API · no tools. Retried after the original timeout, using the same problem, prompt, model, and sampling settings. The original error record is preserved separately.
This is a Korean CSAT mathematics numerical-answer problem. Solve it and put only the final integer value inside \boxed{}.
Problem 30 Letf(x) be an increasing continuous function on the entire real line, and letf −1(x) be its inverse function. The functionf −1(x) satisfies the following conditions: (A) For|x| ≤1, 4 � f −1(x) 2 =x 2(x2 −5) 2. (B) For|x|>1,
f −1(x)
=e |x|−1 + 1. For a real numberm, letg(m) denote the number of intersection points between the line of slopempassing through (1,0) and the curvey=f(x). Suppose thatg(m) is discontinuous atm=aandm=bwitha < b. Evaluate g(a)× lim m→a+ g(m) +g(b)× lnb b 2 . (Here, you may use lim x→∞ lnx x = 0.) [4 points] Mathematics (Geometry)
\boxed{11}
Comparative Analysis (P37) (Average across runs)
Averages use reported measurements only. Missing usage is not treated as zero.