I’m interested why this is true: $$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$ The following holds: $$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$ This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be […]

Translating a graph in the $xy$ plane given by, $$f(x,y)=c$$ With $c \in \mathbb{R}$, $k$ units right and $h$ units up one gets, $$f(x-k,y-h)=c$$ The best explanation as to why this is the one I got in Algebra, to see why this is we can consider, the origin. Suppose $f(0,0)=c$. Translating the origin $k$ units […]

On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the minute hand had been when he started, and the minute hand was exactly where the hour hand had been where the […]

While trying to solve this problem, I stumbled upon the following equality $$ \sqrt{\sqrt{2x}+\sqrt{x+k}}+\sqrt{\sqrt{2x}+\sqrt{x-k}}=(\sqrt2+1) \left( \sqrt{\sqrt{2x}-\sqrt{x+k}}+\sqrt{\sqrt{2x}-\sqrt{x-k}} \right) $$ It seems to hold for any $x$ and $k$ in $\mathbb R$ such that all the square roots are well defined. If it can be proven, the proof of the above mentioned problem is immediate. However, I […]

Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?

The proofs I have come across on showing that $\cos \theta$ is even is something like this: In a unit circle, $\cos\theta$ gives you the $x$ coordinate after traveling $\theta$ radians counterclockwise. Since, moving $\theta$ radians counterclockwise and $\theta$ radians clockwise i.e $-\theta$ will give you the same x coordinate, we have: $\cos(\theta)=\cos (-\theta)$ It […]

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+…..+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using the product-to-sum formula for $\cos(x)+\cos(y)$?

i cant see why we have : $$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$$ $$\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$$ Even i see the notion of Double factorial this question is related to that one : Behaviour of the sequence $u_n = \frac{\sqrt{n}}{4^n}\binom{2n}{n}$ For $\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$ $\dfrac{(2n)!}{4^n n!^2}=\dfrac{(2n)!}{2^{2n} n!^2}=\dfrac{(2n)\times (2n-1)!}{2^{2n} (n\times (n-2)!)^2}$ For $\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$ note that $n!! = \prod_{i=0}^k […]

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I get that: $$\sum_{i=1}^{2^{n+1}} \frac{1}{i} = (\sum_{i=1}^{2^n} \frac{1}{i} ) +(\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} )$$ using the inductive hypothesis. […]

For example the equation $$ n2^n = 8 $$ Is true for $n=2$ which can be guessed, but more complicated examples would require some sort of approach. Also with trigonometric functions, $$ x\sin(x) + B\cos(x) = A $$ I read that solutions of these kinds of equations can not be expressed in closed form, why […]

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