## Search found 20 matches

Fri Apr 16, 2021 6:16 pm
Forum: Geometry
Topic: Prove triangle is isosceles
Replies: 1
Views: 13356

### Re: Prove triangle is isosceles

The incenter has three perpendiculars of equal length to each of the sides, so if the sum equals half the total area, AB + FC equals also half the perimeter of the triangle. The angle bisector theorem says AB/BC = AF/FC, meaning AB/BC = AF/(AF+BC-AB). Letting AB = X, BC = Y and AF = Z: X/Y = Z/(Y+Z-...
Fri Apr 16, 2021 12:56 pm
Forum: Geometry
Topic: Four color theorem - why is this considered wrong?
Replies: 0
Views: 531

### Four color theorem - why is this considered wrong?

I'd like to think I found a proof for the four color theorem, but I also know that it took far smarter people than me a computer simulation to prove. Still, I don't see why this logic should be flawed. If you'd explain to me plainly, I'd love it: 1- The best case scenario for two differently-colored...
Thu Apr 16, 2015 5:30 pm
Forum: Recreational
Topic: When is Cheryl's birthday?
Replies: 3
Views: 3836

### Re: When is Cheryl's birthday?

Others know it too, but it's still better to spoiler it: 1. Since Albert is sure that neither of them knows, he must have realized Bernard must have been given another day than 18 or 19, which means Cheryl first told Albert that her birthday is in a month that rules out these numbers (either July or...
Tue Feb 17, 2015 9:48 am
Forum: Number
Topic: A Real Number Puzzle
Replies: 9
Views: 13275

### Re: A Real Number Puzzle

For any existing lower bound, an even lower one can be found and it should converge to 0, which gives a contradiction. Don't see what's wrong with this kind of logic.
Sat Feb 14, 2015 3:37 pm
Forum: Number
Topic: A Real Number Puzzle
Replies: 9
Views: 13275

### Re: A Real Number Puzzle

ggoyo wrote:What if the lower bound of $\{ |x| : f(x)-x \neq 0\}$ is 0 ?
Why? It can't be 0, because f(0)=0.
Mon Jan 26, 2015 12:30 pm
Forum: Number
Topic: A Real Number Puzzle
Replies: 9
Views: 13275

### Re: A Real Number Puzzle

(i) f(-x) = - f(x) (ii) f(x+1) = f(x) + 1 (iii) f(1/x) = f(x)/(x^2) for x not equal to 0. Old topic, but that's an interesting puzzle. - According to I, f(0) = - f(0), so f(0) must be 0. - Based on I and II, f(x) definitely equals x if x is an integer. Letting x be any real number and n an integer ...
Sun Dec 07, 2014 6:47 pm
Forum: Number
Topic: A number of digits puzzle
Replies: 5
Views: 5426

### Re: A number of digits puzzle

Find all possible solutions to the system of equations: (1) n(p) + n(q) = p; (2) p+q+ n(r) = r; (3) n(p) + n(q) + n(r) = q - 4; where, each of p, q and r are positive integers and n(p), n(q) and n(r) respectively denote the number of digits of the integers p, q and r. n(p) + n(q) + q + n(r) = r; n(...
Sun Dec 07, 2014 3:45 pm
Forum: Number
Topic: help with a proof of a curious GCD formula
Replies: 2
Views: 3628

### Re: help with a proof of a curious GCD formula

Let $m$ be a positive integer, and $ef\equiv1\pmod{m}$. Then for any integers $n_1$ and $n_2$, $\gcd(mn_1,n_2+n_1f)=\gcd(mn_2,n_1+n_2e)$ First of all, $e$ and $f$ must be co-primes with $m$. In addition, it can be argued that $n_2$ ≡ -$n_1f$ (mod first GCD). Therefore, it can't NOT be that $n_1$ ≡ ...
Thu Aug 28, 2014 6:26 pm
Forum: Recreational
Topic: A problem for every age
Replies: 20
Views: 12433

### Re: A problem for every age

Make the most "obvious" connections first, then go on with more and more difficult ones.
Mon Jul 22, 2013 10:39 pm
Forum: Resources
Topic: Paper/Pencil Problems
Replies: 71
Views: 80012

### Re: Paper/Pencil Problems

#336 seems to be one.
Tue Jul 16, 2013 12:12 pm
Forum: Number
Topic: Proof that 1 + 1/2 + 1/3 + ... is infinity.
Replies: 2
Views: 4107

### Re: Proof that 1 + 1/2 + 1/3 + ... is infinity.

It may have been avoided for a time while the concepts of absolute convergence were being worked out, and that therefore only proofs without rearrangements were considered any good. This could be why the classic proof is preferred over this one, but that is just speculation. Good point. Thanks for ...
Mon Jul 15, 2013 8:00 pm
Forum: News, Suggestions, and FAQ
Topic: Errors/Warnings/Bugs
Replies: 645
Views: 225904

### Re: Errors/Warnings/Bugs

I had to enter the same answer for #175 several times before the site could register it as correct.
Mon Jul 15, 2013 2:53 pm
Forum: Number
Topic: Proof that 1 + 1/2 + 1/3 + ... is infinity.
Replies: 2
Views: 4107

### Proof that 1 + 1/2 + 1/3 + ... is infinity.

Few years ago, a mathematician I visited said this proof had never been found before: Let 1 + 1/2 + 1/3 + ... = x. 2 + 2/2 + 2/3 + ... = 2x 2 + 2/3 + ... + 2/(2n-1) + ... + 2/2 + 2/4 + ... + 2/2n + ... = 2x Note that the terms with even denominators can be simplified into 1/n, n being any positive i...
Sun Jul 14, 2013 3:57 pm
Forum: Recreational
Topic: A farmer can take himself, and one of these objects...
Replies: 5
Views: 4766

### Re: A farmer can take himself, and one of these objects...

1. The first move must be taking the chicken with you, because it either eats or is eaten after that otherwise. 2. Return to the starting place alone. 3. There is a symmetry between the fox and the grain in that they both cause such situations together with only the chicken, so it doesn't matter if...
Fri Jul 12, 2013 1:59 pm
Forum: Clarifications on Project Euler Problems
Topic: Problem 033
Replies: 30
Views: 14330

### Re: Problem 033

By reading the problem and the posts in this topic I really have to say that there is a little too much ambiguity in the problem. I recommend editing it and make a better explanation about what you consider trivial/non-trivial examples. That part can be really confusing, I had to come here (on the ...
Sat Jul 06, 2013 12:17 am
Forum: Resources
Topic: Paper/Pencil Problems
Replies: 71
Views: 80012

### Re: Paper/Pencil Problems

Problem 169 (View Problem) with the right formula.
Wed Jul 03, 2013 7:25 am
Forum: Recreational
Topic: Which PE problems were the most satisfying for you to solve?
Replies: 13
Views: 11038

### Re: Which PE problems were the most satisfying for you to so

358. I like problems that become much easier when you figure out what to do, including proving a (minor) theorem.
Tue Apr 30, 2013 2:52 pm
Forum: Recreational
Topic: Grue & Bleen
Replies: 0
Views: 2725

### Grue & Bleen

Let's call anything blue "grue" before the year 2000, and "bleen" from then on. The opposite order applies to green objects. Then why don't grue/bleen objects ‘change’ their color after 1999? What do you think? My two cents: Before 2000 (r) After 2000 (r’) Blue (p) Grue (s) Bleen...
Wed Oct 12, 2011 4:54 pm
Forum: Resources
Topic: Paper/Pencil Problems
Replies: 71
Views: 80012

### Re: Paper/Pencil Problems

I'd say Problem 33 (View Problem) is a fine example too.
Sun Sep 25, 2011 4:00 pm
Forum: News, Suggestions, and FAQ