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Questions and Answers
Consider the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, and y = nx + 1. What is the area of this parallelogram?
Positive real numbers x, y not equal to 1 satisfy log₂ x = log_y 16 and xy = 64. What is (log₂ xy)²?
What is the inverse of the function f(x) = 2x(x−1)?
Given A = (sin x)² + (cos x)⁴, what can be said about the value of A for all real values of x?
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Circles with radii 1, 2, and 3 are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
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The equations representing two pairs of opposite sides of a parallelogram are x² − 5x + 6 = 0 and y² − 6y + 5 = 0. What are the equations of its diagonals?
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What is the value of the expression involving square roots and logarithms?
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How many solutions of the equation tan x + sec x = 2 cos x lie in the interval [0, 2π]?
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What is the area of the region enclosed by the graph of the equation x² + y² = |x| + |y|?
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What is the number of terms with rational coefficients among the 1001 terms in the expansion of (x^(3/2) + y^(2/3))¹⁰⁰⁰?
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For sequences of positive real numbers of the form x, 2000, y... For how many different values of x does the term 2001 appear somewhere in the sequence?
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Suppose that |x + y| + |x − y| = 2. What is the maximum possible value of x² − 6x + y²?
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For how many integers n between 1 and 50, inclusive, is (n² −1)! / (n!)ⁿ an integer?
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What is D(96), where D(n) denotes the number of ways of writing the positive integer n as a product of integers greater than 1?
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Let f₁(x) = 1 − x, and for integers n ≥ 2, let fₙ(x) = fₙ₋₁(n² − x). If N is the largest value of n for which the domain of fₙ is nonempty, what is N + c where the domain of fₙ is [c]?
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Study Notes
Parallelogram Area
- The area of the parallelogram formed by the lines ( y = mx ), ( y = mx + 1 ), ( y = nx ), and ( y = nx + 1 ) is given by ( |m - n| ).
Logarithmic Equations
- Positive real numbers ( x ), ( y ) (not equal to 1) satisfy ( \log_2 x = \log_y 16 ) and ( xy = 64 ); the value of ( ( \log_2 xy )^2 ) is 20.
Inverse Function
- The function ( f(x) = 2x(x-1) ) has an inverse in the domain and range of real values ( \geq 1 ).
Sine and Cosine Functions
- Given ( A = (\sin x)^2 + (\cos x)^4 ), the bounds for ( A ) are established as ( 0 \leq A \leq \frac{4}{3} ).
Triangle Area from Circles
- The area of the triangle formed by the points of tangency of circles with radii 1, 2, and 3 is ( \frac{5}{4} ).
Parallelogram Diagonals
- For the parallelogram defined by equations ( x^2 - 5x + 6 = 0 ) and ( y^2 - 6y + 5 = 0 ), the diagonals are represented by ( 4x + y = 13 ) and ( y = 4x - 7 ).
Mathematical Evaluation
- The mathematical expression given evaluates to 2.
Solutions to Trigonometric Equation
- The equation ( \tan x + \sec x = 2 \cos x ) has 2 solutions within the interval ( [0, 2\pi] ).
Area Enclosed by Graph
- The area enclosed by the graph of ( x^2 + y^2 = |x| + |y| ) is ( 2\pi + 2\sqrt{2} ).
Coefficient Counts in Expansion
- In the expansion of ( (x^{3/2} + y^{2/3})^{1000} ), there are 501 terms with rational coefficients.
Sequence Values
- The number of different values of ( x ) such that ( 2001 ) appears in the sequence formed by positive real numbers ( x, 2000, y ) is 2.
Maximum Expression Value
- For ( |x + y| + |x - y| = 2 ), the maximum value of the expression ( x^2 - 6x + y^2 ) is 9.
Integer Conditions
- For integers ( n ) between 1 and 50, inclusive, the condition ( \frac{(n^2 - 1)!}{(n!)^n} ) results in 34 valid integers.
Factorization Count
- The number of distinct ways to express ( n = 96 ) as a product of integers greater than 1 is 128.
Function Domain
- Establishing ( f_1(x) = 1 - x ) leads to ( N ) as the largest integer where the domain of ( f_N ) remains nonempty; ( N + c ) is defined.
Integral Calculation
- Integral calculations involving the fractional part of positive real numbers ( x ) are implied but not completed in the text.
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Description
Test your understanding of areas, functions, and logarithmic equations with this comprehensive quiz. Covering topics such as parallelogram area calculation, inverse functions, and properties of trigonometric functions, this quiz will challenge your mathematical skills. It's perfect for students seeking to solidify their knowledge in geometry and algebra.
