Kyoto Entrance Mathematics cheat sheet

Difference of squares: a² − b² = (a − b)(a + b); sum/difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Binomial expansion: (a + b)ⁿ = Σₖ C(n,k) aⁿ⁻ᵏ bᵏ, with C(n,k) = n!/(k!(n−k)!).

For nested radicals, write √(a + 2√b) = √m + √n when m + n = a and mn = b, with m, n ≥ 0.

Rationalize a denominator by multiplying by the conjugate: 1/(√a − √b) = (√a + √b)/(a − b).

Absolute value rules: |xy| = |x||y|, |x/y| = |x|/|y|, and |x| = c (c > 0) gives x = ±c.

Double-check a factorization by expanding; a polynomial with integer coefficients and leading coefficient 1 has integer roots dividing the constant term.

Simplify compound fractions by multiplying numerator and denominator by the least common multiple of all inner denominators.

When a problem fixes the order of operations, reduce powers first, then products and quotients left to right, then sums and differences.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); discriminant D = b² − 4ac: D > 0 two real roots, D = 0 double root, D < 0 no real roots.

Vieta's formulas for ax² + bx + c = 0: sum of roots = −b/a, product = c/a.

Multiplying or dividing an inequality by a negative number reverses the inequality sign.

|f(x)| < g(x) is equivalent to −g(x) < f(x) < g(x); |f(x)| > g(x) splits into f(x) > g(x) or f(x) < −g(x).

For √(f(x)) = g(x), require g(x) ≥ 0 and f(x) = g(x)²; squaring without the sign condition can add extraneous roots.

A quadratic inequality ax² + bx + c > 0 (a > 0) holds outside the roots and fails between them; reverse for '< 0'.

Solve simultaneous equations by substitution or by elimination; a linear–quadratic system has at most two solutions.

Linear inequalities in two variables describe half-planes; the feasible region is the intersection, checked with a test point.

De Morgan's laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

Inclusion–exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|; for three sets subtract pairwise intersections and add the triple intersection.

A ⊂ B means every element of A lies in B; the empty set is a subset of every set.

The contrapositive of P ⇒ Q is ¬Q ⇒ ¬P and is logically equivalent to it; the converse Q ⇒ P is not.

P ⇒ Q is false only when P is true and Q is false; a single counterexample disproves a universal statement.

Negation of 'for all x, P(x)' is 'there exists x such that not P(x)'; negate quantifiers by swapping ∀ and ∃ and negating the predicate.

The number of subsets of an n-element set is 2ⁿ; the number of proper subsets is 2ⁿ − 1.

A necessary condition for P is implied by P; a sufficient condition for P implies P. 'P only if Q' means P ⇒ Q.

Vertex form y = a(x − p)² + q gives vertex (p, q); complete the square from y = ax² + bx + c with p = −b/(2a), q = c − b²/(4a).

The axis of symmetry is x = −b/(2a); the parabola opens upward for a > 0 and downward for a < 0.

Maximum or minimum on a closed interval [α, β] occurs at the vertex if it lies inside, otherwise at an endpoint.

Discriminant of ax² + bx + c = 0 equals b² − 4ac; the graph meets the x-axis twice, once, or never according to its sign.

Shifting y = f(x) by (h, k) gives y = f(x − h) + k; a negative h shifts right, a positive k shifts up.

Number of real solutions of f(x) = k equals the number of intersections of y = f(x) with the horizontal line y = k.

For y = ax² + bx + c, the sign of a controls width: larger |a| gives a narrower parabola.

A quadratic takes values of one sign everywhere on ℝ exactly when a and the discriminant condition agree: a > 0 and D < 0 gives always positive.

Pythagorean identity: sin²θ + cos²θ = 1; also tanθ = sinθ/cosθ and 1 + tan²θ = 1/cos²θ.

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

Cosine rule: a² = b² + c² − 2bc cos A; area of triangle = (1/2)bc sin A.

Reduction formulas: sin(θ + 180°) = −sinθ, cos(θ + 180°) = −cosθ, sin(90° − θ) = cosθ.

Addition formulas: sin(α ± β) = sinα cosβ ± cosα sinβ; cos(α ± β) = cosα cosβ ∓ sinα sinβ.

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

Synthesis: a sinθ + b cosθ = √(a² + b²) sin(θ + φ) with cos φ = a/√(a² + b²), sin φ = b/√(a² + b²).

General solutions: sinθ = s gives θ = α + 360°k or 180° − α + 360°k; cosθ = c gives θ = ±α + 360°k; tanθ = t gives θ = α + 180°k.

Laws of exponents: aᵐaⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ for a > 0.

Definition of logarithm: log_a b = c ⇔ aᶜ = b, requiring a > 0, a ≠ 1, b > 0.

Logarithm laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a xⁿ = n log_a x.

Change of base: log_a b = log_c b / log_c a; consequently log_a b = 1/log_b a.

y = aˣ with a > 1 is increasing; y = log_a x with a > 1 is increasing and has domain x > 0, vertical asymptote x = 0.

Solve aˣ = b by taking logarithms: x = log b / log a; solve log equations by converting to exponential form and checking the domain.

Substitution t = aˣ turns equations like a²ˣ − 5aˣ + 6 = 0 into quadratics; t > 0 always.

Graphs of y = aˣ and y = log_a x are reflections in the line y = x.

Derivative definition: f′(x) = lim_{h→0} (f(x + h) − f(x))/h.

Rules: (xⁿ)′ = nxⁿ⁻¹, (sin x)′ = cos x, (cos x)′ = −sin x, (eˣ)′ = eˣ, (log x)′ = 1/x.

Product rule (fg)′ = f′g + fg′; quotient rule (f/g)′ = (f′g − fg′)/g²; chain rule dy/dx = dy/du · du/dx.

Tangent line at x = a: y − f(a) = f′(a)(x − a); normal line has slope −1/f′(a) when f′(a) ≠ 0.

f is increasing where f′ > 0 and decreasing where f′ < 0; f′ = 0 with sign change gives a local extremum.

Second derivative test: f′(a) = 0 and f″(a) > 0 gives a local minimum; f″(a) < 0 gives a local maximum.

Concavity: f″ > 0 means concave up, f″ < 0 concave down; an inflection point requires a change in concavity.

For optimisation, express the target quantity in one variable, differentiate, solve f′ = 0, and verify against the interval endpoints.

Indefinite integral: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫(1/x) dx = log|x| + C.

Standard integrals: ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫eˣ dx = eˣ + C.

Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a), where F′ = f.

Area between curves: ∫_a^b (upper − lower) dx; split the interval wherever the curves cross.

Volume of revolution about the x-axis: V = π∫_a^b {f(x)}² dx; about the y-axis use V = π∫ x² dy.

Substitution: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x); change the limits when the integral is definite.

Even/odd symmetry: ∫_{−a}^{a} f dx = 2∫_0^a f dx for even f, and 0 for odd f.

Integration by parts: ∫u v′ dx = uv − ∫u′ v dx; choose u to simplify on differentiation.

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

Geometric sequence: aₙ = a₁rⁿ⁻¹; sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1; infinite sum a₁/(1 − r) when |r| < 1.

Sum formulas: Σ_{k=1}^{n} k = n(n + 1)/2; Σk² = n(n + 1)(2n + 1)/6; Σk³ = {n(n + 1)/2}².

Telescoping: write each term as a difference, e.g. 1/(k(k + 1)) = 1/k − 1/(k + 1), so the sum collapses.

Recurrence aₙ₊₁ = paₙ + q (p ≠ 1) solves via fixed point α = q/(1 − p): aₙ − α is geometric with ratio p.

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

A series of differences satisfies aₙ = a₁ + Σ_{k=1}^{n−1}(aₖ₊₁ − aₖ); apply when the recurrence gives differences.

For Σ k·rᵏ, subtract r times the sum from the sum to convert it into a geometric series.

Dot product: a·b = |a||b|cosθ = a₁b₁ + a₂b₂ (+ a₃b₃ in space); a ⊥ b ⇔ a·b = 0.

Magnitude |a| = √(a₁² + a₂² + a₃²); unit vector is a/|a|.

Component form: AB→ = (b₁ − a₁, b₂ − a₂); position vectors satisfy AB→ = OB→ − OA→.

In space, cross product a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) is perpendicular to both a and b.

Line through point A with direction d: r = a + t d; plane through A with normal n: n·(r − a) = 0.

Distance from point P to plane n·r = k is |n·p − k|/|n|.

Area of triangle with sides a, b is (1/2)|a × b|; volume of parallelepiped spanned by a, b, c is |a·(b × c)|.

Angle between vectors: cosθ = a·b/(|a||b|); parallel vectors satisfy a = kb for some scalar k.

i² = −1; for z = a + bi, conjugate z̄ = a − bi and modulus |z| = √(a² + b²), with z z̄ = |z|².

Division: (a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²).

Polar form z = r(cosθ + i sinθ) = re^{iθ}, where r = |z| and θ = arg z; product multiplies moduli and adds arguments.

De Moivre: {r(cosθ + i sinθ)}ⁿ = rⁿ(cos nθ + i sin nθ).

Roots of zⁿ = w: n distinct roots with modulus |w|^{1/n} and arguments (arg w + 2kπ)/n for k = 0, …, n − 1.

Euler relation e^{iθ} = cosθ + i sinθ; consequently cosθ = (e^{iθ} + e^{−iθ})/2 and sinθ = (e^{iθ} − e^{−iθ})/(2i).

Geometric meaning: |z − α| = r is a circle centred at α; |z − α| = |z − β| is the perpendicular bisector of αβ.

Triangle inequality: |z + w| ≤ |z| + |w|, with equality only when the arguments coincide.

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

Probability: P(A) = |A|/|Ω| for equally likely outcomes; P(Aᶜ) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Conditional probability P(A|B) = P(A ∩ B)/P(B); independence means P(A ∩ B) = P(A)P(B).

Repeated independent trials: P(exactly r successes in n trials) = C(n,r) pʳ(1 − p)ⁿ⁻ʳ.

Expected value E(X) = Σ xₖ pₖ; variance V(X) = E(X²) − {E(X)}²; E(aX + b) = aE(X) + b.

For data transformed as y = ax + b, the mean and standard deviation transform the same way: ȳ = a x̄ + b, s_y = |a| s_x.

Standard deviation s = √{(1/n)Σ(xᵢ − x̄)²}; correlation coefficient r = S_xy/√(S_xx S_yy) lies in [−1, 1].

Circular arrangements of n distinct objects number (n − 1)!; arrangements with repeated items divide by the factorials of repetition counts.

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

Euclidean algorithm: gcd(a, b) = gcd(b, a mod b), repeated until the remainder is 0; the last nonzero remainder is the gcd.

Bézout identity: gcd(a, b) = ax + by for some integers x, y; the extended Euclidean algorithm produces x and y.

Congruence a ≡ b (mod m) means m divides a − b; it is preserved under addition, subtraction, and multiplication.

Linear congruence ax ≡ b (mod m) is solvable exactly when gcd(a, m) divides b, and then has gcd(a, m) solutions mod m.

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

A number is divisible by 3 or 9 exactly when its digit sum is; by 11 when the alternating digit sum is divisible by 11.

If gcd(a, m) = 1, the inverse of a mod m exists and a^{φ(m)} ≡ 1 (mod m), where φ is Euler's totient.