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 quadrilateral inscribed in a circle with center 
and 
be the intersection of segments 
and 
. Let 
be the circumcircle of 
and 
be the circumcircle of 
. The tangent to at 
and the tangent to at 
meet at 
. The tangent to at 
and the tangent to at 
meet at 
. Show that 
.
, let 
denote the number of 
s in the binary representation of . Prove that for any positive integer 
,![\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]](http://latex.artofproblemsolving.com/d/e/9/de975cc2632685d7a3f71c023115535d0aeb5dbb.png)
Holden Mui
+2 w
grid (
), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly black cells.Detemine the number of balanced colorings.
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at . Let 
be the projection of onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at . Prove that 
is an isosceles triangle.
for which there exist real numbers 
satisfying 
, 
and
for 
.
be an integer. Find the smallest integer 
with the property that there exists a set of distinct real numbers such that each of its elements can be written as a sum of other distinct elements of the set.
be a regular 
-gon. Assign integers between and to the vertices of such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of . Find the smallest integer such that one can achieve every other assignment from a given one with no more than operations., the point is the midpoint of and is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.
is also an integer and equals 
.
, some of which consisting of Fibonacci numbers.
such that for all positive real numbers 
and 
:
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 . 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 
.
and 
be the incircle and incenter of triangle . Let , , be the tangency points of to 
, 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points and . Show that 
.
,
.
.
.
,
from the perpendiculars.
be the unit circle
to get that 
and 
angles sum up to 
we prove that 
and using the fact that 
we get that 
