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Y by math90, NguyenVanHoa29, Zuyong, TNKT
Problem. Let 
Prove that
Prove that
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Lower and Upper Bounds of x
Darealzolt 3
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by HAL9000sk
Let 
fulfill
![\[
x=\frac{1}{\frac{1}{770}+\frac{1}{771}+\frac{1}{772}+\dots+\frac{1}{799}+\frac{1}{800}}
\]](//latex.artofproblemsolving.com/c/d/d/cdd8a054709169890c55672cc6300eb1f5719bb2.png)
Hence find the sum of the greatest integer value that is less than and the smallest integer value that is greater than .
fulfill![\[
x=\frac{1}{\frac{1}{770}+\frac{1}{771}+\frac{1}{772}+\dots+\frac{1}{799}+\frac{1}{800}}
\]](http://latex.artofproblemsolving.com/c/d/d/cdd8a054709169890c55672cc6300eb1f5719bb2.png)
Hence find the sum of the greatest integer value that is less than and the smallest integer value that is greater than .
3 replies
Math Relay
evt917 8
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by elizhang101412
if it's a marathon thread i don't want to know lol
Alright, this is the math relay!
I will start with a simple math question. Let the answer to that problem be
. Then, you make a problem using .
Try to solve the above problem without looking at the hide tag.
First problem: Find the area of a square with diagonal
Alright, this is the math relay!
I will start with a simple math question. Let the answer to that problem be
. Then, you make a problem using .Try to solve the above problem without looking at the hide tag.
First problem: Find the area of a square with diagonal
8 replies
polynomial division vs simplify
Miranda2829 9
N
by aops-g5-gethsemanea2
the question is
4x² - 4x +1 divide by 2x + 1 write in fraction format
the answer is 2x-3 + 4/2x+1
so my question is can we simiplfy this -4x with 2x become -2x
then answer 4x² -2x+1?
Im confused , when do usual fraction division we can simplify, but in above question doesn't seem to work.
many thanks
4x² - 4x +1 divide by 2x + 1 write in fraction format
the answer is 2x-3 + 4/2x+1
so my question is can we simiplfy this -4x with 2x become -2x
then answer 4x² -2x+1?
Im confused , when do usual fraction division we can simplify, but in above question doesn't seem to work.
many thanks
9 replies
I found this
TennisChampion 13
N
by AR17296174
A ball and a bat cost $1.10
the bat costs one more dollar than the ball
how much does the ball cost?
I'll post the answer when someone gets it
the bat costs one more dollar than the ball
how much does the ball cost?
I'll post the answer when someone gets it
13 replies
Worst math problems
LXC007 63
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by Andrew2019
What is the most egregiously bad problem or solution you have encountered in school?
63 replies
Personal Modulo Problem (Title: Encrypted Code)
PikaVee 2
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by Yihangzh
3 Pokemon each have they own number to use on a part of a lock. Star the Pikachu has the number 87, Luna the Eevee has the number 92, and Aero the Mew has the number 79. A part of their code is found in the expression 
.
To double check if it is right then it needs to follow
and 
. If it does follow it then you need to find the sum of the coefficients with the code the code itself, but if not replace the sum of the coefficients with the lowest possible sum of coefficients where 
in 
where X, Y, and Z are positive integers more than 0.
.To double check if it is right then it needs to follow
and 
. If it does follow it then you need to find the sum of the coefficients with the code the code itself, but if not replace the sum of the coefficients with the lowest possible sum of coefficients where 
in 
where X, Y, and Z are positive integers more than 0.2 replies
An annoying math problem
Wolfpierce 9
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by Yihangzh
Okay so what is 2/((√3+1)((3 to the 1/4)+1)((3 to the 1/8)+1)((3 to the 1/16)+1)) to the power of 32?
9 replies
3D Geometry problem
ReticulatedPython 2
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by Yihangzh
Three cones of base radius 
and slant height 
rest on the same plane.and have their bases tangent to each other. A sphere is tangent to the plane and to the three cones. Find the radius of this sphere, and give your answer in simplest radical form.
Source: Own
and slant height 
rest on the same plane.and have their bases tangent to each other. A sphere is tangent to the plane and to the three cones. Find the radius of this sphere, and give your answer in simplest radical form.Source: Own
2 replies
9 Prodigy AoPS or Khanacadamy
ZMB038 109
N
by skyleaf
Hey everyone just was wondering what everybody prefers? Try not to fight so this doesn't get locked!
109 replies
450th post!
JohannIsBach 10
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by ShrewdBunny
this is my 450th post! the last time i posted one of these was when i had 100! wow iv gon so far... what should my next goal be?
10 replies
Non-negative real variables inequality
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Source: own
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#2
Apr 25, 2025, 3:31 AM
Y by
Are there any easier methods?
After multiplying both sides by
, we need to show that
![\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]](//latex.artofproblemsolving.com/2/4/3/243941650f929368784625dffef0acd7bc2c2e2a.png)
where 
is a fifth degree polynomial:
![\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]](//latex.artofproblemsolving.com/a/b/f/abf1e0242ef9a8fa9a5338cf5ab57065c22e79b0.png)
By the 
method, it is not hard to show that 
is true.
After multiplying both sides by
, we need to show that![\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]](http://latex.artofproblemsolving.com/2/4/3/243941650f929368784625dffef0acd7bc2c2e2a.png)
where 
is a fifth degree polynomial:![\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]](http://latex.artofproblemsolving.com/a/b/f/abf1e0242ef9a8fa9a5338cf5ab57065c22e79b0.png)
By the 
method, it is not hard to show that 
is true.
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Y by arqady
I think it is a concave function according to w^3 and the rest is easy checking.
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