O3
AM-GM Inequality
算术-几何平均不等式
Optimization / 最值Grades G7 - G8
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What It Looks Like
Recognition signals — when you see these, think of this structure:
- 1Optimization with fixed product or sum
- 2Questions about maximum/minimum of expressions
- 3Inequality problems
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What It Really Tests
The core mathematical idea behind this structure:
For non-negative numbers, arithmetic mean ≥ geometric mean, with equality when all numbers are equal.
对于非负数，算术平均≥几何平均，当所有数相等时等号成立。
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Why Students Get Stuck
Common mistakes to watch out for:
- ⚠Not recognizing when AM-GM applies
- ⚠Forgetting equality condition
- ⚠Applying it when numbers can be negative
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Your First Step
How to begin thinking about problems with this structure:
When product is fixed, find when sum is minimum (or vice versa) by making values equal.
当乘积固定时，通过使值相等找到和的最小值（或相反）。
Related Structures
These structures share similar patterns or thinking approaches: