📏 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

1. Memorize core formulas — Quadratic, Vieta’s, series
2. Practice applications — Work through micro-examples
3. Build speed — Time yourself on formula recall
4. 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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