,
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
and 
onto the internal bisectors from 
is parallel to 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers 
with 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
,
.
is a permutation of the elements of the set 
}
and 
be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers 
and 
occurs. Show that the numbers 
![\[
\text{Given that } 0 < \theta < 90^\circ,\ \text{solve the equation: } \sin(\theta - 60^\circ)\sin\theta + \sin(54^\circ - \theta)\sin 54^\circ = 0
\]](http://latex.artofproblemsolving.com/b/2/4/b24276c1fbddbcc97e50cef52db03ea9b264d7ee.png)
![\[
\text{What is the value of } \theta\ (\text{in degrees})\ \text{that satisfies the equation?}
\]](http://latex.artofproblemsolving.com/0/0/f/00fbcd5839fedced0e5636c1def5360d65fc0444.png)
and circumcenter 
.
.
be the incenter of triangle 
,
be a point on the arc 
of 
.
be the circumcenter of triangle 
.
and 
meet on 
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an 
.
respectively.
and 
as the reflections of 
.
with 
with 
can be inscribed within a circle.
where the 
are positive integers such that each 
and none of two consecutive terms when summed are equal to 
.
Y by
Let 
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major 
arc midpiont.
i)Show that points
and are coconic on a conic 
ii) If intersects 
again at 
, not equal to 
or 
, then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major 
arc midpiont.i)Show that points
and are coconic on a conic 
ii) If
intersects 
again at 
, not equal to 
or 
, then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
This post has been edited 1 time. Last edited by TestX01, Feb 25, 2025, 11:35 PM
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