Useful identities, counting techniques, geometry theorems, number theory tricks and inequalities common in AMC-style contests.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- Sophie Germain Identity: factor a^4 + 4b^4
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(a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)- How to factor a^3 - b^3
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(a - b)(a^2 + ab + b^2)- Sum of arithmetic sequence: a + (a+d) + (a+2d) + ... with n terms
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na + n(n-1)d/2- Sum of geometric series a + ar + ar^2 + ... + ar^(n-1) where r ≠ 1
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a(1 - r^n)/(1 - r)- Sum of 1 + 2 + 3 + ... + n
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n(n+1)/2- Sum of 1^2 + 2^2 + 3^2 + ... + n^2
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n(n+1)(2n+1)/6- Sum of 1^3 + 2^3 + 3^3 + ... + n^3
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[n(n+1)/2]^2- AM-GM Inequality: for positive numbers a and b
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(a + b)/2 >= sqrt(ab), with equality iff a = b- Cauchy-Schwarz Inequality: express (a_1 b_1 + ... + a_n b_n)^2
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<= (a_1^2 + ... + a_n^2)(b_1^2 + ... + b_n^2)- Rearrangement Inequality: when is a_1 b_1 + a_2 b_2 + ... + a_n b_n maximized given sorted sequences?
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When both sequences are sorted in the same order- Chebyshev's Inequality: if a_1 <= a_2 <= ... <= a_n and b_1 <= b_2 <= ... <= b_n
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n(a_1 b_1 + a_2 b_2 + ... + a_n b_n) >= (a_1 + ... + a_n)(b_1 + ... + b_n)- QM-AM-GM-HM chain: order these means from largest to smallest
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Quadratic Mean >= Arithmetic Mean >= Geometric Mean >= Harmonic Mean- Pigeonhole Principle: if n items go into m containers with n > m
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At least one container has more than one item- Stars and Bars: distribute n identical objects into k distinct bins
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C(n + k - 1, k - 1)- Inclusion-Exclusion for two sets: |A union B|
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|A| + |B| - |A intersect B|
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