[/quote]
,
be non-negative real numbers such that 
Prove that
over 
.
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
,
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by Adventure10, Mango247
Let 
be a trapezoid of bases 
and 
. Let 
be the intersection point of its diagonals 
and 
. If the area of triangle ABC is 
and the area of triangle 
is 
. Calculate the area of triangle 
.
be a trapezoid of bases 
and 
. Let 
be the intersection point of its diagonals 
and 
. If the area of triangle ABC is 
and the area of triangle 
is 
. Calculate the area of triangle 
.This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2022, 1:41 AM
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(8cm);
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 = -5.662313783896716, xmax = 7.687379225634272, ymin = -3.1319901589181595, ymax = 4.821795913366269; /* image dimensions */
/* draw figures */
draw((-4,0)--(-2,3), linewidth(1));
draw((2,3)--(3,0), linewidth(1));
draw((3,0)--(-2,3), linewidth(1));
draw((-4,0)--(2,3), linewidth(1));
draw((-2,3)--(2,3), linewidth(1));
draw((3,0)--(-4,0), linewidth(1));
draw((-0.18181818181818182,0)--(-0.18181818181818182,3), linewidth(1));
/* dots and labels */
dot((-4,0),dotstyle);
label("$A$", (-3.9570610272048014,0.12017045562999089), NE * labelscalefactor);
dot((3,0),dotstyle);
label("$B$", (3.0466556520655614,0.12017045562999089), NE * labelscalefactor);
dot((2,3),dotstyle);
label("$C$", (2.04786475171744,3.1165431566743544), NE * labelscalefactor);
dot((-2,3),dotstyle);
label("$D$", (-1.9472988496750454,3.1165431566743544), NE * labelscalefactor);
dot((-0.18181818181818182,1.9090909090909091),dotstyle);
label("$O$", (-0.1324227014815078,2.0324896184916374), NE * labelscalefactor);
dot((-0.18181818181818182,0),dotstyle);
label("$E$", (-0.1324227014815078,0.12017045562999089), NE * labelscalefactor);
dot((-0.18181818181818182,3),dotstyle);
label("$F$", (-0.1324227014815078,3.1165431566743544), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]](http://latex.artofproblemsolving.com/1/1/a/11a1805592bcb3948987eacebb9b97fb84f2b722.png)
![$[ABC] = \frac{1}{2} AB \cdot EF = 150$](http://latex.artofproblemsolving.com/6/6/2/662d6db4ce98b6dc05516863e7a1343df68fd3b0.png)
![$[ACD] = \frac{1}{2} CD \cdot EF = 120$](http://latex.artofproblemsolving.com/4/c/3/4c32a09289737d38b376ab784c201ee8e51a908c.png)
,
.![$[ABO] = \frac{1}{2} AB \cdot OE = \frac{1}{2} AB \cdot \frac{5EF}{9} = \frac{750}{9}$](http://latex.artofproblemsolving.com/7/1/6/716e947348ee474c061f6db4dda01e2150cd564e.png)
![$[CDO] = \frac{1}{2} CD \cdot OF = \frac{1}{2} AB \cdot \frac{4EF}{9} = \frac{480}{9}$](http://latex.artofproblemsolving.com/8/d/7/8d78ee6bb0f58253ef4057ffa6fa5170d9110d79.png)
![$[BCO] = [ADO]$](http://latex.artofproblemsolving.com/5/2/f/52fabea6becf883be8da174889bfe42f7fcb060a.png)
.![$[BCO] = \frac{1}{2} ([ABCD] - [ABO] - [CDO]) = \frac{200}{3}$](http://latex.artofproblemsolving.com/2/0/8/208ffb60fcfec9052794bf792c03cc6cc4deb13b.png)
.
