AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
be two isogonal conjugate with respect to triangle 
. Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
, respectively. Prove that 
and 
bisect two arcs 
containing 
and not containing , respectively, of 
.
be a triangle such that 
and 
. The tangent at to the circumcircle 
of triangle meets the line at 
. Let 
be the reflection of in the line , let 
be the foot of the perpendicular from to 
, and let 
be the midpoint of the segment 
. Let the line 
intersect the circle again at 
.
is tangent to the circumcircle of triangle 
.![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
, 
, 
, each different from 
, and satisfying 
., , , each different from , and satisfying .
+2 w
there exists an -digit number divisible by 
all of whose digits are odd.
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer 
.
such that for all real numbers 
and 
, ![\[f(f(xy)+y)=(x+1)f(y).\]](http://latex.artofproblemsolving.com/d/1/f/d1f369109456464bc8b0102763211802770d983d.png)
such that for all positive real numbers and :
edit oops sorry I misinterpreted that the original poster had solved it, solvable tho
be positive real numbers such that 
. Show that![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
formed by using the digits 
once each such that the number 
is divisible by 
for 
.
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
be a set of 
positive integers with the property that the product of any two distinct members of will not exceed 1994. Show that there are numbers 
and 
in the set such that the gcd of and is greater than 1.
and : 
, knowing that the median 
makes an angle of 
with side 
and an angle of 
with side 
.
such that
; and are relatively prime;
.
is a random variable reprasenting the number of girls in the teams combined. Clearly, 
.
is independent of 
, 
. and are hyper-geometric, that is
. Consider the ratio![\[\frac{\mathrm{P}(X_A = k+1)}{\mathrm{P}(X_A = k)} = 1 \implies k = \left\lfloor \frac{(m_2+1)(p+1)}{m_1+m_2+2} \right\rfloor\]](http://latex.artofproblemsolving.com/a/e/8/ae8dbf4a70f1768a3ca248770089b7a8bbd15cff.png)
has 
and otherway round for 
. to get the solution that 
is maximised at![\[
\boxed{
k = \left\lfloor \frac{(m_2+1)(p+1)}{m_1+m_2+2} \right\rfloor + \left\lfloor \frac{(n_2+1)(p+1)}{n_1+n_2+2} \right\rfloor
}
\]](http://latex.artofproblemsolving.com/a/3/4/a34bf0aad5c1305d2e1997825baf7af02cdfb47c.png)
