rt，最近学校有项目，需要用llm批量做高难数学题（IMO预赛难度），之前用的都是cli反代的Gemini 3.1 Pro，效果不错，但是最近谷歌的配额缩紧，经常429且限额，因此来问问各位佬友们有没有什么做题能力强的模型推荐（公益站能白嫖最好，付费渠道的话也能接受，OpenRouter的Gemini太贵了实在用不起 
）
参考难度
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Suppose a, b, c are positive real numbers such that \frac{1}{a} + \frac{1}{b} + \frac{2}{c} = 1. Find the minimum value of a + b + c + \sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{c^2 + a^2}.
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For p_0, \dots, p_d \in \mathbb{R}^d, let
S(p_0, \dots, p_d) = \left\{ \alpha_0 p_0 + \cdots + \alpha_d p_d : \alpha_i \le 1, \sum_{i=0}^d \alpha_i = 1 \right\}.
Let \pi be an arbitrary probability distribution on \mathbb{R}^d, and choose p_0, \dots, p_d independently with distribution \pi. Calculate the minimum possible value of the expectation of \pi(S(p_0, \dots, p_d)) over all such distributions \pi.
