Bulgaria DZI Mathematics cheat sheet

For sets A and B: A ∪ B = {x | x ∈ A or x ∈ B}; A ∩ B = {x | x ∈ A and x ∈ B}; A \ B = {x | x ∈ A and x ∉ B}.

De Morgan's laws: (A ∪ B)' = A' ∩ B'; (A ∩ B)' = A' ∪ B'.

The implication P ⇒ Q is false only when P is true and Q is false; its contrapositive ¬Q ⇒ ¬P is logically equivalent.

The negation of ∀x P(x) is ∃x ¬P(x); the negation of ∃x P(x) is ∀x ¬P(x).

For real numbers a and b: |a + b| ≤ |a| + |b| (triangle inequality); |a·b| = |a|·|b|.

The distance between real numbers a and b is |a − b|; the midpoint is (a + b)/2.

Absolute value inequality |x| < a (a > 0) is equivalent to −a < x < a; |x| > a is equivalent to x < −a or x > a.

Interval notation: (a, b) = {x | a < x < b}; [a, b] = {x | a ≤ x ≤ b}; (a, b] = {x | a < x ≤ b}.

For any real x, √(x²) = |x|; if x ≥ 0 then (√x)² = x.

Quadratic formula: roots of ax² + bx + c = 0 (a ≠ 0) are x = [−b ± √(b² − 4ac)] / (2a).

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots; D = 0 gives one real double root; D < 0 gives no real roots.

Vieta's formulas: for ax² + bx + c = 0, x₁ + x₂ = −b/a and x₁x₂ = c/a.

Solving |ax + b| = c (c ≥ 0) gives ax + b = c or ax + b = −c.

Solving a rational inequality: bring all terms to one side, factor numerator and denominator, and use a sign chart with critical points where factors are zero or undefined.

Linear system a₁x + b₁y = c₁, a₂x + b₂y = c₂ has unique solution when a₁b₂ − a₂b₁ ≠ 0.

Exponential equation a^f(x) = a^g(x) (a > 0, a ≠ 1) is equivalent to f(x) = g(x).

Logarithmic equation log_a f(x) = log_a g(x) requires f(x) > 0 and g(x) > 0, then f(x) = g(x).

Inequality x² < a (a > 0) is equivalent to −√a < x < √a; x² > a is equivalent to x < −√a or x > √a.

A function f: A → B assigns to each x ∈ A exactly one f(x) ∈ B; the domain is A, the range is {f(x) | x ∈ A}.

f is even if f(−x) = f(x) for all x in its domain (graph symmetric about the y-axis); f is odd if f(−x) = −f(x) (graph symmetric about the origin).

For f(x) = ax² + bx + c (a ≠ 0), the vertex is at x = −b/(2a); the axis of symmetry is x = −b/(2a).

The graph of y = f(x − h) + k is the graph of y = f(x) translated h units horizontally and k units vertically.

The graph of y = a·f(x) is a vertical stretch by factor |a| when |a| > 1, or compression when 0 < |a| < 1; if a < 0 it is also reflected across the x-axis.

If f is strictly increasing on an interval, then f(x₁) < f(x₂) whenever x₁ < x₂; strictly decreasing means f(x₁) > f(x₂).

The inverse function f⁻¹ exists iff f is one-to-one; its graph is the reflection of the graph of f across the line y = x.

For f(x) = a^x (a > 0, a ≠ 1): domain ℝ, range (0, ∞), horizontal asymptote y = 0; increasing if a > 1, decreasing if 0 < a < 1.

Laws of exponents: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); (ab)^n = a^n b^n; a^0 = 1 (a ≠ 0); a^(−n) = 1/a^n.

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

Laws of logarithms: log_a (xy) = log_a x + log_a y; log_a (x/y) = log_a x − log_a y; log_a (x^p) = p log_a x.

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

Identities: a^(log_a x) = x (x > 0); log_a a = 1; log_a 1 = 0.

To solve a^f(x) = b (b > 0), take logarithms of both sides: f(x) = log_a b.

To solve log_a f(x) = c, rewrite as f(x) = a^c and check f(x) > 0.

The equation a^(2x) + p·a^x + q = 0 is solved by substituting t = a^x (t > 0) to obtain a quadratic in t.

The function f(x) = log_a x (a > 1) is increasing with domain (0, ∞) and range ℝ; its graph passes through (1, 0) and has vertical asymptote x = 0.

In a right triangle: sin α = opposite/hypotenuse; cos α = adjacent/hypotenuse; tan α = opposite/adjacent.

Fundamental identity: sin² α + cos² α = 1; consequently 1 + tan² α = 1/cos² α (cos α ≠ 0).

Reduction formulas: sin(π − α) = sin α; cos(π − α) = −cos α; sin(π + α) = −sin α; cos(π + α) = −cos α.

Addition formulas: sin(α ± β) = sin α cos β ± cos α sin β; cos(α ± β) = cos α cos β ∓ sin α sin β.

Double-angle formulas: sin 2α = 2 sin α cos α; cos 2α = cos² α − sin² α = 1 − 2 sin² α = 2 cos² α − 1.

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; equivalently cos A = (b² + c² − a²) / (2bc).

Area of triangle: S = (1/2)ab sin C = √(p(p−a)(p−b)(p−c)), where p = (a + b + c)/2.

General solution: sin x = sin α ⇔ x = α + 2kπ or x = π − α + 2kπ; cos x = cos α ⇔ x = ±α + 2kπ (k ∈ ℤ).

Arithmetic progression: a_n = a₁ + (n − 1)d; sum S_n = n(a₁ + a_n)/2 = n[2a₁ + (n − 1)d]/2.

Geometric progression: a_n = a₁ q^(n−1); sum S_n = a₁(1 − q^n)/(1 − q) for q ≠ 1; S_n = n a₁ for q = 1.

Sum of infinite geometric series: S = a₁/(1 − q) when |q| < 1; the series diverges when |q| ≥ 1.

Characteristic property of arithmetic progression: a_n = (a_{n−1} + a_{n+1})/2 for n ≥ 2.

Characteristic property of geometric progression: a_n² = a_{n−1} · a_{n+1} for n ≥ 2 (all terms positive or all negative).

Sum of first n natural numbers: 1 + 2 + … + n = n(n + 1)/2.

Sum of squares: 1² + 2² + … + n² = n(n + 1)(2n + 1)/6.

A sequence is bounded if there exist m and M such that m ≤ a_n ≤ M for all n; it is monotone increasing if a_{n+1} ≥ a_n for all n.

For a geometric progression with |q| < 1, the limit of the partial sums equals a₁/(1 − q).

Permutations of n distinct objects: P_n = n!; permutations of n objects taken k at a time: P(n, k) = n!/(n − k)!.

Combinations: C(n, k) = n! / [k!(n − k)!]; C(n, k) = C(n, n − k); C(n, 0) = C(n, n) = 1.

Addition rule: if events A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Conditional probability: P(A | B) = P(A ∩ B)/P(B), provided P(B) > 0; A and B are independent iff P(A ∩ B) = P(A)P(B).

Probability of complementary event: P(A') = 1 − P(A).

Classical probability: P(A) = (number of favourable outcomes) / (total number of equally likely outcomes).

Mean of data x₁, …, x_n: x̄ = (x₁ + … + x_n)/n; variance σ² = [(x₁ − x̄)² + … + (x_n − x̄)²]/n; standard deviation σ = √σ².

Median: for ordered data, the middle value if n is odd; the average of the two middle values if n is even.

Area of triangle: S = (1/2) · base · height = (1/2)ab sin C = √(p(p−a)(p−b)(p−c)), p = (a + b + c)/2.

Area of parallelogram: S = base · height; area of trapezoid: S = (a + b)/2 · h, where a and b are the parallel sides.

Circle: circumference C = 2πr; area S = πr²; arc length l = rθ (θ in radians); sector area S = (1/2)r²θ.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c².

Angle bisector theorem: in triangle ABC, the bisector of angle A meets BC at D such that BD/DC = AB/AC.

Thales' theorem: an angle inscribed in a semicircle is a right angle; the measure of an inscribed angle is half the measure of its intercepted arc.

Tangent-secant theorem: if a tangent from point P touches the circle at T and a secant through P meets the circle at A and B, then PT² = PA · PB.

Similar triangles: if two triangles are similar with ratio k, then their areas are in ratio k² and their perimeters in ratio k.

Sum of interior angles of a convex n-gon: (n − 2)·180°.

Volume of a prism: V = B · h, where B is the base area and h is the height; volume of a pyramid: V = (1/3)B · h.

Volume of a cylinder: V = πr²h; lateral surface area: S_lat = 2πrh; total surface area: S = 2πr² + 2πrh.

Volume of a cone: V = (1/3)πr²h; lateral surface area: S_lat = πrl, where l is the slant height; l² = r² + h².

Sphere: volume V = (4/3)πr³; surface area S = 4πr².

Volume of a frustum of a cone: V = (1/3)πh(R² + Rr + r²), where R and r are the radii of the bases.

A line is perpendicular to a plane if it is perpendicular to every line in the plane passing through its foot; a line perpendicular to two intersecting lines in a plane is perpendicular to the plane.

The angle between a line and a plane is the angle between the line and its orthogonal projection onto the plane.

The angle between two planes is the angle between two lines, one in each plane, perpendicular to the line of intersection at the same point.

For a rectangular parallelepiped with dimensions a, b, c: V = abc; space diagonal d = √(a² + b² + c²).

Distance between points A(x₁, y₁) and B(x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope of line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁) (x₁ ≠ x₂); parallel lines have equal slopes; perpendicular lines have slopes m₁m₂ = −1.

Equation of line: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); general Ax + By + C = 0.

Distance from point (x₀, y₀) to line Ax + By + C = 0: d = |Ax₀ + By₀ + C| / √(A² + B²).

Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r²; general form x² + y² + Dx + Ey + F = 0 with centre (−D/2, −E/2) and radius √[(D/2)² + (E/2)² − F].

Parabola y² = 2px has focus (p/2, 0) and directrix x = −p/2; parabola x² = 2py has focus (0, p/2) and directrix y = −p/2.

Ellipse x²/a² + y²/b² = 1 (a > b > 0) has foci (±c, 0) with c² = a² − b²; eccentricity e = c/a.

Hyperbola x²/a² − y²/b² = 1 has foci (±c, 0) with c² = a² + b²; asymptotes y = ±(b/a)x.

Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): S = (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.