[/quote]
,
'is a perfect square.
Prove that
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
and 
.
Let 
.
such that 
that satisfy the equation
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
In triangle 
, the median drawn from vertex 
, splits it in two triangle of the same perimeter. Is it possible to draw a ray from the same vertex that breaks the original triangle into two such that the perimeter of one is twice the perimeter of the other?
(S. Usov)
, the median drawn from vertex 
, splits it in two triangle of the same perimeter. Is it possible to draw a ray from the same vertex that breaks the original triangle into two such that the perimeter of one is twice the perimeter of the other?(S. Usov)
![[asy]
import graph; size(4cm);
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 = -4.629676912413191, xmax = 16.890083672273658, ymin = -4.705230884777879, ymax = 9.217085539332018; /* image dimensions */
/* draw figures */
draw((0,6)--(0,0), linewidth(2));
draw((0,0)--(6,0), linewidth(2));
draw((6,0)--(0,6), linewidth(2));
draw((0,0)--(3,3), linewidth(2));
/* dots and labels */
dot((0,0),linewidth(4pt) + dotstyle);
label("$A$", (0.08073846711226249,-0.3), SW * labelscalefactor);
dot((0,6),dotstyle);
label("$B$", (0.08073846711226249,6.197101485534646), NE * labelscalefactor);
dot((6,0),dotstyle);
label("$C$", (6.0827193539269535,0.19512059874238485), NE * labelscalefactor);
dot((3,3),linewidth(4pt) + dotstyle);
label("$M$", (3.081728910519608,3.1581238213360328), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/5/9/7/597e5a5ba7d0dfea86568c90b26d96fa50e6872b.png)
,
is the midpoint of the hypotenuse 
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(4cm);
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 = -4.914581068432877, xmax = 16.605179516253976, ymin = -3.280710104684779, ymax = 10.641606319425119; /* image dimensions */
pen yqqqyq = rgb(0.5019607843137255,0,0.5019607843137255); pen ttttff = rgb(0.2,0.2,1);
/* draw figures */
draw((0,6)--(0,0), linewidth(2));
draw((0,0)--(6,0), linewidth(2));
draw((0,0)--(4.101419168668149,1.8985808313318513), linewidth(2));
draw((0,6)--(4.101419168668149,1.8985808313318513), linewidth(2) + yqqqyq);
draw((4.101419168668149,1.8985808313318513)--(6,0), linewidth(2) + ttttff);
/* dots and labels */
dot((0,0),linewidth(4pt) + dotstyle);
label("$A$", (0.0807384671122614, -0.3), SW * labelscalefactor);
dot((0,6),dotstyle);
label("$B$", (0.0807384671122614,6.197101485534646), NE * labelscalefactor);
dot((6,0),dotstyle);
label("$C$", (6.082719353926953,0.19512059874238485), NE * labelscalefactor);
dot((4.101419168668149,1.8985808313318513),dotstyle);
label("$D$", (4.183358313795721,2.094481638866518), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/e/3/2/e321369d2dd0a41456e095b605e6f6b4528e134c.png)
is twice the perimeter of 
,
and 
,
But triangle inequality tells us that
Which implies that we 
draw the ray.
