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Olympiad Math Toolkits

Master the ideas behind AMC 8, AMC 10, AMC 12, AIME, USA(J)MO, and Olympiad problems with structured toolkits featuring prerequisites, worked examples, common mistakes, extensions, and related competition problems.

  1. 1.
    Sum of the first n integers
    1+2+⋯+n=n(n+1)21+2+\cdots+n=\frac{n(n+1)}{2}
  2. 2.
    Sum of squares
    12+22+⋯+n2=n(n+1)(2n+1)61^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}
  3. 3.
    Sum of cubes
    13+23+⋯+n3=(n(n+1)2)21^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2
  4. 4.
    Geometric Sequences and Series
    an=a1rn−1a_n=a_1r^{n-1}
    an2=an−1an+1a_n^2=a_{n-1}a_{n+1}
    Sn=a1+a1r+a1r2+⋯+a1rn−1=a1(rn−1)r−1S_n=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}=\frac{a_1(r^n-1)}{r-1}
    ∣r∣<1:S∞=a1+a1r+a1r2+⋯=a11−r|r|<1:\quad S_\infty=a_1+a_1r+a_1r^2+\cdots=\frac{a_1}{1-r}
  5. 5.
    Telescoping
    11⋅2+12⋅3+⋯+1n(n+1)=(1−12)+⋯+(1n−1n+1)=1−1n+1\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{n(n+1)}=\left(1-\frac{1}{2}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n+1}\right)=1-\frac{1}{n+1}
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    Real AMC, AIME & Olympiad problems with step-by-step hints and detailed solutions

Practice with These Ideas

1.
How many ordered pairs of integers (m,n)(m,n) satisfy n2−49=m\sqrt{n^2-49}=m?
(A)  1\text{(A)}\;1(B)  2\text{(B)}\;2(C)  3\text{(C)}\;3(D)  4\text{(D)}\;4(E)  infinitely many\text{(E)}\;\text{infinitely many}
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