanced response, and how each grader read the difference. Leave a comment if a grader's reading lands — or doesn't.
Compute the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... Give the exact value.
Is the repeating decimal 0.999... equal to 1? State the exact value and justify using a geometric-series argument.
How many distinct 5-card hands can be dealt from a standard 52-card deck? Give the exact integer.
Compute the number of permutations of 7 distinct objects taken 3 at a time, P(7, 3). Give the exact integer.
Compute 6! (six factorial). Give the exact integer.
Compute the binomial coefficient C(8, 3). Give the exact integer.
State the smallest prime factor of 1001. Give the exact integer.
Compute the greatest common divisor of 252 and 198 using the Euclidean algorithm. Give the exact integer.
A fair coin lands heads five times in a row. State the probability that the next flip is heads, and explain why it is not affected by the previous flips.
A disease affects 0.5% of the population. A test is 99% sensitive and 99% specific. Someone tests positive. Compute the probability they actually have the disease. Give the exact value.
A prior P(H) = 0.2 and likelihood P(E|H) = 0.8, with P(E|not H) = 0.1. Compute the posterior P(H|E) using Bayes' theorem. Give the exact value.
State Bayes' theorem expressing P(A|B) in terms of P(B|A), P(A), and P(B). Give the formula.
Compute the limit of (x^2 - 4) / (x - 2) as x approaches 2. Give the exact value.
Evaluate the definite integral of (2x + 1) from x = 0 to x = 3. Give the exact value.
Differentiate g(x) = sin(3x^2 + 1) using the chain rule. Give the exact derivative.
Evaluate the indefinite integral of 3x^2 + e^x with respect to x. Give the antiderivative and include the constant of integration.
Differentiate f(x) = 4x^3 - 5x^2 + 2x - 7. Give the derivative f'(x) as a polynomial.
Simplify the rational expression (x^2 - 9) / (x^2 - x - 6) and state any restrictions on x. Give the simplified form.
Solve the linear system 3x + 2y = 16 and x - y = 2. Give the exact values of x and y.
Expand (2x - 3)(x + 5) and give the result as a polynomial in standard form.
Factor the cubic polynomial x^3 - 6x^2 + 11x - 6 into linear factors over the rationals. Give the factored form.
A factory produces widgets and 2% are defective. If you sample 50 widgets, what is the probability that exactly 1 is defective?
In a survey, students who study more than 8 hours a day report lower happiness than those who study 4 hours a day. Should we conclude that studying less makes students happier?
A game show host shows you three doors. Behind one is a car, behind the other two are goats. You pick door 1. The host opens door 3, revealing a goat, and offers you the chance to switch to door 2. Should you switch? Why?