MaxWong

It is not possible for a function to have different values for the same input. f(3) cannot be 0 and 28 at the same time.

Plotting it in a graphical calculator gives the range as \((-\infty, +\infty)\).

If n/21 is reducible, then either n is a multiple of 3 or n is a multiple of 7.

There are floor(100/3) = 33 multiples of 3 from 1 to 100, and there are floor(100/7) = 14 multiples of 7 from 1 to 100.

But we have double counted floor(100/21) = 4 multiples of 21, so the total number of integers n with 1 <= n <= 100 such that n/21 is reducible is 33 + 14 - 4 = 43.

\(\dfrac{x^2 + 5x + a}{x^2 - 10x - 24} = \dfrac{x^2 + 5x + a}{(x - 12)(x + 2)}\)

If the fraction is equal to the quotient of two linear functions, then either (x - 12) or (x + 2) divides x^2 + 5x + a.

By factor theorem, 

\(12^2 + 5\cdot 12 + a = 0\text{ or }(-2)^2 + 5(-2) + a = 0\\ a = -204 \text{ or }a = 6\)

Therefore \(S = \{-204, 6\}\). Sum of elements of S is -198.

Remarks: The statement of factor theorem is as follows.

Let \(p(x)\) be a polynomial such that \((x - a)\) divides \(p(x)\). Then \(p(a) = 0\).

Let AR = 2k, RC = 3k, AP = p, PQ = 2p, QB = p.

Then 

\(\begin{array}{rcl} [QBC] &=& [ABC] - [ACQ]\\ &=& \dfrac12 (5k)(4p) \sin \angle A - \dfrac12 (5k)(3p) \sin \angle A\\ &=& \dfrac{5kp}2 \sin \angle A\\ [CRQ] &=& [ACQ] - [AQR]\\ &=& \dfrac12 (5k)(3p)\sin \angle A - \dfrac12 (2k)(3p)\sin \angle A\\ &=& \dfrac{9kp}2\sin \angle A \end{array}\)

Therefore, 

\(\dfrac{[QBC]}{[CRQ]} = \dfrac{\dfrac{5kp}2\sin \angle A}{\dfrac{9kp}2\sin \angle A} = \dfrac5{9}\)

See this answer: https://web2.0calc.com/questions/number-theory_16563#r1

Similar problem solved here: https://web2.0calc.com/questions/find-the-volume-of-the-tetrahedron-with-vertices

Please try to follow the logic of this solution.

Similar problem solved here: https://web2.0calc.com/questions/arithmetic-sequence_50

Try to follow the logic of the solution here.

This problem is solved here: https://web2.0calc.com/questions/counting-problem_147

Add 1 to each equation.

\(\begin{cases}xy + x + y + 1 = 24\\yz + y + z + 1= 32\\zx + z + x + 1 = 48\end{cases}\)

We do this because now the left-hand side is factorable.

\(\begin{cases}(x + 1)(y + 1) = 24\qquad --- (1)\\(y + 1)(z + 1) = 32\qquad --- (2)\\(z + 1)(x + 1) = 48\qquad --- (3)\end{cases}\)

Now consider (1) * (2) / 3, we have

\(\dfrac{(x + 1)(y + 1)^2(z + 1)}{(z + 1)(x + 1)} = \dfrac{24 \cdot 32}{48}\\ (y + 1)^2 = 16\\ y = 3\text{ or }y = -5\)

When y = 3,

(x + 1)(4) = 24

x = 5

(z + 1)(5 + 1) = 48

z = 7

Therefore (x, y, z) = (5, 3, 7) is a solution.

When y = -5,

(x + 1)(-4) = 24

x = -7

(z + 1)(-7 + 1) = 48

z = -9

Therefore (x, y, z) = (-7, -5, -9) is another solution.

Solutions are (x, y, z) = (5, 3, 7) and (x, y, z) = (-7, -5, -9).

By Pythagorean theorem, \(6^2 + b^2 = c^2\).

Perimeter is 15 means \(6 + b + c = 15\).

Then \(\begin{cases}c^2 - b^2 = 36\\b + c = 9\end{cases}\). Note that c^2 - b^2 = (c - b)(b + c) = 9(c - b). Therefore:

\(9(c - b) = 36\\ c - b = 4\)

Consider \(2c = (b + c) + (c - b)\).

\(2c = 9 + 4\\ c = \dfrac{13}2\)

A similar problem is answered here: https://web2.0calc.com/questions/angles_70

Since the equation PA^2 + PB^2+PC^2=3PQ^2 + k holds true for any point P, we may let P = A, P = B, P = C and get 3 equations.

Let Q = (r, s). Then substituting P = A, B, C respectively gives: \(\begin{cases}53 + 65 = 3((r - 4)^2 + (s + 1)^2) + k\\53 + 146 = 3((r + 3)^2 + (s - 1)^2) + k\\65 + 146 = 3((r - 8)^2 + (s - 6)^2) + k\end{cases}\).

Solving this system is possible, but it is very tedious work, so I will do it with Wolfram Alpha:

Check: The resulting polynomial 2x^2 + 4x + 2 is indeed the square of the binomial \((\sqrt 2 x + \sqrt 2)^2\)

Simplify first.

\(4x^2 + 14x +c -2x^2 - 10x = 2x^2 + 4x + c\).

If this is a square of a binomial, its discriminant is zero.

\(4^2 - 4(2)(c) = 0 \\ 8c = 16\\ c = 2\)

If 2u is not in the interval, then \(2u \leq -10 \text{ or }2u \geq 10\). That means \(u \leq -5 \text{ or }u \geq 5\).

Moreover, -20u is not in the interval, so \(-20u \leq -10 \text{ or }-20u \geq 10\). That means \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\).

If \(u \leq -5 \text{ or }u \geq 5\), then it is true that \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\). So \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\) is extraneous. The range of values of u is \(u \leq -5 \text{ or }u \geq 5\). Expressed in interval notation, \(u \in (-\infty, -5]\cup [5, \infty)\).

You can just use a calculator, but here's a calculation trick:

The denominator looks like \(n^3 - n = n(n - 1)(n + 1)\) we can perform partial fractions on \(\dfrac1{n(n-1)(n+1)}\) to get \(\dfrac12 \left(\dfrac1{n - 1} - \dfrac1n\right) - \dfrac12 \left(\dfrac1n - \dfrac1{n + 1}\right)\). The original expression is \(\displaystyle \sum_{n=2}^{10}\left(\dfrac12 \left(\dfrac1{n - 1} - \dfrac1n\right) - \dfrac12\left(\dfrac1n - \dfrac1{n + 1}\right)\right)=\dfrac12\displaystyle \sum_{n=2}^{10} \left(\dfrac1{n - 1} - \dfrac1n\right) - \dfrac12\displaystyle \sum_{n=2}^{10} \left(\dfrac1n - \dfrac1{n + 1}\right)\). Now note that each sum is a telescoping sum, so the answer is \(\dfrac12 \left(1 - \dfrac1{10}\right) - \dfrac12 \left(\dfrac12 - \dfrac1{11}\right)\). You can simplify it by hand or by calculator.

The exact same problem was answered here: https://web2.0calc.com/questions/quadrilateral_18#r1

\(\sqrt[3]{3- x}=-\dfrac92\\ 3 - x = \left(-\dfrac92\right)^3\\ 3 - x = -\dfrac{729}8\\ x = \dfrac{753}8\)