📐 ALGEBRA

Algebra — Key Concepts & Formulas

Must-Know Formulas
Vieta’s: \(r_1+r_2=-\frac{b}{a},\; r_1 r_2=\frac{c}{a}\)
Quadratic: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Infinite GP: \(S=\dfrac{a}{1-r},\;|r|<1\)
Discriminant: \(\Delta=b^2-4ac\)
\((a+b)^2=a^2+2ab+b^2\)
Logs: \(\log_b(xy)=\log_b x+\log_b y\)
Quick Example If \(x+y=5\) and \(xy=6\), find \(x^2+y^2\).
Solution: \(x^2+y^2=(x+y)^2-2xy=25-12=\mathbf{13}\)
01
Algebra
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Let \(p\) and \(q\) be the two roots of \(x^2 - 6x + 7 = 0\). What is the value of \(p^2 + q^2\)?
02
Algebra
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The sum of an infinite geometric series is \(12\), and the first term is \(4\). What is the common ratio \(r\)?
03
Algebra
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For how many real values of \(k\) does \(x^2 + kx + 9 = 0\) have exactly two equal real roots?
04
Algebra
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If \(\log_2 x + \log_2(x-2) = 3\), what is the value of \(x\)?
Geometry
◢ GEOMETRY

Geometry — Key Concepts & Formulas

Must-Know Formulas
Circle: \(A=\pi r^2\), \(C=2\pi r\)
Triangle: \(A=\tfrac{1}{2}bh\)
Heron’s: \(A=\sqrt{s(s-a)(s-b)(s-c)}\)
Law of Cosines: \(c^2=a^2+b^2-2ab\cos C\)
Similar triangles: sides proportional, areas in ratio of squares
Power of a Point: \(PA\cdot PB=PC\cdot PD\)
Quick Example A right triangle has legs 3 and 4. Hypotenuse \(=\sqrt{9+16}=5\). Area \(=\tfrac{1}{2}(3)(4)=\mathbf{6}\).
05
Geometry
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A triangle has sides \(7\), \(24\), and \(25\). What is its area?
06
Geometry
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A rectangle has a diagonal of length \(10\) and a perimeter of \(28\). What is its area?
07
Geometry
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Two circles, each of radius \(5\), are externally tangent to each other. What is the length of a common external tangent between the two points of tangency?
Number Theory
🔢 NUMBER THEORY

Number Theory — Key Concepts

Must-Know Facts
Divisors: if \(n=p_1^{a_1}p_2^{a_2}\cdots\), then \(\tau(n)=(a_1+1)(a_2+1)\cdots\)
GCD·LCM: \(\gcd(a,b)\cdot\text{lcm}(a,b)=ab\)
Fermat’s Little: \(a^{p-1}\equiv1\pmod{p}\) for prime \(p\nmid a\)
Euler’s \(\phi\): \(\phi(p^k)=p^{k-1}(p-1)\)
Quick Example Divisors of \(360=2^3\cdot3^2\cdot5^1\): \((3+1)(2+1)(1+1)=\mathbf{24}\).
08
Number Theory
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How many positive divisors does \(2^4 \cdot 3^2 \cdot 5^1\) have?
09
Number Theory
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How many positive integers less than \(100\) are divisible by neither \(3\) nor \(5\)?
10
Number Theory
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What is the remainder when \(7^{100}\) is divided by \(5\)?
Combinatorics & Probability
🎲 COMBINATORICS & PROBABILITY

Combinatorics & Probability — Key Concepts

Must-Know Formulas
Permutation: \(P(n,r)=\dfrac{n!}{(n-r)!}\)
Combination: \(\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}\)
Complement: \(P(A)=1-P(A^c)\)
Inclusion-Exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|\)
Quick Example \(\binom{8}{3}=\dfrac{8\cdot7\cdot6}{3!}=56\).   \(P(\text{both red from 4R,6B})=\dfrac{4}{10}\cdot\dfrac{3}{9}=\dfrac{2}{15}\).
11
Combinatorics
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A committee of 3 is chosen from 8 people. How many different committees are possible?
12
Probability
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A bag contains 4 red and 6 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are red?
13
Combinatorics
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How many 4-digit positive integers have all four digits distinct and sum to 10, with no leading zero?
Trigonometry
◢ TRIGONOMETRY

Trigonometry — Key Identities

Must-Know Identities
\(\sin^2\theta+\cos^2\theta=1\)
\(\sin(2\theta)=2\sin\theta\cos\theta\)
\(\cos(2\theta)=\cos^2\theta-\sin^2\theta\)
Law of Sines: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=2R\)
Law of Cosines: \(c^2=a^2+b^2-2ab\cos C\)
Sum formula: \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)
Quick Example \(\sin75^\circ=\sin(45^\circ+30^\circ)=\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}=\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
14
Trigonometry
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If \(\sin\theta = \dfrac{3}{5}\) and \(\theta\) is in the first quadrant, what is \(\sin(2\theta)\)?
15
Trigonometry
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In triangle \(ABC\), sides \(a=7\), \(b=8\), and \(\angle C=60^\circ\). What is the length of side \(c\)?
Complex Numbers & Functions
∞ COMPLEX NUMBERS & FUNCTIONS

Complex Numbers & Functions — Key Concepts

Must-Know Facts
Modulus: \(|a+bi|=\sqrt{a^2+b^2}\)
\(|z_1 z_2|=|z_1|\cdot|z_2|\)
De Moivre: \((r e^{i\theta})^n=r^n e^{in\theta}\)
AM-GM: \(\dfrac{a+b}{2}\geq\sqrt{ab}\), equality iff \(a=b\)
Quick Example \(|(2+i)(3-2i)|=|2+i|\cdot|3-2i|=\sqrt{5}\cdot\sqrt{13}=\sqrt{65}\).
16
Complex Numbers
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What is the modulus of \((3+4i)(1-2i)\)?
17
Functions
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Let \(f(x) = \dfrac{x+3}{x-1}\). What is \(f(f(x))\) in simplest form?
Advanced Topics
⭐ ADVANCED TOPICS

Polynomials, 3D Geometry & Mixed

Must-Know Facts
Sphere: \(V=\frac{4}{3}\pi r^3\), \(SA=4\pi r^2\)
Remainder Thm: remainder of \(f(x)\div(x-a)\) is \(f(a)\)
Sum of squares: \(\sum_{k=1}^n k^2=\dfrac{n(n+1)(2n+1)}{6}\)
AM-GM: for positive reals, \(\dfrac{x+y}{2}\geq\sqrt{xy}\)
Quick Example Remainder when \(x^3-2x^2+3\) is divided by \((x-2)\): \(f(2)=8-8+3=\mathbf{3}\).
18
Polynomials
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A polynomial \(P(x)\) satisfies \(P(1)=3\), \(P(2)=5\), and \(P(3)=9\). What is the minimum possible degree of \(P(x)\)?
19
3D Geometry
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A sphere is inscribed in a cube with side length \(6\). What is the ratio of the volume of the sphere to the volume of the cube?
20
Parabolas
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Let \(f(x) = x^2 - 4x + 3\). For how many integer values of \(c\) with \(-10 \le c \le 10\) does the equation \(f(x) = c\) have exactly two distinct real solutions?
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Step-by-step explanations for all 20 problems