AMC 10 topics and formulas

• Difference of squares: $a^2-b^2=(a-b)(a+b)$.

• Arithmetic series: $1+2+\cdots +n=\frac{n(n+1)}{2}$, and in general the number of terms times the average of the first and last.

• Geometric series: $a+ar+\cdots +a{r}^{n-1}=a\frac{r^n-1}{r-1}$ for $r\ne 1$.

• Vieta: for $x^2+bx+c=0$ the roots add to $-b$ and multiply to $c$.

• Simon’s trick: $xy+ax+by=(x+b)(y+a)-ab$, which turns a whole-number equation into a factor search.

• Choosing $k$ from $n$: $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$.

• Sharing $n$ identical items among $k$ people: $\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)$ ways.

• Complementary counting: count what you do not want and subtract from the total.

• Inclusion–exclusion: $|A\cup B|=|A|+|B|-|A\cap B|$.

• Expected value: the sum of each outcome times its probability; it adds across separate parts.

• Number of divisors: if $n=p^a{q}^b{r}^c$ then $n$ has $(a+1)(b+1)(c+1)$ divisors.

• $\gcd (a,b)\times \mathrm{lcm}(a,b)=ab$.

• Divisibility by 3 and by 9: the digit sum. By 11: the alternating digit sum.

• Remainders multiply and add: work with the remainders, not the numbers.

• Trailing zeros of $n!$: $\lfloor n/5\rfloor +\lfloor n/25\rfloor +\dots$

• Pythagorean triples worth recognising: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples.

• Area of a triangle: $\frac{1}{2}ab\sin C$, or $rs$ with $r$ the inradius and $s$ the semi-perimeter, or Heron: $\sqrt{s(s-a)(s-b)(s-c)}$.

• Special right triangles: sides $1:1:\sqrt{2}$ and $1:\sqrt{3}:2$.

• Inscribed angle: half the central angle on the same arc.

• Power of a point: for two chords through a point, $PA\cdot PB=PC\cdot PD$.

• Shoelace formula for a polygon with vertices $(x_i,y_i)$: area $=\frac{1}{2}\left|\sum (x_i{y}_{i+1}-x_{i+1}{y}_i)\right|$.

• Similar figures: lengths scale by $k$, areas by $k^2$, volumes by $k^3$.

Original practice tests written by Math Webb in the style of the contest, free to read, each question with its answer and a full worked solution.