Let a and b be rational numbers and suppose that (a-b)=4 and a*b=1 . Show that (a+b)^2 = 20.

cross-posted from: https://sopuli.xyz/post/22688165

Random thought on magic squares:

If I view the smallest possible non-trivial magic square

2 7 6
9 5 1
4 3 8

since its rows and diagnoals sum up to 2+5+8 = 2+7+6 = 4+5+6 = 2+9+4 = … = 15

Lets view it as a 3x3 Matrix, its determinant is Δ = -360 . Its inverse:

-37/360 19/180 23/360
17/90 1/45 -13/90
-7/360 -11/180 53/360

note how this is a magic square, rows and diagonals sum up to 1/15.

https://matrix.reshish.com/inverse.php

Now if you are really bored (I can not do this): proof that for any non trivial magic squares the inverse …

• exists (i.e. every non-trivial magic square has an inverse)
• is a magic square.

I have ten meters of mesh fence. I want to make two enclosures for my rabbits who can't stand each other (they can share a wall). What is the shape and the area of the largest enclosure (in terms of area) I can build? Each rabbit needs to have access to the same area. The shape can be arbitrary (although it would be nice if it were continuous or smooth to some extent, and each area contiguous).

Examples:

A square of 2x2m, divided by a 2m wall. Area: 4 square meters

A circle of radius 1.21m, divided by a wall. Area: ~4.58 square meters.

Is it possible to do better?

Question

• Find the function f such that f'' = -f
• Show that the function can be written in format f = A sin(x + B)

Extension

Find a function g such that g'' = C * -g and

• g(0) = D
• g'(0) = 0 where C and D are arbitrary constants.

• Contour lines joins up points on a surface that are of the same height.
• Field lines points towards the direction of the steepest descent.

As you may have noticed, whenever a field line and contour line crosses each other, they are perpendicular.

Prove it.

Yeah I can't lie, there is no calling this a daily challenge now.

Anyhow, have a go at proving this, I don't want any unrigorous "imagine zooming in until the line is straight" nonsense.

Difficulty: not a lot

Would appreciate if u put ur proofs or attempts below, I got a proof but it's like kinda mediocre.

Draw a hypocycloid using a graphical calculator (such as Desmos or Geogebra).

Your hypocycloid should include

• Inner circle of radius `a
• Outer circle of radius `b
• As time t increases the point on the inner circle should trace out the pattern, you can animate the graph using t.

Below is the link to a Desmos graph:

https://www.desmos.com/calculator/vzgog7xqrz

• Given n and m are coprime, show that there exist integer n' such that nn' mod m=1.
• The extended Euclid's algorithm is given below without proof, which may be useful in your proof.

(I'm too lazy to type out the algorithm again, so look at the image yourself)

• Prove that z(x mod y) = (zx) mod (zy)

Be rigorous

(trust me bro im gonna daily post trust me bro)

EDIT: assume all variables are integers

I recently started reading TAOCP, in other words you can expect daily posts from me again, because I'll just take some of the cooler questions from there and repost them here.

S=sum of (-1)^n/n from 1 to infty

For why I named the post as so, here's why

spoiler

This is related to the the May 16 post, but takes only the prime indexed terms. Does it still diverge?

• Show that cosθ=(u⋅v)/(|u||v|) for 2D vectors u and v.

(it is quite hard to come up with these challenges, so if you got any ideas, please post them)

We're playing a game. I flip a coin. If it lands on Tails, I flip it again. If it lands on Heads, the game ends.

You win if the game ends on an even turn, and lose otherwise.

Define the following events:

A: You win the game

B: The game goes on for at least 4 turns

C: The game goes on for at least 5 turns

What are P(A), P(B), and P(C)? Are A and B independent? How about A and C?

It is not

Consider the function defined by y = x^(sin(x)^sin(x)). Observe its graph. Find an increasing function which passes through each of its local maximums, and another increasing function which passes through each of its local minimums.

Extra credit: You'll notice the graph isn't drawn for x-values which make sin(x) negative. This is because most of those values make the function undefined - though it is defined for infinitely many points in those intervals, it just also has infinitely many holes. Since it lacks continuity here, it has no true local maxes or local mins, and doesn't impact the original problem. We can nonetheless cheat and fill in the holes by expanding the function to these regions with y = x^|sin(x)|^sin(x) (Using x^-|sin(x)|^sin(x) should also be technically valid, but is being ignored because it's discontinuous with the rest of the graph and not as pretty, but will be mentioned in my solution). Doing so adds more local maxes and local mins. The new local mins should line up with your function that finds the local maxes for the original function - but, find a new function which hits all of the new local maxes.

I've even got a starter question to get you guys into the scenario.

Once you've completed the starter question, under the solution comment attaches the main question, which is unsolved.

• Show that the infinite multiplication (1+1/1)(1+1/2)(1+1/3)... does not converge.

• Show that if a function is differentiable for an interval, it is continuous over that interval.
• A function is continuous if lim_x->a f(x) = f(a)

(x/5)^log_b(5) - (x/6)^log_b(6) = 0

• Express y in terms of x for differential equation dy/dx=ylny

(I'm officially out of ideas again)

• Show that it's possible a^b=c where a and b are irrational, and c is rational.

Sry for the gap I ran out of ideas.

• Show that the sum of the first n squares is n(n+1)(2n+1)/6.
• I know this is often in the textbook for proof by induction, which is why proof by induction is not allowed.

This is a relatively hard one, take your time.