Armenia Mathematics Unified Exam flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Divisibility definition

An integer a divides b (written a | b) if there exists an integer k with b = ak. If a | b and a | c, then a divides any combination mb + nc.

Euclidean algorithm

A method for finding gcd(a, b): replace the pair by (b, a mod b) and repeat until the remainder is zero. The last nonzero remainder is the greatest common divisor.

Fundamental theorem of arithmetic

Every integer greater than 1 can be written as a product of primes, and this prime factorization is unique up to the order of the factors.

Divisor-count formula

If n = p1^e1 · p2^e2 ··· pk^ek is the prime factorization, the number of positive divisors equals (e1 + 1)(e2 + 1)···(ek + 1).

Difference of squares

The identity a^2 − b^2 = (a − b)(a + b). It factors a difference of two perfect squares into a product of the sum and difference of the bases.

Quadratic discriminant

For ax^2 + bx + c = 0, the discriminant is D = b^2 − 4ac. Two distinct real roots occur when D > 0, one double root when D = 0, no real roots when D < 0.

Vieta's formulas

For the quadratic ax^2 + bx + c = 0 with roots x1 and x2, the sum of roots is −b/a and the product of roots is c/a.

Absolute value inequality

For a ≥ 0, the inequality |x| ≤ a means −a ≤ x ≤ a, while |x| ≥ a means x ≤ −a or x ≥ a. The two cases split at the center point.

Quadratic formula

The roots of ax^2 + bx + c = 0 with a ≠ 0 are x = (−b ± √(b^2 − 4ac)) / (2a), obtained by completing the square.

Interval sign method

To solve a rational inequality, find all zeros of numerator and denominator, mark them on a number line, and test one sample point per interval to determine the sign.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Adding or subtracting any number never changes the direction.

Even and odd functions

A function is even when f(−x) = f(x) for all x in its domain, giving symmetry about the y-axis. It is odd when f(−x) = −f(x), giving symmetry about the origin.

Parabola vertex formula

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

Graph shift rule

Replacing x by (x − h) shifts the graph of f horizontally by h, and adding k shifts it vertically by k. The transformed graph is y = f(x − h) + k.

Arithmetic progression formula

An arithmetic progression has constant difference d, so a_n = a_1 + (n − 1)d. The sum of the first n terms is S_n = n(a_1 + a_n)/2.

Geometric progression sum

A geometric progression has constant ratio q, so b_n = b_1 q^{n−1}. For q ≠ 1, the sum of the first n terms is b_1(q^n − 1)/(q − 1).

Infinite geometric series

The series b_1 + b_1 q + b_1 q^2 + … converges exactly when |q| < 1, and its sum equals b_1/(1 − q). Otherwise the series diverges.

Pythagorean trigonometric identity

For every angle x, sin^2 x + cos^2 x = 1. Dividing this identity by cos^2 x gives 1 + tan^2 x = 1/cos^2 x wherever cosine is nonzero.

Law of cosines

In any triangle with sides a, b, c and angle C opposite side c, the relation c^2 = a^2 + b^2 − 2ab cos C holds, generalizing the Pythagorean theorem.

Law of sines

In any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle around the triangle.

Sine double-angle formula

For any angle x, sin 2x = 2 sin x cos x. The companion formula is cos 2x = cos^2 x − sin^2 x = 1 − 2 sin^2 x.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc. An angle subtended by a diameter is therefore a right angle of 90 degrees.

Triangle area by sine

The area of a triangle equals one half the product of two sides times the sine of the included angle: S = (1/2)ab sin C.

Similarity area ratio

If two figures are similar with linear scale factor k, their areas are in ratio k^2 and, for solids, their volumes are in ratio k^3.

Sphere volume and area

A sphere of radius R has volume V = (4/3)πR^3 and surface area S = 4πR^2. Both formulas are expressed purely in terms of the radius.

Cone and pyramid volume

A cone or pyramid with base area S and height h has volume V = (1/3)S·h, exactly one third of the prism or cylinder with the same base and height.

Combination formula

The number of ways to choose k objects from n distinct objects without regard to order is C(n, k) = n!/(k!(n − k)!), which equals C(n, n − k).

Binomial theorem

For any positive integer n, (a + b)^n = Σ C(n, k) a^{n−k} b^k over k from 0 to n, where C(n, k) are binomial coefficients.

Classical probability

When all outcomes are equally likely, the probability of an event equals the number of favorable outcomes divided by the total number of outcomes, a value between 0 and 1.

Conditional probability formula

For events A and B with P(B) > 0, the conditional probability P(A|B) = P(A ∩ B)/P(B) measures the chance of A given that B occurred.

Derivative as limit

The derivative f'(x) = lim_{h→0} (f(x + h) − f(x))/h, when this limit exists, equals the slope of the tangent line to the graph at x.

Newton–Leibniz formula

If F is any antiderivative of a continuous function f on [a, b], then the definite integral of f from a to b equals F(b) − F(a).