[/quote]
,
be odd natural number and 
be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum.
)
,
,
terms is divisible by 
,
,
positive real numbers. Prove that:![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]](http://latex.artofproblemsolving.com/f/b/5/fb5a179f427615c8bc2da46164ecd9a3484d6625.png)
![$a,b,c\in{[2-\sqrt{3},2+\sqrt{3}]},$](http://latex.artofproblemsolving.com/0/c/9/0c9d55a20ca1a60e4d0cd51e44171923fc44abf1.png)
t![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+3\]](http://latex.artofproblemsolving.com/4/c/0/4c092e0e8e46d1014b2cc80caee0f5959003c36f.png)
![$a,b,c,d\in{[\frac{1}{2},2]},$](http://latex.artofproblemsolving.com/b/6/9/b69e17212ca1f383bf3f1abf908169d012a8e12b.png)
t![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}+4\]](http://latex.artofproblemsolving.com/9/6/c/96cc857aeceb419f95f7fea36fcd44428fe26d55.png)
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
be the set of vertices of 
be the set of triangles with vertices all in 
of 
be the set of triangles of this type.
be the set of triangles of this type.
,
the area of ![\[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]](http://latex.artofproblemsolving.com/5/e/d/5ed261e1e812466ea3e78f3b681f6bbdb7ebccd1.png)
and orthocenter 
intersect 
respectively. Lines through 
intersects 
respectively. Point 
is a parallelogram. Prove that lines 
and 
.
![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Y by chessgocube
An acute-angled triangle 
and its Euler circle are given. Two different circles are drawn through vertices 
and 
, tangent to Euler's circle at points 
and 
. Points 
are obtained by similar constructions. Prove that three straight lines 
, 
, 
intersect at one point belonging to the Euler line.
(D. Soskin)
and its Euler circle are given. Two different circles are drawn through vertices 
and 
, tangent to Euler's circle at points 
and 
. Points 
are obtained by similar constructions. Prove that three straight lines 
, 
, 
intersect at one point belonging to the Euler line.(D. Soskin)
to the opposite sides be 
are cyclic, so 
is on the radical axis of 
must be the polar of 
similarly. 
on the radical axis of 
