# Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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### Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
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### Struggling with BIDMAS.

This question came up an I'm not sure about it: You need to simplify leaving the answer in standard form: $\dfrac{2((3-3^2)^2)}{3+\sqrt{4^2-7}}$ I struggle to work it out myself. When I used a ...
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### Solve equation $\frac{1}{x}+\frac{1}{y}=\frac{2}{101}$ in naturals

My try was $$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=\frac{2}{101}\\x+y=2k,xy=101k\\x=2k-y\\y(2k-y)=101k\\2ky-y^2=101k\\y^2-2ky+101k=0\\y=k+\sqrt{k^2-101k}\\x=k-\sqrt{k^2-101k}$$ Now $\sqrt{k^2-101k}$ ...
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### Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
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### Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
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### What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
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### Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...
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### Proving that $\cos(\frac{\arctan(\frac{11}{2})}{3}) = \frac{2}{\sqrt{5}}$

I am trying to solve the cubic equation $x^3-15x-4=0$ using Cardano's formula. I already know that the solutions are $x=4$, $x= \sqrt{3}-2$ and $x= -\sqrt{3}-2$ and that using the formula in this ...
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### Show that $x^3y^3(x^3+y^3) \leq 2$

If $x,y$ are positive reals such that $x+y = 2$ show that $x^3y^3(x^3+y^3) \leq 2$. I thought about working backwards on this one. We can try to prove that $x^3y^3(x^3+y^3) \leq 2$ is true by ...
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### Distance between a point $X$ and a line $2x-y = 1$

I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a ...
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### Solve the equation $7t+[2t] =52$ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+\left\lfloor 2t\right\rfloor =52$. My effort Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional ...
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### Comparing the asymptotic growth of certain functions.

Consider some nonconstant functions $f(x)$ and $g(x)$, and suppose $\lim_ {x \to a} f(x) > \lim_{x \to a} g(x)$ for some nonzero $a$. Can we somehow conclude that $f(x) - g(x)$ is of constant sign ...
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### Expanding logarithm of function

Is there a way (there has to be), I can expand an expression like this? $$\log_2 (3f(n)^n)$$ P.S. This part of an assignment I'm working on, please do not give solutions
### Find $a,b,c \in \{1,2,..,9\}$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}$ [on hold]
Find $a,b,c \in \{1,2,..,9\}$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}.$$ It seems to be easy but I want a smart solution.