DAPO: An Open-Source LLM Reinforcement Learning System at Scale — part 3/3

Cleaned-text code points [31730, 33962). Bibliography omitted here; source.md includes it.

The original answer to the problem is an algebraic expression 11 − 2 ​ 6 . 11-2\sqrt{6}. We need to transform this into an integer form. First, note that the answer consists of an integer part and a square root term. According to the given rule, we replace the integer and square root components with undetermined coefficients. The original problem uses x x and y y as variables, so we choose other letters as undetermined coefficients. We express the answer in the form k − m ​ n k-m\sqrt{n} and require the student to find k + m + n k+m+n .

#### Answer Format

k − m ​ n k-m\sqrt{n}

#### Transformed Problem

Let x x and y y be real numbers such that x 2 + y 2 − 22 ​ x − 16 ​ y + 113 = 0 . x^{2}+y^{2}-22x-16y+113=0. Determine the smallest possible value of x x . The original answer is in the form k − m ​ n k-m\sqrt{n} , where k k , m m , and n n are integers. Please find the value of k + m + n k+m+n .

#### Answer Calculation for the Transformed Problem

• The original answer is: 11 − 2 ​ 6 11-2\sqrt{6} .

• The required format is: k − m ​ n k-m\sqrt{n} .

• Matching the coefficients: k = 11 k=11 , m = 2 m=2 , n = 6 n=6 .

• The final result is: k + m + n = 11 + 2 + 6 = 19 . k+m+n=11+2+6=19.

#### Final Answer

19 19

## Appendix B Supplementary Case

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