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 positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by ?
+1 w
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 
.
's are positive reals with sum 
, they are all strictly less than , so 
is positive. Now note that by weighted AM-GM inequality 
Summing this over all 
(taking 
) we see that 
This completes the proof. 
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 
.
be an integer, and suppose 
are positive real numbers such that 
Prove that 
.
,
![\[LHS=\sum{((x_i-1)+1)^{1-x_{i+1}}}\leq \sum{(1-x_{i+1})(x_i-1)+1}=2\sum{x_i}-\sum{x_ix_{i+1}}<2\]](http://latex.artofproblemsolving.com/1/8/2/182f7f4f78cf16d059abb87078ab2b15ea9c6119.png)
As desired
Summing over 
yields the conclusion.