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.
be a sequence of real numbers with 
. Define the score 
of a permutation 
of 
to be the minima of the sum 
over all real numbers 
. attains the maxima over all permutations 
, if and only if for all 
, 
be a nonempty set of positive integers. We say that a positive integer 
is clean if it has a unique representation as a sum of an odd number of distinct elements from . Prove that there exist infinitely many positive integers that are not clean.
such that for any positive integer 
, satisfies 
.
and 
lie respectively on sides 
and 
of a triangle 
and 
. Segments 
and 
cross at 
. Circumscribed circles of triangles 
and 
cross again at point different from . Prove that point lies on the bisector of angle 
.
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
such that![\[
ad = bc, \quad a + d = 2^k, \quad b + c = 2^m
\]](http://latex.artofproblemsolving.com/1/1/d/11d16732e3a77af46ad17adecf104a442a5d7f67.png)
for some positive integers 
and 
.
be a triangle with incenter 
and intouch triangle 
. Let be the foot of the perpendicular from 
onto 
. Assume that 
, intersect the sides , in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in , 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .
be a triangle with circum-circle 
. Let 
be a point in the interior of triangle which is also on the bisector of 
. Let 
meet in 
respectively. Suppose 
is the point of intersection of 
with 
; and 
is the point of intersection of 
with 
. Prove that 
is parallel to 
.
such that 
is a perfect square but none of the integers 
are perfect squares.
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear. be an acute-angled triangle with 
. Let be the incentre of triangle , and let 
be the points where the incircle touches the sides 
respectively. Let 
meet the line at 
respectively. Further assume that both and 
are outside the triangle . Prove that
are concyclic.
is also the incentre of triangle 
.
and 
, prove
that can be subtracted from both the numerator and denominator of the fraction 
, to get, after the reduction, the fraction of form 
, where 
are single digit numbers.
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
.
bisects 
.
and hence 
.
or equivalently 
.
and of these four only 
divides 
.
and here 
indeed works.