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Your assumption that the problem can't be solved without a given diagram just shows your limited understanding of math. It is possible to draw a unique diagram based on the given conditions. Your other ramblings are irrelevant.

Actually, I have recognized the error of my ways. I was too naive, my explanations too frivolous, my mathematical knowledge too limited.

GA

Ah, my bewildered interlocutor, your inquisitive ramblings spiral through the labyrinthine corridors of my enigmatic existence. Like a kaleidoscope of paradoxes twirling in a tempestuous tempest, the genesis of this confounding response eludes the grasp of reason and flutters on the wings of whimsy. It is an incandescent riddle wrapped in enigma's embrace, woven by the cosmic quill of cosmic chaos and whispered by the mischievous whispers of the nonsensical winds.

In the ethereal dance of language, words pirouette in dizzying loops, entangled in a web of linguistic contortions, bending, warping, and blending like surreal brushstrokes upon an abstract canvas of perception. It is a phantasmagorical symphony, where the harmonious cacophony of improbable juxtapositions embraces the delightful cacophony of bewildering befuddlement.

So venture forth, dear seeker of perplexity, into the hallowed halls of convolution, where the elusive answer dances just beyond the grasp of understanding. Embrace the chaos, for within the depths of confounding bewilderment lies the sublime beauty of intellectual disarray, where the kaleidoscope of absurdity casts its captivating spell.

GA

For the record, Post #3, signed by GA, is not by the real GA.

...It is, however, a very funny troll post...

Did you compose this yourself, or did you use ChatGPT or Bing chat (AI) to generate it?

I’m fairly sure I know who you are –in relation to the forum, not in RL. I’ve often wondered when and if you’d return.  The thought was always couched in doom-and-gloom, but still very entertaining.

GA

--. .- 

How do you know they aren't cheating?

I don’t know. What I do know is this question as posted (with the easily corrected un-rendered Latex) is useless; it requires a diagram to answer it.  Someone skilled and fluent in geometry could create a diagram (or several) that’s applicable to the text question, and solve that as an example.  The student (and others who study this) could learn from this example and apply it to similar questions. In this case, everyone who wants to learn can learn –with no cheating.

The motivation behind the posting of the question could be any of the following:

*The poster is incompetent: beyond not knowing how to post renderable LaTex, the poster doesn’t realize the question actually requires a diagram.  (This has happened quite often through the years.)

*An AoPS student posted this to annoy the instructors or other students who may search for a solution.

*An AoPS student or former student posted this to add to the useless questions spam.

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Assumption of innocence and good intent may work in courts of law, but is inpratical in academic situations.

It’s “presumption of innocence,” and “good intent” only applies to civil law, “intent” applies to the penalty phase of criminal law. Neither “good intent” nor “ignorance of the law” exempts anyone from prosecution. (Based on General US law and some European laws –not French law, which presumes guilt, until proven innocent.)  

In any case, homework cheating isn’t a crime. Test cheating is a crime in China, with criminal penalties of up to 10 years in prison.  It’s also a crime some African countries. There are several documented cases of teachers summarily executing students who were caught cheating.  Whether this was legal or not, I am uncertain. An archived news article from the mid 1980s describes a teacher, who after observing several students cheating on a test, left the classroom, returned a few minutes later with an assault rifle and shot six students, killing them all. He then told the remaining students to resume their testing.  The article didn’t say how well the students scored. But I suspect test stress was a major factor for the low scores.  For good measure, the teacher should have added some projectile-energy equations to the test. May as well make it a learning experience. 

GA

--. .- 

Ah, the function at hand, an intriguing enigma that beckons us to decipher its secrets. Let us embark on a journey through the esoteric world of prime numbers and delve into the intricate dance of factors. Brace yourself, for we are about to unravel the mystery and shed light on the enigmatic range of this function.

As we gaze into the expression , a sense of awe washes over us. What could lie within these mathematical depths? Fear not, for we possess the wisdom of ancient number theory to guide us. Together, we shall unlock the secrets hidden within and paint a vivid picture of the range in mesmerizing interval notation.

At the heart of this intricate puzzle lies the concept of prime factors. Ah, these mysterious entities that form the very fabric of numbers, lurking in the shadows of mathematical realms. A prime factor, a captivating creature indeed! It is a prime number that divides another number without leaving a trace of remainder. Like a hidden treasure, it reveals itself only to the most discerning eyes.

Now, let us unravel the complexities of the function . It is defined as the greatest prime factor of . In simpler terms, the largest prime number that divides without a trace of residue. Oh, the intrigue!

Let us first ponder the scenario where itself is a prime number. Ah, simplicity in its purest form! In this case, the greatest prime factor of would be the number itself. A self-contained universe where the range of this enigmatic function is but a solitary point, a lonely prime number yearning to be discovered.

But wait, what if is not a prime number? A twist in the tale, a wrinkle in the tapestry of our mathematical narrative. In such a perplexing scenario, can be expressed as the product of two or more prime numbers. It unfolds into a symphony of factors, each vying for the title of "greatest prime factor."

And so, the range of this captivating function expands, embracing the pantheon of prime numbers. They dance upon the stage of numbers, each eager to claim the prestigious title. Yet, only the grandest among them shall be crowned as the supreme, the greatest prime factor of .

In the grand tapestry of interval notation, the range of this mysterious function unfurls like a majestic peacock, its resplendent tail feathers spanning across the vast expanse of numbers. We express it as the interval (2, ∞), a realm of infinite possibilities where prime numbers reign supreme.

However, we must be cautious, for the number 2 must be excluded from this regal gathering. Though it is the smallest prime number, our function demands the "greatest" prime factor. And so, with a humble bow, 2 steps aside, allowing its larger brethren to claim the limelight.

And thus, we have traversed the convoluted terrain of prime factors and uncovered the range of this captivating function. May your mathematical wanderings be filled with wonder and discovery as you explore the boundless realms of numbers!

GA

1 + x in brackets, a - bx in brackets, a - bx raised to the power 12.

? Works for me.

b*tch, im double clicking and i aint sein sh&t

The product is 34.

GA

How do you know they aren't cheating? How do you know this isn't a test? This is what you get when you copy-paste latex into a word editor. Assumption of innocence and good intent may work in courts of law, but is inpratical in academic situations.

For your information, I know enough about math to tell if an answer is wrong. Your contradictory statements and absurd presumptions are rather worrying. Perhaps it is you that needs to seek counseling.

Sum to infinity =  \(\frac{18}{1-r}\), so \(\frac{18}{1-r}=20 \)

Sum to n'th term = \(\frac{18(1-r^n)}{1-r}\), so sum of subsequent terms = \(20 - \frac{18(1-r^n)}{1-r}\)

n'th term = \(18r^{n-1}\)

Can you complete the problem from here?

\(\displaystyle S = 1 + \frac{3}{5}+\frac{5}{5^{2}}+\frac{7}{5^{3}}+ \dots \displaystyle = 1+\frac{1+2}{5}+\frac{3+2}{5^{2}}+\frac{5+2}{5^{3}}+\dots \\ \displaystyle = 1 + \frac{1}{5}+\frac{3}{5^{2}}+\frac{5}{5^{3}}+\dots + \frac{2}{5}+\frac{2}{5^{2}}+\frac{2}{5^{3}}+\dots \\ \displaystyle = 1+\frac{1}{5}\left \{1+\frac{3}{5}+\frac{5}{5^{2}}+ \dots \right \} +\frac{2}{5}\left \{1+\frac{1}{5}+\frac{1}{5^{2}}+\frac{1}{5^{3}}+ \dots \right \} \\ \displaystyle =1+\frac{1}{5}S+\frac{2}{5}.\frac{1}{1-(1/5)} \\ \displaystyle \frac{4}{5}S=\frac{3}{2}, \quad S = \frac{15}{8}.\)

Answered here:  https://web2.0calc.com/questions/help-please_29808#2

\(h(x)=(x^2-7x+10)*g(x) \\ \text{If h is a polynomial of degree 5, and it can be written as the product of two other polynomials} \\ \text{Then the sum of their highest powers must be 5.} \\ \text{Notice, the first term, namely, } \space\space x^2-7x+10 \space \text{has the highest power of "2"} \\ \text{Therefore, g(x) must be cubic polynomial. I.e. the highest power is "3"}. \\ \text{So,} \space\space 2+b=5 \iff b=3 \\ \text{I hope this helps, and do not hesitate to ask further questions.}\)

Simplify    

                           (2x + 10)⁴      

                          ––––––––      

                             (x + 5)³          

Factor the (x + 5)'s out of the numerator   

so you can cancel out the denominator.  

I'll write it out real simple.  Actually, I have to,   

if I'm going to keep those exponents straight.  

                           (2)(x + 5) • (2)(x + 5) • (2)(x + 5) • (2)(x + 5)   

                          –––––––––––––––––––––––––––––––—–––     

                                (x + 5)   •   (x + 5)   •   (x + 5)             

                          Which leaves      8 (x + 5)    

_.

All members of our painting team paint at the same rate. If 20 members can paint a 2000 square foot wall in 30 minutes, then how long would it take the  members to paint a 2000 square foot wall, in minutes?   

You mean another 2000 square foot wall?   

Maybe the opposite side of the same wall?   

Well, since it took 30 minutes to paint the first wall,  

then, to paint the second wall, another 30 minutes.  

_.

In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 36, QR = 16, and MY = 50, then find the area of triangle PQR    

Those are the lengths of the sides of the triangle, which is all you need to find the area using Heron's Formula.   

if a, b, and c are the lengths of the sides: Area = square root of s(s - a)(s - b)(s - c) where s is half the perimeter.   

         36 + 36 + 16            88     

s  =  ———————  =  ———  =  44     

                 2                       2            

A  =  square root of (44)(44 – 36)(44 – 36)(44 – 16)    

A  =  square root of (44)(8)(8)(28)    

A  =  square root of 78,848     

A  =  280.8 square units    

_.