[/quote]
,
satisfy that: 
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
such for any 
:
.
and 
be nonzero real numbers satisfying![\[
a_1^2 b_1^2 (a_1 + b_1) + a_2^2 b_2^2 (a_2 + b_2) + \cdots + a_n^2 b_n^2 (a_n + b_n) \leq 7,
\]](http://latex.artofproblemsolving.com/a/7/e/a7ea7af24ac529090f3b2e305be43e2728b79eb4.png)
![\[
\frac{1}{a_1} + \cdots + \frac{1}{a_n} = \frac{1}{4}, \quad \frac{1}{b_1} + \cdots + \frac{1}{b_n} = \frac{1}{3}.
\]](http://latex.artofproblemsolving.com/e/f/3/ef341b850979a9b780f5daaa49f46f1d1324cdbd.png)
Find the maximum value of![\[
a_1 b_1 + a_2 b_2 + \cdots + a_n b_n.
\]](http://latex.artofproblemsolving.com/d/3/3/d338bf5be4f1a218e150bfd84173e988ee92ddad.png)
![\[
P(x) = a_0 + a_1 x^2 + a_2 x^4 + \cdots + a_{10} x^{20}
\]](http://latex.artofproblemsolving.com/0/d/c/0dcd08456f253de08f3b880c7a1fddc07b335932.png)
be a polynomial of degree 20 with only even powers of 
.
be 
.![\[
(x_1^2 + 1)(x_2^2 + 1) \cdots (x_{20}^2 + 1) = 2025,
\]](http://latex.artofproblemsolving.com/b/a/4/ba4bc5f526565e56a9a685fe01572c1366241084.png)
find the **minimum value** of 
.
?
by the recurrence![\[
a(n) = \left| a(n-1) - a(n-2) \right|
\]](http://latex.artofproblemsolving.com/e/7/9/e7981725c3dfd9fd1a0e22c17eded1a4cde84e9f.png)
for all 
,
,
,
.
,
such that![\[
a(i) = a(i+3)
\]](http://latex.artofproblemsolving.com/d/c/7/dc75144ca27873b2420e79b5e8f045be864a4c2e.png)
for all integers 
.
and 
Prove that
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
,
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
.
Y by Adventure10, Mango247
In an acute scalene triangle 
the bisector of the acute angle between the altitudes 
and 
meets the sides 
and 
at 
and 
respectively. The bisector of the angle 
intersects the segment joining the orthocenter of and the midpoint of 
at point 
. Prove that , , , lie on a circle.
the bisector of the acute angle between the altitudes 
and 
meets the sides 
and 
at 
and 
respectively. The bisector of the angle 
intersects the segment joining the orthocenter of and the midpoint of 
at point 
. Prove that , , , lie on a circle.
- diametrically opposite to 
in the circle 
,
- its orthocenter. Obviously, 
and 
are symmetrical about the angle bisector 
,
( 
is a parallelogram ).
,
( 1 ), so 
( 2 ). But 
,
is angle bisector, hence 
,
and 
,
is cyclic.
,
; 
; 
,
,
,
,so 
is concylic
and 
.
be the midpoint of 
be the reflection of 
across point 
and 
,
is a parallelogram. Furthermore, 
,
is cyclic. But 
,
is a diameter of 
.
is isogonal to 
and 
.
gives us 
by AA similarity. Therefore, 
.
bisects 
,
,
.
bisects 
,
.
,
.
,
is cyclic, as desired. //
and 
are coaxial. Let 
be the midpoints of the arcs 
and 
.
we conclude 
so 
From angle bisector theorem, we have 
we conclude that 
as desired.
denote the intersection of circle with diameter 
henceforth) and 
.
meet 
.
the antipode of 
.
bisects the segment 
.
as well. Now, observe that 
.![\[C_1P/PA = HC_1/HA = HA_1/HC = A_1Q/QC\]](http://latex.artofproblemsolving.com/2/3/f/23f80303ef884e03871a49246c3288b0ef827d3b.png)
which implies that 
.
are all concylic. First, observe that from the spiral sim, 
is cyclic. Next, remark that 
due to 
.![\[\angle PXH = \angle PXC_1+\angle C_1XH = \angle W'XB+\angle C_1XH = \angle HXW+\angle C_1XH = \angle C_1XW = \angle C_1BW = \angle PBR\]](http://latex.artofproblemsolving.com/0/b/2/0b23b3f178024e0e4521fe8dc72edd6438f48a5c.png)
where we used 
due to the pairs 
being antipodes in 
,
and thus in particular points 
are concylic, as required.
