[/quote]
,
of the set 
,
such that:
,
,
,
.
?
and circumcenter 
.
the orthocenter of triangle 
. The incircle of 
at 
respectively. Suppose that 
and 
intersects 
at 
and 
respectively (differ from 
lies on a circle.
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
is given. Determine the smallest natural number 
,
,
)![\[
\tau(m^k) \tau(m^k + 1) \cdots \tau(n^k - 1) \tau(n^k)
\]](http://latex.artofproblemsolving.com/b/8/6/b863243eaace4276de6b12b7c5a449f1306b0145.png)
is a perfect square, where 
denotes the number of positive divisors of 
.
,
,
intersect sides 
,
,
,
,![\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \]](http://latex.artofproblemsolving.com/b/6/f/b6f492b82d7b43d9a5a0a7f9e81205cb8e732bc9.png)
![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
denotes the area of triangle 
.
,
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing 
be a circle and 
a line in its plane. Let 
be a point on 
and 
be the tangents from 
are distinct points on 
and respectively 
and 
intersect the second time circle 
and 
and 
be the roots of the equation 
where 
is an odd integer. Let 
is an integer
)
Y by Adventure10, Mango247
On sides 
and 
of a non-isosceles triangle 
are selected points 
and 
such that the quadrilateral 
is cyclic. Lines 
and 
intersect at point 
. Line 
intersects the circumscribed circle of triangle at the point 
. Prove that the lines 
and 
, where 
is the midpoint of 
, intersect at the circumscribed circles of triangle .
and 
of a non-isosceles triangle 
are selected points 
and 
such that the quadrilateral 
is cyclic. Lines 
and 
intersect at point 
. Line 
intersects the circumscribed circle of triangle at the point 
. Prove that the lines 
and 
, where 
is the midpoint of 
, intersect at the circumscribed circles of triangle .This post has been edited 2 times. Last edited by parmenides51, Jul 29, 2019, 2:52 PM
Reason: source
Reason: source
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.
and 
.
. Then we'll prove that 
is the median in 
we need to prove that its the 
since they are oppositely similiar so the median and the symmedian get swapped. So we can restate the problem as follows.
Let 
.
and let 
.
and finally let 
.
.
. So we just need to prove 
collinear. Note that by radical axis theorem we are left to prove 
cyclic. Now we have that since 
cyclic. Now note that 
cyclic as desired
meet 
and 
are similar so we need to prove 
.
is cyclic.
is cyclic so 
is cyclic.
