Arithmetic and Algebra

Arithmetic

• 2¹⁵ + 2¹⁵ equals 2¹⁶
• Considering the numbers 91, 100, 231, 440, 1003, only one of them is a prime number.
• In the ternary numeral system, the digits used are 0, 1, 2.

Ternary Numeral System Example

• 010 = 03, 110 = 13, 210 = 23, 310 = 103, 410 = 113, 510 = 123
• 91210 is represented as 10202103 in ternary form.
• Given positive integers a and b where the product ab is a multiple of 10, then a is even or b is even.

Algebra

• If polynomial P(x) is divisible by x² - 4, then 2 and -2 are certainly roots of P(x).
• Given x < 2, then √4(x-2)² equals 4 - 2x.
• The greatest common divisor (GCD) for polynomials P(x) = x³ - x², Q(x) = x³ - 2x² + x, and R(x) = x³ - x is x² - x.
• Given m as a real parameter, and the remainder of the division of 2x⁴ - mx³ + x² - 7 by x + 2 is 5, then m must be 0.
• The algebraic equation ax⁴ + bx² + c = 0, where a > 0, b is real, and c < 0, admits two real roots.
• Given K is a real parameter, for the equation x² - (K - 2)x + K - 1 = 0 to have opposite solutions, K must equal 1.
• The equation (4x² - 25)(x + 9) = 0 admits three distinct solutions in rational numbers.
• The inequality (x+2)/(x+1) ≥ 1 is satisfied for x < -1 or x ≥ -2.

Functions

• Given a = 0.2, b = 1/5, and c = 1 / log₂5, then c < a < b
• If a = 2^(log₂7 + log₅3), then a = 21.
• Given x is an angle in radians between 0 and 2π, the equation sin x + cos x = 0 has two solutions.
• The expression 7^(2+log₇x) is equal to 49x.
• The minimum period for the function y = cos²x is π.
• The inequality 2 - |log₃x| > 0 is verified for 1/9 < x < 9.
• Considering x as an angle whose measure in radians varies between 0 and 2π, the equation sin x - cos²x = 1 admits two solutions.
• The equation 8((3x-1)/3) = 4((3x+1)/2) has infinite solutions for the real values of variable x.
• Given π/4 < α < β ≤ π, then cos α > cos β.

Geometry

• When crossing a rectangular flowerbed along the diagonal instead of walking along its two sides, the maximum distance saved is approximately 30%.
• A right triangle has a perimeter of 12 cm and its legs are 3 cm and 4 cm.
• Among the geometric figures listed, a semi-plane is not convex.
• Considering a square Q and a regular hexagon E inscribed in the same circle, with A(Q) and A(E) as their respective areas, A(E) > A(Q).
• Given a Cartesian coordinate system Oxy, the distance from the point (2, 1) to the line x + y + 1 = 0 is 2√2.
• In a Cartesian plane Oxy, consider lines r: 2x + 3y + k = 0 and s: 6x + ky + 9 = 0, where k is a real parameter, for a suitable choice of k the lines r and s are parallel and distinct.
• Considering an equilateral triangle, S is the area of the circumscribed circle and s is the area of the inscribed circle, S = 4s.
• A right circular cone has a base radius r and height h. If the base radius is doubled and the height is halved, the volume of the cone doubles.
• Given a Cartesian coordinate system Oxy, and two circles with equations x² + y² = 9 and (x - 1)² + y² = 1, there are two common tangent lines to c and c'.
• Using a Cartesian coordinate system Oxy, the locus of points whose coordinates (x, y) satisfy the equation x(2x + y - 1) = 0 is a pair of lines.

Logic

• Aldo, Bruno, and Carlo are three friends where at least one of them is a graduate; if Aldo is a graduate, so is Bruno; if Carlo is a graduate, so is Aldo; only one between Bruno and Carlo is a graduate, then it is deduced that Aldo and Bruno are graduates.
• The assertion "There is no skyscraper without an elevator" means every skyscraper has at least one elevator.
• In a soccer championship, 60% of penalties were in favor of the home team, and 40% were in favor of the away team 80% of the penalties awarded to the home team were successful, while only 75% of those awarded to the away team were successful. The overall percentage of successful penalties was 78%.
• In three years, Aldo will be twice the age that Sara was three years ago, now the quadruple of his age is equal to the quintuple of her age. It can be deduced that Sara is older than Aldo.
• A box contains 10 cubes where each face of each cube is colored green, white, or red. Six cubes have at least one green face, 7 have at least one white face, and 9 have at least one red face. No cube has all faces of the same color, and bhere are two cubes in the box have faces of all three colors.
• From the proposition "A sequence of real numbers, if increasing and bounded, is convergent", it can be deduced that if the conditions of increasing and boundedness are sufficient for the convergence of a real sequence.
• In a cube, two vertices are said to be opposite if the segment connecting them passes through the center of the cube. No face contains both of them.
• Given two fractions, it is false to say that it is not necessary for them to have an equal numerator and denominator for them to be equal.

Statistics

• Given 5 flags (3 yellow, 2 blue) being raised on a pole, there are 10 possible different signals that Aldo can obtain.
• Given all 4-letter "words" are taken once from the 6 letters A, B, C, D, E, and F, the percentage of such words that do not contain a vowel is approximately 7%.
• With a set of 6 elements, the amount of three element subsets that exist within it are 20.
• Carlo has passed 9 university exams and if the average of their grades is 24 (out of 30), and what will the minimum grade Carlo will need to take in the next exam in order to raise their average to 24.5 is a 29.

Trigonometry

• By launching a regular coin in the air, then the odds of it landing on heads or tails each time is 1/2, launching three coins the frequency of 2 heads and 1 tails occurring is 3/8.
• With an angle measuring 15°, it's radians in measurement is less than 0.25 radians.
• With the measurement of the shadow of a bell tower is half its height, the angle formed by the sun 𝛼° is less than 30°.
• Within a circumference of r radius, the length of a chord being taken to produce an arch, is 2r sin0,5.
• Given the expression sin 31° + sin 29°, then it is equal to cos 1.