Armenia Mathematics Unified Exam cheat sheet

Divisibility: a | b iff b = ak for some integer k; if a | b and a | c then a | (mb + nc) for all integers m, n.

Euclid's lemma: gcd(a, b) = gcd(b, a mod b); iterate until remainder 0, the last nonzero remainder is the gcd.

a = bq + r with 0 ≤ r < b is unique; a and b are coprime iff there exist integers x, y with ax + by = 1 (Bezout).

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

Number of divisors: if n = p1^e1 ··· pk^ek then d(n) = (e1+1)(e2+1)···(ek+1).

Divisibility tests: by 3 or 9 iff digit sum is; by 4 iff last two digits form a multiple of 4; by 11 iff alternating digit sum is a multiple of 11.

Modular arithmetic: a ≡ b (mod m) implies a^n ≡ b^n (mod m); congruences ax ≡ b (mod m) are solvable iff gcd(a, m) | b.

lcm(a, b) · gcd(a, b) = |ab|, and gcd(a, b) = 1 implies lcm(a, b) = |ab|.

(a ± b)^2 = a^2 ± 2ab + b^2; (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

Difference of squares: a^2 − b^2 = (a − b)(a + b); sum/difference of cubes: a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2).

For ax^2 + bx + c, the discriminant D = b^2 − 4ac determines the number of real roots: D > 0 two, D = 0 one, D < 0 none.

Vieta's formulas: for ax^2 + bx + c = 0, x1 + x2 = −b/a and x1·x2 = c/a.

AM–GM inequality: for positive reals, (a + b)/2 ≥ √(ab), with equality iff a = b.

|x| ≤ a (a ≥ 0) is equivalent to −a ≤ x ≤ a; |x| ≥ a is equivalent to x ≤ −a or x ≥ a.

Rationalizing: multiply numerator and denominator by the conjugate √a ∓ √b to remove radicals from a denominator.

Logarithm laws: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x − log_a y, log_a(x^k) = k log_a x, log_a b = ln b / ln a.

Linear equation ax + b = 0 has unique solution x = −b/a when a ≠ 0; no solution when a = 0 and b ≠ 0; infinitely many when a = b = 0.

Quadratic formula: x = (−b ± √(b^2 − 4ac)) / (2a) for ax^2 + bx + c = 0 with a ≠ 0.

Sign of a quadratic with a > 0 is positive outside the roots and negative between them; reverse for a < 0.

Interval method: mark zeros of numerator and denominator on a number line, test one point per interval to fix signs.

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

A system of two linear equations has a unique solution iff the coefficient determinant a1b2 − a2b1 ≠ 0.

Exponential equation a^f(x) = a^g(x) with 0 < a ≠ 1 implies f(x) = g(x); for a > 1, a^u > a^v iff u > v.

Extraneous roots can appear after squaring; every candidate must be checked in the original equation.

A function is even if f(−x) = f(x) (symmetric about the y-axis), odd if f(−x) = −f(x) (symmetric about the origin).

y = f(x − h) + k shifts the graph of f right by h (if h > 0) and up by k.

The domain of f(g(x)) requires x in the domain of g and g(x) in the domain of f.

A function has an inverse iff it is injective on its domain; graphs of f and f^{-1} are mirror images across y = x.

For y = ax^2 + bx + c with a ≠ 0, the vertex is at x = −b/(2a) and the axis of symmetry is x = −b/(2a).

y = k/x (k ≠ 0) has vertical asymptote x = 0 and horizontal asymptote y = 0; branches lie in opposite quadrants.

y = √x is defined for x ≥ 0; y = log_a x requires x > 0 and 0 < a ≠ 1.

f is increasing on an interval if x1 < x2 implies f(x1) < f(x2) there; decreasing if the inequality reverses.

Arithmetic progression: a_n = a_1 + (n − 1)d, and S_n = n(a_1 + a_n)/2 = n(2a_1 + (n − 1)d)/2.

Geometric progression: b_n = b_1 q^{n−1}, and S_n = b_1(q^n − 1)/(q − 1) for q ≠ 1.

Infinite geometric series converges iff |q| < 1, with sum S = b_1/(1 − q).

Characteristic property: a, b, c are in arithmetic progression iff 2b = a + c; in geometric progression iff b^2 = ac (b ≠ 0).

Each term of an arithmetic progression is the arithmetic mean of its neighbors: a_n = (a_{n−1} + a_{n+1})/2.

Each term of a geometric progression is the geometric mean of its neighbors: b_n^2 = b_{n−1}·b_{n+1} for positive terms.

Sum of the first n natural numbers: 1 + 2 + ··· + n = n(n + 1)/2; sum of squares: n(n + 1)(2n + 1)/6.

Method of differences: if a_{n+1} − a_n = d for all n, the sequence is arithmetic with common difference d.

Pythagorean identity: sin^2 x + cos^2 x = 1; also 1 + tan^2 x = 1/cos^2 x and 1 + cot^2 x = 1/sin^2 x.

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: sin 2x = 2 sin x cos x; cos 2x = cos^2 x − sin^2 x = 1 − 2sin^2 x = 2cos^2 x − 1.

Law of sines: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

Law of cosines: c^2 = a^2 + b^2 − 2ab cos C.

Triangle area: S = (1/2)ab sin C, and S = abc/(4R) = pr, where p is the semiperimeter and r the inradius.

Reduction: sin(π − x) = sin x, cos(π − x) = −cos x, sin(x + 2πk) = sin x for integer k.

Equation sin x = a with |a| ≤ 1 has solutions x = (−1)^k arcsin a + πk; cos x = a gives x = ±arccos a + 2πk.

Triangle angle sum is 180°; an exterior angle equals the sum of the two non-adjacent interior angles.

Pythagorean theorem: in a right triangle a^2 + b^2 = c^2, where c is the hypotenuse.

Similar triangles: corresponding sides are proportional and the ratio of areas equals the square of the similarity ratio.

Circle: inscribed angle equals half the intercepted arc; an angle subtended by a diameter is 90°.

Chord and tangent: the tangent is perpendicular to the radius at the point of tangency; tangent segments from one external point are equal.

Area formulas: triangle S = (1/2)ah; trapezoid S = (a + b)h/2; circle S = πR^2, circumference C = 2πR.

Parallelogram area S = ab sin α, where α is the angle between sides a and b.

Median to the hypotenuse of a right triangle equals half the hypotenuse, so the midpoint of the hypotenuse is equidistant from all three vertices.

Volume of a prism and of a cylinder: V = S_base · h; lateral surface of a cylinder is 2πRh.

Pyramid and cone: V = (1/3)S_base · h; cone lateral surface is πRl where l is the slant height, and l^2 = R^2 + h^2.

Sphere: V = (4/3)πR^3, surface area S = 4πR^2.

For similar solids with ratio k, areas scale as k^2 and volumes as k^3.

Rectangular box: V = abc, surface area S = 2(ab + bc + ac), space diagonal d = √(a^2 + b^2 + c^2).

Plane sections of a sphere are circles; the largest is the great circle with radius R, smaller sections have radius √(R^2 − d^2) at distance d from the center.

Volume of a frustum: V = (h/3)(S1 + S2 + √(S1·S2)), where S1 and S2 are the base areas.

A line perpendicular to a plane is perpendicular to every line in that plane passing through its foot.

Rule of product: if a choice can be made in m ways and then another in n ways, the pair can be made in mn ways; the rule of sum applies to disjoint alternatives.

Permutations of n distinct objects: P(n) = n!; arrangements of k from n: A(n, k) = n!/(n − k)!.

Combinations: C(n, k) = n!/(k!(n − k)!), with C(n, k) = C(n, n − k) and C(n, 0) = C(n, n) = 1.

Permutations with repetition: the number of distinct arrangements of a multiset with groups n1, n2, … is n!/(n1!·n2!·…).

Pascal's identity: C(n, k) = C(n − 1, k − 1) + C(n − 1, k).

Binomial theorem: (a + b)^n = Σ C(n, k) a^{n−k} b^k for k = 0 to n.

Circular arrangements of n distinct objects number (n − 1)!; with reflections identified, (n − 1)!/2.

Inclusion–exclusion for two sets: |A ∪ B| = |A| + |B| − |A ∩ B|.

Classical probability: P(A) = (number of favorable outcomes)/(number of equally likely outcomes), with 0 ≤ P(A) ≤ 1.

Complement rule: P(not A) = 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) for P(B) > 0.

Total probability: P(A) = Σ P(H_i)·P(A|H_i) over a complete set of mutually exclusive hypotheses H_i.

Mean of a data set is the sum of values divided by their count; median is the middle value of an ordered set (average of two middles if n is even).

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

For a binomial distribution with n trials and success probability p: expected value np, variance np(1 − p).

Adding a constant c to every value shifts the mean by c but leaves the variance and standard deviation unchanged.

Derivative definition: f'(x) = lim_{h→0} (f(x + h) − f(x))/h; it equals the slope of the tangent line at x.

Rules: (x^n)' = n x^{n−1}, (sin x)' = cos x, (cos x)' = −sin x, (e^x)' = e^x, (ln x)' = 1/x.

Product and quotient rules: (uv)' = u'v + uv'; (u/v)' = (u'v − uv')/v^2.

Chain rule: (f(g(x)))' = f'(g(x))·g'(x).

A function has a local extremum only at critical points (f'(x) = 0 or f' undefined); sign change of f' from + to − gives a local maximum.

On [a, b], a continuous function attains its greatest and least values either at critical points or at the endpoints.

Newton–Leibniz formula: ∫_a^b f(x) dx = F(b) − F(a), where F is any antiderivative of f.

The definite integral equals the signed area between the graph and the x-axis; the area between two curves is ∫_a^b (f(x) − g(x)) dx when f ≥ g on [a, b].

L'Hôpital's rule: for a 0/0 or ∞/∞ indeterminate form, lim f/g = lim f'/g' when the latter limit exists.