📏 Essential Formulas — Quick Reference
Compact formula bank for quick contest reference and memorization.
🎯 How to Use This Section
Before Contests
- Review all formulas — Ensure you know them by heart
- Practice micro-examples — Understand how to apply each formula
- Time yourself — Can you recall formulas quickly?
During Contests
- Quick lookup — Use as reference when stuck
- Pattern matching — Match problem to formula
- Verification — Double-check your work
Study Strategy
- Memorize core formulas — Quadratic, Vieta’s, series
- Practice applications — Work through micro-examples
- Build speed — Time yourself on formula recall
- Review regularly — Keep formulas fresh
📚 Formula Categories
🔢 Basic Algebra
- Factoring patterns — $(a \pm b)^2$, $a^2 - b^2$, $a^3 \pm b^3$
- Order of operations — PEMDAS rules
- Fraction operations — Addition, subtraction, multiplication, division
🎯 Quadratics
- Quadratic formula — $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
- Discriminant — $\Delta = b^2 - 4ac$
- Vertex form — $y = a(x-h)^2 + k$
- Vertex coordinates — $h = -\frac{b}{2a}$, $k = \frac{4ac-b^2}{4a}$
🧮 Polynomials
- Vieta’s formulas — Sum and product of roots
- Remainder theorem — Remainder when $f(x) \div (x-a)$ is $f(a)$
- Factor theorem — $(x-a)$ is a factor if and only if $f(a) = 0$
📊 Series & Sequences
- Arithmetic sequences — $a_n = a_1 + (n-1)d$
- Arithmetic sums — $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
- Geometric sequences — $a_n = a_1 \cdot r^{n-1}$
- Geometric sums — $S_n = a_1 \frac{1-r^n}{1-r}$
🔢 Exponents & Logs
- Exponent rules — $a^x \cdot a^y = a^{x+y}$, $(a^x)^y = a^{xy}$
- Logarithm rules — $\log_a(xy) = \log_a x + \log_a y$
- Change of base — $\log_a x = \frac{\log_b x}{\log_b a}$
⚖️ Inequalities
- AM-GM inequality — $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b > 0$
- Cauchy-Schwarz — $(a^2+b^2)(c^2+d^2) \geq (ac+bd)^2$
🧮 Complex Numbers
- Imaginary unit — $i^2 = -1$
- Complex conjugate — $\overline{a+bi} = a-bi$
- Modulus — $|a+bi| = \sqrt{a^2+b^2}$
⚡ Quick Tips
- Memorize by category — Group related formulas together
- Practice with examples — Don’t just memorize, understand
- Build associations — Connect formulas to problem types
- Review regularly — Keep formulas fresh in memory
See Essential Formulas for the complete formula bank with micro-examples!
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