# 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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### Prove that $f(A)\leq max(f(P),f(Q),f(R))$

Consider any $\bigtriangleup PQR$ in the $x-y$ plane. Let $f(x,y)=ax+by+c$ , where $a,b,c\in\mathbb{R}$. Let $A\in\mathbb{R^2}$ be any point in the interior or on the $\bigtriangleup PQR$. Prove that ...
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### $f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
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### Number of digits of the number of digits of the number of digits of $2014^{2014}$

How would you solve that problem : What is the number of digits of the number of digits of the number of digits of $2014^{2014}$ ? (for instance the number of digits of $12345678901234567890$ is ...
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### Systems of equations with three variables

Consider: $a + b = 5$ $2a + b + c = 4$ $a - b - c = 5$ I like to use substituiton for solving systems of equations, so I firstly look at equation 1 and solve for $a$ $a + b = 5$ $a = 5 - b$ I ...
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### How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$?

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$? I thought about it but I could not find a proof. Please give me some hints?
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### Having trouble with fractions

If I want to solve the following equation for $a$: $$\frac{3a}{12} - \frac{4a + 8} { 12} = 9$$ What should my first steps be? If I should cross multiply the two fractions on the LHS, but, if I do ...
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### Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
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### Given an ellipse's center, focus and point, find its equation.

Given an ellipse's center is $(2,1)$, focus is (2,4) and point is (3,-3), we have Plug in center: $\frac{(x-2)^2}{a^2}+\frac{(y-1)^2}{b^2} = 1$ Use focus: $4^2=a^2-b^2$ $16=a^2-b^2$ Use point: ...
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### Given equation of parabola, find vertex and directrix

Given that $x^2-bx+17-ay=0$ has vertex $(3,2)$, find the directrix and focus. My attempt is to make it into the form $(x-h)^2=4a(y-k)$ which has focus $(h,k+a)$ and directrix $y=k-a$. Is this right? ...
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### Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...