CEMC cheat sheet

Quadratic formula: roots of ax²+bx+c=0 are x = (−b ± √(b²−4ac))/(2a); discriminant b²−4ac decides 2, 1, or 0 real roots.

Vertex of y=ax²+bx+c occurs at x=−b/(2a), with value y=c−b²/(4a).

Factor/remainder theorem: (x−r) divides P(x) iff P(r)=0; remainder on dividing by (x−r) is P(r).

Log laws: log(ab)=log a+log b, log(a/b)=log a−log b, log(aⁿ)=n log a, valid for positive arguments.

For f∘g, domain excludes x with g(x) outside domain of f; composition is generally not commutative.

A function is invertible on its domain iff it is strictly monotonic (one-to-one) there.

Absolute value: |x|<k means −k<x<k; |x|>k means x<−k or x>k, for k>0.

AM-GM: for positive reals, (a+b)/2 ≥ √(ab), with equality iff a=b; extends to n terms.

Pythagorean identity sin²θ+cos²θ=1; also 1+tan²θ=sec²θ and 1+cot²θ=csc²θ.

Law of cosines: c²=a²+b²−2ab cos C; law of sines: a/sin A=b/sin B=c/sin C=2R.

Area of triangle = ½ab sin C = √(s(s−a)(s−b)(s−c)) with s the semiperimeter.

Circle: area πr², circumference 2πr; arc length rθ and sector area ½r²θ for central angle θ in radians.

Angle sum of an n-gon is (n−2)·180°; each interior angle of a regular n-gon is (n−2)·180°/n.

Sum/difference: sin(A±B)=sin A cos B ± cos A sin B; cos(A±B)=cos A cos B ∓ sin A sin B.

Double angle: sin 2θ=2 sin θ cos θ; cos 2θ=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1.

Volume: prism/cylinder Bh; pyramid/cone Bh/3; sphere 4πr³/3 with surface area 4πr².

Division algorithm: for integers a, b>0 there exist unique q, r with a=bq+r and 0≤r<b.

gcd(a,b)·lcm(a,b)=|ab|; Euclidean algorithm computes gcd via repeated remainders.

A number is divisible by 3 (or 9) iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Fundamental theorem of arithmetic: every integer >1 factors uniquely into primes up to order.

Modular arithmetic: if a≡b and c≡d (mod m), then a±c≡b±d and ac≡bd (mod m).

Fermat's little theorem: if p is prime and p∤a, then a^(p−1)≡1 (mod p).

Linear Diophantine ax+by=c has integer solutions iff gcd(a,b) divides c.

A graph has an Eulerian circuit iff it is connected and every vertex has even degree.

Arithmetic sequence: aₙ=a₁+(n−1)d; sum Sₙ=n(a₁+aₙ)/2=n(2a₁+(n−1)d)/2.

Geometric sequence: aₙ=a₁r^(n−1); sum Sₙ=a₁(1−rⁿ)/(1−r) for r≠1, and infinite sum a₁/(1−r) when |r|<1.

Σk=n(n+1)/2; Σk²=n(n+1)(2n+1)/6; Σk³=[n(n+1)/2]².

A sequence converges iff it is monotonic and bounded; bounded monotone sequences have a limit.

Squeeze theorem: if aₙ≤bₙ≤cₙ and aₙ,cₙ→L, then bₙ→L.

lim(n→∞)(1+1/n)ⁿ=e; lim(x→0) sin x/x=1 and lim(x→0)(eˣ−1)/x=1.

Ratio test for series: converges if lim|aₙ₊₁/aₙ|<1, diverges if >1; inconclusive at 1.

A series converges only if its terms tend to 0; the converse fails (harmonic series diverges).

Permutations P(n,r)=n!/(n−r)!; combinations C(n,r)=n!/(r!(n−r)!), with C(n,r)=C(n,n−r).

Multiplication principle: sequential independent choices multiply; addition principle applies to disjoint cases.

Pigeonhole principle: placing n+1 objects into n boxes forces some box to hold at least two.

Inclusion–exclusion: |A∪B|=|A|+|B|−|A∩B|; extends to alternating sums for more sets.

Probability of an event = favorable outcomes / total equally likely outcomes; P(Aᶜ)=1−P(A).

P(A∪B)=P(A)+P(B)−P(A∩B); independent events satisfy P(A∩B)=P(A)P(B).

Conditional probability P(A|B)=P(A∩B)/P(B); Bayes' rule inverts conditionals.

Binomial probability: P(k successes in n trials)=C(n,k)pᵏ(1−p)^(n−k).

Proof by contradiction: assume the negation and derive an impossibility to establish the original claim.

Mathematical induction: verify the base case, assume the statement for n=k, then prove it for n=k+1.

Invariant principle: identify a quantity unchanged by each operation to rule out reaching a target state.

Extremal principle: examine a maximal or minimal element of a set to force a structural conclusion.

Working backwards from the target condition often reveals the necessary form of a solution.

Symmetry and substitution (e.g., set u=x+1/x) reduce symmetric expressions to lower degree.

Check parity, modular residues, or bounds first to eliminate impossible cases before full computation.

Counterexample reasoning: one counterexample disproves a universal statement but never proves one.