ЦТ Mathematics cheat sheet

Divisibility: by 3 or 9 when the digit sum is divisible by 3 or 9; by 4 when the last two digits form a multiple of 4; by 8 when the last three digits form a multiple of 8; by 11 when the alternating digit sum is divisible by 11.

Core identities: (a ± b)² = a² ± 2ab + b², a² − b² = (a − b)(a + b), a³ ± b³ = (a ± b)(a² ∓ ab + b²).

For a ≥ 0 and b ≥ 0, √(ab) = √a · √b and √(a/b) = √a / √b; in general √(a²) = |a|.

Power laws: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, aᵐ/ⁿ = ⁿ√(aᵐ), with a > 0 for fractional exponents.

A real number is rational exactly when its decimal expansion terminates or eventually repeats; non-terminating non-repeating decimals are irrational.

Proportion a/b = c/d is equivalent to ad = bc; a percent is a ratio with denominator 100, so p% = p/100.

Percent change: new value = old value × (1 ± p/100); percent of a whole = (part/whole) × 100%.

To rationalize a denominator containing √a ± √b, multiply numerator and denominator by the conjugate √a ∓ √b.

Linear equation ax + b = 0 with a ≠ 0 has the unique solution x = −b/a; if a = 0 and b ≠ 0 there is no solution, and if a = b = 0 every x is a solution.

Quadratic formula: for ax² + bx + c = 0, a ≠ 0, x = (−b ± √(b² − 4ac))/(2a). The discriminant D = b² − 4ac gives two real roots if D > 0, one if D = 0, none if D < 0.

Vieta: for ax² + bx + c = 0, x₁ + x₂ = −b/a and x₁x₂ = c/a; factoring gives ax² + bx + c = a(x − x₁)(x − x₂).

The equation |x| = a has solutions x = ±a when a > 0, x = 0 when a = 0, and no solutions when a < 0.

For 2×2 systems, elimination or substitution reduces the system to one variable; Cramer's rule gives x = Δx/Δ, y = Δy/Δ when the determinant Δ ≠ 0.

An identity holds for every admissible value of the variables, while an equation holds only for its solution set; distinguish them when simplifying or proving statements.

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

Squaring both sides can create extraneous roots, so every candidate solution of a radical equation must be checked in the original equation.

Adding or subtracting the same number preserves an inequality; multiplying or dividing by a negative number reverses its sign.

For a > 0: |x| < a ↔ −a < x < a; |x| > a ↔ x < −a or x > a; non-strict versions include the corresponding endpoints.

Quadratic inequality a(x − x₁)(x − x₂) > 0 is solved by the sign of the product on the intervals cut by x₁ and x₂; the sign alternates at simple roots.

Rational inequalities are solved by moving everything to one side, factoring numerator and denominator, and using a sign chart; denominator zeros are always excluded.

The solution of a system is the intersection of the individual solution sets; a compound statement joined by “or” gives their union.

A quadratic x² ≥ 0 holds for all real x, while x² ≤ 0 holds only for x = 0; x² < 0 has no real solutions.

Interval notation: (a, b) excludes endpoints, [a, b] includes them, and (−∞, a] ∪ [b, ∞) denotes two rays.

For non-strict inequalities, test whether endpoints satisfy the original inequality before including them in the answer.

Domain restrictions: denominator ≠ 0, radicand of an even root ≥ 0, logarithm argument > 0; the domain is the intersection of all restrictions.

Transformations: f(x) + c shifts up by c, f(x + c) shifts left by c, −f(x) reflects across the x-axis, f(−x) reflects across the y-axis, and a·f(x) stretches vertically by |a|.

A function is even when f(−x) = f(x) and odd when f(−x) = −f(x); even graphs are symmetric about the y-axis, odd graphs about the origin.

For y = ax² + bx + c, the vertex is at x = −b/(2a); it is a minimum for a > 0 and a maximum for a < 0, giving the range [y_vertex, ∞) or (−∞, y_vertex].

A differentiable function is increasing where f′(x) > 0 and decreasing where f′(x) < 0; critical points occur where f′(x) = 0 or is undefined.

A function and its inverse swap domain and range; the graph of f⁻¹ is the reflection of the graph of f across the line y = x.

A rational function may have vertical asymptotes at denominator zeros and a horizontal asymptote determined by comparing degrees of numerator and denominator.

Monotonicity on an interval permits solving f(x) = c by uniqueness: a strictly monotone function attains each value in its range at most once.

Fundamental identities: sin²x + cos²x = 1, tan x = sin x/cos x, 1 + tan²x = sec²x, 1 + cot²x = csc²x.

Addition formulas: 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 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1; tan 2x = 2tan x/(1 − tan²x).

General solutions: sin x = a gives x = (−1)^k arcsin a + πk; cos x = a gives x = ±arccos a + 2πk; tan x = a gives x = arctan a + πk, k ∈ ℤ.

Inverse ranges: arcsin x ∈ [−π/2, π/2], arccos x ∈ [0, π], arctan x ∈ (−π/2, π/2).

Reduction formulas: use the quadrant to choose the sign and the reference angle to express a trigonometric value through an acute angle.

The expression a sin x + b cos x equals √(a² + b²) sin(x + φ), where cos φ = a/√(a² + b²) and sin φ = b/√(a² + b²).

Trigonometric inequalities are solved by marking solution points on the unit circle and selecting the arcs where the required coordinate has the correct sign.

Arithmetic progression: aₙ = a₁ + (n − 1)d, where d = aₙ₊₁ − aₙ is the common difference.

Sum of the first n terms of an arithmetic progression: Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n − 1)d)/2.

Geometric progression: bₙ = b₁qⁿ⁻¹, where q = bₙ₊₁/bₙ is the common ratio; q may be negative or fractional.

Sum of the first n terms of a geometric progression: Sₙ = b₁(qⁿ − 1)/(q − 1) for q ≠ 1, and Sₙ = nb₁ for q = 1.

Infinite geometric series converges exactly when |q| < 1, and then S = b₁/(1 − q).

Characteristic properties: arithmetic terms satisfy aₙ = (aₙ₋₁ + aₙ₊₁)/2; positive geometric terms satisfy bₙ² = bₙ₋₁bₙ₊₁.

The number of terms in an arithmetic progression is n = (aₙ − a₁)/d + 1 when d ≠ 0.

Triangle area: S = ½ab sin C, Heron's formula S = √(p(p − a)(p − b)(p − c)) with p = (a + b + c)/2, and S = pr = abc/(4R).

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

Angle sums: triangle angles total 180°; a convex n-gon has interior angle sum (n − 2)·180° and exterior angle sum 360°.

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

Similar figures with linear scale factor k have area ratio k² and volume ratio k³; corresponding angles are equal.

Areas: parallelogram bh, trapezoid (a + b)h/2, rhombus d₁d₂/2, where diagonals are perpendicular.

Vectors: a·b = |a||b|cos θ = x₁x₂ + y₁y₂; perpendicular vectors have zero dot product.

An inscribed angle equals half the central angle subtending the same arc; an angle inscribed in a semicircle is 90°.

Prism and cylinder volume V = Bh, where B is base area and h is perpendicular height; pyramid and cone volume V = ⅓Bh.

Sphere: volume 4πr³/3, surface area 4πr²; hemisphere adds base area πr² to half the spherical surface.

Cylinder lateral area 2πrh and total area 2πr(h + r); cone lateral area πrl and total area πr(l + r), where l is slant height.

A line is perpendicular to a plane when it is perpendicular to two intersecting lines in that plane.

If two parallel planes are cut by a transversal, the intercepted segments on any two transversals are proportional.

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

Euler's formula for a convex polyhedron: V − E + F = 2, relating vertices, edges and faces.

Similar solids with linear scale factor k have surface area ratio k² and volume ratio k³.

Product rule: if a choice has m options and a second independent choice has n options, the number of combined outcomes is mn.

Permutations: P(n, k) = n!/(n − k)! counts ordered selections of k objects from n; combinations: C(n, k) = n!/(k!(n − k)!) counts unordered subsets.

Classical probability: P(A) = number of favorable equally likely outcomes / total number of outcomes; always 0 ≤ P(A) ≤ 1.

Complement rule P(Ā) = 1 − P(A); addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Independent events satisfy P(A ∩ B) = P(A)P(B); conditional probability is P(A|B) = P(A ∩ B)/P(B), P(B) > 0.

Mean = sum of values divided by count; median = middle value of ordered data; mode = most frequent value.

Variance is the mean of squared deviations from the mean; standard deviation is its non-negative square root.

Pascal's identity C(n, k) = C(n − 1, k − 1) + C(n − 1, k) and symmetry C(n, k) = C(n, n − k) simplify binomial counts.

Uniform motion: distance = speed × time; for motion toward each other add speeds, for motion in the same direction subtract speeds.

Work: rate = 1/time for one full job; combined rate is 1/t₁ + 1/t₂, so combined time is t₁t₂/(t₁ + t₂).

Mixtures: concentration = solute amount / total amount; conservation of solute gives c₁m₁ + c₂m₂ = c(m₁ + m₂).

Successive percent changes multiply factors: a 20% rise followed by a 10% fall gives 1.20 × 0.90 = 1.08, a net 8% rise.

Average speed = total distance / total time; it equals the arithmetic mean of speeds only when equal times are spent at each speed.

Optimization with a quadratic model: the maximum or minimum occurs at the vertex x = −b/(2a), subject to the problem's constraints.

Estimation via rounding: compute with rounded values, then state bounds using the maximum possible rounding error in each quantity.

When modeling, define the variable, write an equation for the stated relation, solve, and reject solutions that violate physical constraints such as positive length or time.