if-a-k-gt-0-k-1-5-then-prove-that-exists-i-j-1-5-such-that-0-a-j-a-i-1-a-i-a-j-2-1-

Question Number 162995 by HongKing last updated on 02/Jan/22

if a_{k} > 0; k = \overset{―}{1, 5}

then prove that exists i, j \in \overset{―}{1, 5} such that :

0 ⩽ \frac{a_{j} - a_{i}}{1 + a_{i} a_{j}} ⩽ \sqrt{2} - 1

Answered by mindispower last updated on 03/Jan/22

a_{k} = t g (β_{k}), β_{k} \in] 0, \frac{π}{2} [, x \to t g (x) b i j e c t i o n [0, \frac{π}{2} [\to [0, \infty [

α_{k} - a_{i} = \frac{t g (β_{k}) - t g (β_{i})}{1 + t g (β_{k}) t g (β_{i})} = t g (β_{k} - β_{i})

\Leftrightarrow t o s h o w \exists i, j s u c h

0 ⩽ t g (β_{j} - β_{i}) ⩽ \sqrt{2} - 1

[0, \frac{π}{2} [] = \underset{k = 0}{\overset{3}{\cup}} [\frac{k π}{8}, \frac{k + 1}{2} π [\dots (E)

i f \exists i, j s u c h β_{i} = β_{j} t r u e t g (β_{i} - β_{j}) = 0 \in [0, \sqrt{2} - 1 [

You can't use 'macro parameter character #' in math mode

\Rightarrow \exists (i, j), \exists k \in [0,] s u c h T h a t β_{i}, β_{j} \in [\frac{k π}{8}, \frac{k + 1}{8} π [

5 n u m b e r w i t h e 4 i n t e r v a l^{″}

0 ⩽∣ β_{i} - β_{j} ∣⩽ \frac{π}{8}

0 ⩽ t g ∣ (β_{j} - β_{i}) ∣⩽ t g (\frac{π}{8})

t g (\frac{π}{8}) = \frac{2 {s i n}^{2} (\frac{π}{8})}{s i n (2 . \frac{π}{8})} = \frac{1 - c o s (\frac{π}{4})}{s i n (\frac{π}{4})} = \sqrt{2} - 1

\Rightarrow \exists (i, j) s u c h t h a t 0 ⩽ \frac{t g (β_{i}) - t g (β_{j})}{1 + t g (β_{i}) t g (β_{j})} = \frac{a_{i} - a_{j}}{1 + a_{i} a_{j}} ⩽ \sqrt{2} - 1

Commented by HongKing last updated on 03/Jan/22

Perfect solution my dear Sir than you so much