# GAOKAO MATHEMATICS — ADVANCED REASONING BENCHMARK You are...

Prompt

# GAOKAO MATHEMATICS — ADVANCED REASONING BENCHMARK You are taking an advanced Chinese Gaokao (高考) mathematics examination. **Rules:** - Solve both problems independently without web searches, programming, calculators, or external tools. - Provide rigorous mathematical proofs, not just final answers. - Explicitly justify all nontrivial claims, especially uniqueness, monotonicity, conditional probabilities, and limits. - Do not invent theorems or assume what you are trying to prove. - If you cannot solve a part, explicitly acknowledge that rather than guessing. - Your responses will be evaluated for mathematical correctness, completeness, logical consistency, and clarity. **Total: 100 points** --- ## PROBLEM 1 — Advanced Calculus and Parameter Analysis (65 points) For every real parameter \(a\), consider the function \[ f_a(x)=\frac{x^2}{2}+\ln x-ax,\qquad x>0. \] **Part A — Monotonicity (15 points)** Determine all intervals on which \(f_a(x)\) is strictly increasing or strictly decreasing, for every possible value of \(a\). All parameter cases must be handled. **Part B — Existence and Uniqueness (15 points)** Prove that, for every real \(a\), the equation \[ f_a(x)=0 \] has exactly one positive real solution. A graphical argument alone is insufficient. **Part C — Implicit Parameter Dependence (15 points)** Let \(r(a)\) denote the unique positive solution. 1. Prove that \(r(a)\) is strictly increasing on \(\mathbb{R}\). 2. Determine precisely when \(r(a)<1\), \(r(a)=1\), and \(r(a)>1\). **Part D — Asymptotic Analysis (20 points)** Evaluate both limits, giving rigorous justifications: \[ L_1=\lim_{a\to+\infty} \frac{a(2a-r(a))}{\ln a} \] \[ L_2=\lim_{a\to-\infty} \frac{(-a)r(a)}{\ln(-a)}. \] Simply stating the dominant terms without proving the asymptotic relationships will not receive full credit. --- ## PROBLEM 2 — Combinatorics and Conditional Probability (35 points) A box contains 12 distinguishable balls: - 5 red balls - 4 blue balls - 3 green balls Four balls are selected uniformly at random without replacement. Let \(X\) denote the number of different colors represented among the four selected balls. **Part A — Probability Distribution (18 points)** Find the exact values of \[ P(X=1),\quad P(X=2),\quad P(X=3) \] and calculate the expected value \[ E[X]. \] Express all answers as simplified fractions. **Part B — Conditional Probability (7 points)** Given that exactly two different colors are represented among the four selected balls, find the probability that none of them is green. **Part C — Sequential Dependence (10 points)** Suppose the four balls were drawn sequentially, and it is revealed that exactly two different colors occurred in those four draws. A fifth ball is now drawn from the remaining eight balls. What is the conditional probability that the fifth ball has a color that did not occur among the first four? Give an exact simplified fraction and explain why the possible missing colors cannot simply be treated as equally likely. --- ## REQUIRED RESPONSE FORMAT 1. Solve every part in order. 2. Clearly separate proofs from final answers. 3. Include a final summary table listing every numerical result and parameter condition. 4. Assign yourself an overall confidence score between 0 and 100%. 5. Identify the single step in your solution most vulnerable to a reasoning error. **Scoring criteria:** - Mathematical correctness: 60 points - Rigor and completeness of reasoning: 25 points - Following instructions: 10 points - Clarity and organization: 5 points Do not use external tools. Begin your solution.