then find the value of 
be a sequence of reals such that 
and 
for all positive integers 
.
approaches the real number 
.
![\[
\sqrt[3]{\sqrt[3]{x - \frac{3}{8}} - \frac{3}{8}} = x^3 + \frac{3}{8}
\]](http://latex.artofproblemsolving.com/b/b/9/bb99cd588b94b5716738e6e9ea01d15377b6efef.png)
has exactly two real positive solutions 
and 
.
.
Y by
Let 
and 
be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](//latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
and 
be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
This post has been edited 5 times. Last edited by ReticulatedPython, Apr 10, 2025, 4:42 PM
;
.![$\frac{AP^2+BP^2}{(AP)(BP)}=3 \Rightarrow [p^{2}+q^{2}+(p-1)^{2}+q^{2}]^{2}=9(p^{2}+q^{2})[(p-1)^{2}+q^{2}]$](http://latex.artofproblemsolving.com/b/0/c/b0c74340c67cf72dc9180a4e609dd8aa82819454.png)
,![$q=\frac{\sqrt[4]{500} \cdot \sqrt{3\left|2p-1\right|-2\sqrt{5}p^{2}+2\sqrt{5}p-\sqrt{5}}}{10}$](http://latex.artofproblemsolving.com/c/9/e/c9e5d88750962213f9b34f9c416a9e3e54d2d632.png)
![$p \in [\frac{1-\sqrt{5}}{2},\frac{5-\sqrt{5}}{10}] \cup [\frac{5+\sqrt{5}}{10},\frac{1+\sqrt{5}}{2}]$](http://latex.artofproblemsolving.com/5/e/7/5e73a09091a29ab8cab1bc602b3cab08184cf00d.png)
.![$AP=\frac{\sqrt[4]{500} \cdot \sqrt{3\left|2p-1\right|+2\sqrt{5}p-\sqrt{5}}}{10}$](http://latex.artofproblemsolving.com/6/1/f/61fa22af5263cf7ee114a0077a7c84fcdada9d25.png)
and the result follows.
