the product of all 
'
.![$\sqrt[2024]{p(n)} \in \mathbb{Z}$](http://latex.artofproblemsolving.com/7/0/4/70442d4ec6ae8b7a3cf9717719e862bf3b5d4a28.png)
and is not a perfect power of another integer.
is extended beyond 
to 
so that 
.
is extended beyond 
to 
so that 
.
,
such that 
as 
intersects 
at 
such that 
is equilateral.
where 
are non-negative integers that are not necessarily distinct. Find the smallest positive integer 
Y by Adventure10
Let 
be a given convex quadrilateral in a plane. Prove that there exist a line with four different points 
on it and a square 
such that 
lies on both line 
and 
lies on both line 
and 
lies on both line 
and 
lies on both line 
and 
be a given convex quadrilateral in a plane. Prove that there exist a line with four different points 
on it and a square 
such that 
lies on both line 
and 
lies on both line 
and 
lies on both line 
and 
lies on both line 
and 
This post has been edited 1 time. Last edited by Henry_2001, Jul 31, 2019, 9:49 AM
draw a line from 
making 
the only condition should be 
.![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -8.933393727086195, xmax = 10.719550498276064, ymin = -4.743185416351953, ymax = 7.076485907826339; /* image dimensions */
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/* draw figures */
draw((-5.06,3.03)--(0.48,3.37), linewidth(2) + wrwrwr);
draw((0.48,3.37)--(0.84,-1.71), linewidth(2) + wrwrwr);
draw((0.84,-1.71)--(-5.22,-0.83), linewidth(2) + wrwrwr);
draw((-5.22,-0.83)--(-5.06,3.03), linewidth(2) + wrwrwr);
draw((xmin, 24.125*xmin + 125.10249999999988)--(xmax, 24.125*xmax + 125.10249999999988), linewidth(2) + wrwrwr); /* line */
draw((xmin, -0.14521452145214522*xmin-1.588019801980198)--(xmax, -0.14521452145214522*xmax-1.588019801980198), linewidth(2) + wrwrwr); /* line */
draw((xmin, -0.785596746736963*xmin + 0.7182775889005671)--(xmax, -0.785596746736963*xmax + 0.7182775889005671), linewidth(2) + wrwrwr); /* line */
draw((xmin, 1.2729176949288263*xmin + 10.996904020527792)--(xmax, 1.2729176949288263*xmax + 10.996904020527792), linewidth(2) + wrwrwr); /* line */
draw(circle((0.7072939540428336,0.16262975961779422), 10.92963619383511), linewidth(2) + wrwrwr);
draw((xmin, 1.2729176949288263*xmin-0.7376972299995048)--(xmax, 1.2729176949288263*xmax-0.7376972299995048), linewidth(2) + wrwrwr); /* line */
draw((xmin, 0.5311802879311687*xmin + 4.795095746891031)--(xmax, 0.5311802879311687*xmax + 4.795095746891031), linewidth(2) + rvwvcq); /* line */
draw((xmin, -1.882599981062516*xmin-4.759306736810882)--(xmax, -1.882599981062516*xmax-4.759306736810882), linewidth(2) + rvwvcq); /* line */
draw((xmin, -1.882599981062516*xmin + 4.669067597145297)--(xmax, -1.882599981062516*xmax + 4.669067597145297), linewidth(2) + rvwvcq); /* line */
draw((xmin, 0.5311802879311687*xmin-0.2130708465426529)--(xmax, 0.5311802879311687*xmax-0.2130708465426529), linewidth(2) + rvwvcq); /* line */
/* dots and labels */
dot((-5.06,3.03),dotstyle);
label("$A$", (-4.986859288321765,3.229611429636777), NE * labelscalefactor);
dot((0.48,3.37),dotstyle);
label("$B$", (0.5542345196404137,3.5684552955912983), NE * labelscalefactor);
dot((0.84,-1.71),dotstyle);
label("$C$", (0.9130103777099072,-1.5142026937265174), NE * labelscalefactor);
dot((-5.22,-0.83),dotstyle);
label("$D$", (-5.14631522524154,-0.6371950406677571), NE * labelscalefactor);
dot((-4.993225319959166,4.640939155985119),dotstyle);
label("$P$", (-4.907131319861877,4.844102790949495), NE * labelscalefactor);
dot((3.6014387967358266,-2.111001013387381),dotstyle);
label("$S$", (3.6835572816909963,-1.912842536025954), NE * labelscalefactor);
dot((-3.0960580649995517,3.1505307324729523),linewidth(4pt) + dotstyle);
label("$Q$", (-3.01359206893955,3.3093393980966646), NE * labelscalefactor);
dot((0.7072939540428336,0.16262975961779422),linewidth(4pt) + dotstyle);
label("$R$", (0.7934184250200761,0.31954058085089054), NE * labelscalefactor);
dot((7.4592341234153325,8.757293876312787),linewidth(4pt) + dotstyle);
label("$E$", (5.9358723906828175,6.75757403398679), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */ [/asy]](http://latex.artofproblemsolving.com/2/f/2/2f2d7167e40d39e6866c3f45dbdc53c15cfd476b.png)
,
,
,
.
where 
denotes the length of segment 
exactly once.
be the acute angle formed by the side of the square with the line (not the extension of the side of the square).
where 
and this indeed works.
that is perspective to 
