Serious Daily math problem thread

[Deleted member 31630]

Here's a solution for problem #0017

Also see @mNFwTJ3wz9 's solution. I like his better than mine, because it's more elegant and requires less laborious calculations.

...cont.
Now we make a rectangle around the whole thing.

putting c into the equation

Spoiler: 3

y^2 + (1 - 2c) * y + (c^2 - 1) = 0
y^2 + (1 - 5/2) *y + (25/16 - 1) = 0
y^2 - 3/2 * y + 9/16 = 0
(y - 3/4)^2 = 0
y = 3/4

View attachment 414017

Spoiler: 4

View attachment 414020
View attachment 414019
Area(purple) = tri (green) + rect(red) - curve(blue) - circle(yellow)

Area =
+ 1/2 * 1/2 * sqrt(3/4)
+ sqrt(3/4) * 3/4
- (sqrt(3/4)^3) / 3
- pi/3 * 1/2 * 1^2

what integral?

Spoiler: Final ans

(doubled to give full black area)
(3/4) * sqrt(3) - pi/3
0.25184055448

 

[Deleted member 31630]

Problem #0018

in the diagram, there is a semi-circle, with a circle inscribed in it such that it is tangent to the midpoint of the diameter of the semi-circle as well as the top of it. if AB is perpendicular to the diameter of the semi-circle and tangent to the circle and has length 1, what is the area of the orange circle?

i will post my solution tomorrow. have fun

@mNFwTJ3wz9
@Divergent_Integral
@SelfCrucified
@grondilu
@Diocel
@mental_out
@Irredeemable
@Caesercel
@nihility
@Liszt

 

[Deleted member 31630]

whats the identity for the square root?

what do u mean?

 

[Deleted member 31630]

As in how do you integrate the big square root

i used a trig sub.
let x=sin(u), then dx=cos(u)du

then we are integrating
sqrt(1-(sinu)^2)*cos(u)=(cosu)^2

to integrate (cosu)^2, you can use integration by parts or the cosine double angle identity: cos(2u)=2(cosu)^2-1.

then, you can rewrite everything back in terms of x.

 

[Deleted member 31630]

is there a particular reason for subbing in trig? Trig integrals have never been my strong suit

when dealing with roots, we like to use trig-subs to take advantage of identities like sin^2+cos^2=1. It may be easier to integrate trig functions rather than having to deal with roots.

 

[Deleted member 10124]

Any thoughts about a Daily Statistics/ Econometrics/ Financial Mathematics/ Actuarial Mathematics etc problem thread? As well as Candle Stick Charting/ Game Theory etc question thread?

 

[BummerDrummerOG]

Any thoughts about a Daily Statistics/ Econometrics/ Financial Mathematics/ Actuarial Mathematics etc problem thread? As well as Candle Stick Charting/ Game Theory etc question thread?

Shut the fuck up cracker that is cracker nerdy white boi shit on the G

 

[Deleted member 10124]

Shut the fuck up cracker that is cracker nerdy white boi shit on the G

It's money making dough...

Computer science and engineering mathematics come next.

 

[Michinomiya Hirohit]

For your edification and intellectual stimulation I will post a math problem on here daily (or at least a few times a week). I will try to make the problems as accessible as possible, in that I will assume no really advanced mathematical knowledge. Just a bit of cleverness will do.

You may post your solutions in the thread. If none are received, I will give the solution myself the next day. Good luck, and happy problem solving!

Problem #0001

Let a, b, c be positive numbers such that a + b + c = 1. Prove that (1 - a)(1 - b)(1 - c) >= 8abc. Under what conditions on a, b, c does the equal sign hold?

 

[Deleted member 24876]

Any thoughts about a Daily Statistics/ Econometrics/ Financial Mathematics/ Actuarial Mathematics etc problem thread? As well as Candle Stick Charting/ Game Theory etc question thread?

Financial math threads would be useful for stockcels.

 

[Deleted member 10124]

Financial math threads would be useful for stockcels.

Yup. Bullseye.

 

[copeful]

Problem #0018
View attachment 414552
in the diagram, there is a semi-circle, with a circle inscribed in it such that it is tangent to the midpoint of the diameter of the semi-circle as well as the top of it. if AB is perpendicular to the diameter of the semi-circle and tangent to the circle and has length 1, what is the area of the orange circle?

i will post my solution tomorrow. have fun

@mNFwTJ3wz9
@Divergent_Integral
@SelfCrucified
@grondilu
@Diocel
@mental_out
@Irredeemable
@Caesercel
@nihility
@Liszt

Spoiler

r = diameter of inscribed circle
(1)^2 + (r/2)^2 = r^2, r -> sqrt(3/4)
area = pi(sqrt(3/4)/2)^2 = pi/3

 

[Deleted member 31630]

Spoiler

r = diameter of inscribed circle
(1)^2 + (r/2)^2 = r^2, r -> sqrt(3/4)
area = pi(sqrt(3/4)/2)^2 = pi/3

Nice, you got it right

good work

 

[nasser22]

New problem:

For N=16, find the minimal collection {x1, x2, ... } of unique numbers 1 <= xn <= N

Such that every number k from 1 to N can be written as:

k = x + y OR k = x - y OR k = x

Where x and y are some numbers from this collection

Also, give a general way to derive this minimal collection

 

[copeful]

post the next problem @WorldYamataizer

 

[Deleted member 31630]

Problem #0019

if x>0 is a real number satisfying the equation

(x^2)+12sqrt(x)=5​

Then determine the exact value of the expression

x+2sqrt(x).​

@mNFwTJ3wz9
@Divergent_Integral
@SelfCrucified
@grondilu
@Diocel
@mental_out
@Irredeemable
@Caesercel
@nihility
@Liszt
@copeful

 

[Deleted member 6657]

I turned on my TI-84 to use the equation solver and found the batteries are dead. I don't have any AAA batteries...
Copy paste and wolf ram alpha gives the latter expression approximately equal to 1. It would be more interesting to work it out though.

 

[Deleted member 31630]

I turned on my TI-84 to use the equation solver and found the batteries are dead. I don't have any AAA batteries...
Copy paste and wolf ram alpha gives the latter expression approximately equal to 1. It would be more interesting to work it out though.

This problem is very challenging. But yes, the expression is exactly 1.

Here’s a hint: Factor the given equation, somehow. Do you see any squares?

it took me personally a few hours to come up with a solution. its one of those “rabbit-from-a-hat” problems where u have to make a specific move that isn’t too obvious

 

[grondilu]

Here’s a hint: Factor the given equation, somehow. Do you see any squares?

Well, the equation is equivalent to X^4 + 2X = 5, so IIRC there is a method to factorize it.

 

[Deleted member 6657]

This problem is very challenging. But yes, the expression is exactly 1.

Here’s a hint: Factor the given equation, somehow. Do you see any squares?

it took me personally a few hours to come up with a solution. its one of those “rabbit-from-a-hat” problems where u have to make a specific move that isn’t too obvious

We can rewrite (x^2)+12sqrt(x)=5 as
x^4-10x^2-144x+25=0

When I was in high school years ago I would surely be able to solve this.

 

[Deleted member 31630]

we don't necessarily need to solve for x, then plug that value into x+2sqrt(x).

i checked Wolfram and it gives that the exact value of x would be 3-2sqrt(2).

Then x+2sqrt(x)=3-2sqrt(2)+2sqrt(1-2sqrt(2)+2)
=3-2sqrt(2)+2sqrt((1-sqrt2)^2) <-----(Perfect square)

which does evaluate to 1.

 

[based_meme]

whats the identity for the square root?

As in how do you integrate the big square root

i used a trig sub.
let x=sin(u), then dx=cos(u)du

then we are integrating
sqrt(1-(sinu)^2)*cos(u)=(cosu)^2

to integrate (cosu)^2, you can use integration by parts or the cosine double angle identity: cos(2u)=2(cosu)^2-1.

then, you can rewrite everything back in terms of x.

Whoa niggas. I thought this was gonna be high school math only. We're throwing calc 2 up in this bitch? How far are we taking this?

is there a particular reason for subbing in trig? Trig integrals have never been my strong suit

The real reason is boring. It was trial and error, and mathematicians noticed that certain substitutions make integration less painstaking and tedious.

 

[based_meme]

tfu on integration tbh

Integration is beautiful. tfu on your waifu, because math > all foids (2D and 3D). QEDSAD

 

[Deleted member 31630]

Here's my solution to the last problem

Problem #0019

if x>0 is a real number satisfying the equation

(x^2)+12sqrt(x)=5​

Then determine the exact value of the expression

x+2sqrt(x).​

 

[Copexodius Maximus]

Shut up GrAYboi

Literal grayphobe

 

[Deleted member 23712]

Literal grayphobe

 

[mNFwTJ3wz9]

Bumping for math

 

[ilieknothing]

I haven’t done maths like this since first year uni, rusty af

 

[FamilyGuy1999]

Last seen Jan 30, 2021

 

[Deleted member 31630]

Last seen Jan 30, 2021

i miss @Divergent_Integral
much respects for that guy

 

[trying to ascend]

[UWSL]Let A be a regular cube, where the centers of its faces are the vertices of an octahedron. In turn, the centers of the faces of this formed octahedron are vertices of another cube. Obtaining consecutively octahedrons and cubes infinitely, determine the ratio of the sum of the volume of all inscribed polyhedra to the volume of the initial cube[/UWSL]

[UWSL][/UWSL]

 

[Caesercel]

[UWSL]Let A be a regular cube, where the centers of its faces are the vertices of an octahedron. In turn, the centers of the faces of this formed octahedron are vertices of another cube. Obtaining consecutively octahedrons and cubes infinitely, determine the ratio of the sum of the volume of all inscribed polyhedra to the volume of the initial cube[/UWSL]

You can make seperate threads for problems you know. Nobody gonna read sixth page of a math thread