Questions tagged [alternative-proof]
If you already have a proof for some result but want to ask for a different proof (using different methods).
3,855 questions
- Bountied 0
- Unanswered
- Frequent
- Score
- Trending
- Week
- Month
- Unanswered (my tags)
1
vote
1
answer
88
views
Can an isosceles triangle with a 60ドル^\circ$ angle be proven equilateral independently of the triangle angle sum theorem?
There is a famous theorem in elementary geometry:
Theorem. An isosceles triangle with a 60ドル^\circ$ angle is equilateral.
Two cases of this theorem are depicted below. I consider any (or both) of ...
3
votes
1
answer
184
views
Real function such that $|f'(x)|\le|f(x)|$ has fixed sign
Let a differentiable real function $f$ be such that $\left|f'(x)\right|\le|f(x)|$. Prove that $f(x)$ has the same sign for all $x$. (Let 0ドル$ and any real number have the same sign)
If $f$ has no ...
2
votes
1
answer
167
views
Prove that for all real $a,b,c$ holds $a^2 + b^2 + c^2 + 2 + (abc)^2 \ge 2(ab + bc + ca).$
I recently came across this nice inequality, which looks simple but elegant.
Here’s my short proof — and I’d love to see alternative approaches, preferably using only classical inequalities (Cauchy–...
2
votes
1
answer
95
views
Can Sumner’s theorem also be proved using the Tutte's 1-factor theorem?
The classical result, known as Sumner’s theorem, shows that every connected claw-free graph of even order has a perfect matching(i.e., 1-factor). One of the proofs can be found in Sumner’s original ...
2
votes
1
answer
110
views
Proof of the wheel stud problem
Consider a regular $n$-sided polygon in two-dimensional space (metaphorically: the stud pattern on a typical car wheel), and a sequence to choose each of its sides exactly once. If the choice sequence ...
1
vote
2
answers
132
views
A simpler upper bound of $\sum _{ \quad k \le a_n \\ \gcd(k,6)=1} \frac{1}{k}$?
Let $a_n=4n+1-2 \displaystyle \left \lfloor \frac{n}{2} \right \rfloor$ , for $n \in \mathbb{N}$
i.e : integers $\ge1$ that are odd and not divisble by 3ドル \quad (\star)$
$a_0=4\times 0+1-2\...
4
votes
6
answers
220
views
Proving $ \angle MAN = 45^\circ$ in an isosceles right triangle
Regional Mathematical Olympiad 2003 (India)
Let $ABC$ be a triangle in which $AB =AC$ and $\angle CAB = 90^{\circ}$. Suppose that $M$ and $N$ are points on the hypotenuse $BC$ such that $BM^2 + CN^2 = ...
2
votes
2
answers
102
views
In a trapezium $ABCD,ドル by trigonometric relations find the value of $\angle CBA$.
Let $ABCD$ be a trapezium with sides $AD$ and $CB$ parallel. If $\angle BCA = \angle DCA$ and $XA=DC, DX=CB$ where $DB\cap CA = X$.
Find the value of $\angle CBA$
Place triangle $BCD$ and apply the ...