[/quote]
,
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
lies on side 
lies on side 
.
and 
be points on segments 
and 
,
is parallel to 
.
be a point on line 
,
.
be the point on line 
,
.
,
and 
be positive real numbers satisfying 
Prove that
be a random point on the smaller arc 
,
be the intersection point of segments 
.
are 
is the center of the square. Prove that points 
,
,
such that:
is strictly increasing,
such that 
for all 
,
,
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
is written on the board. A move consists of adding either 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
,
all of whose digits are odd.
and 
,
,
,
are integers such that 
.
and 
of 
such that for each integer 
with 
,![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
satisfy 
.
be an odd prime congruent to 2 modulo 3. Prove that at most 
members of the set 
are divisible by 
,
,
intersects lines 
and 
at 
and 
be distinct points on side 
.
be the circumcircle of 
.
.
is divisible by 
Y by MathLuis, mpcnotnpc, JollyEggsBanana, Rounak_iitr
Let 
be a triangle with incenter 
, and let 
be a point on side 
. Points 
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
be a triangle with incenter 
, and let 
be a point on side 
. Points 
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
and 
,
be midpoint of arc 
on 
,
be 
respectively now let reflection of 
over 
and reflection of 
over 
.
and 
are diameters on 
respectively, now let 
be the 
excenters of 
which implies 
cyclic and similarily 
is cyclic however since 
we also get that 
is cyclic and similarily 
is cyclic, however it doesn't end here...
so 
on 
however 
are symetric in 
are reflections of 
over 
are colinear, and similarily 
are colinear.
which means 
is cyclic and similarily we have 
cyclic, but also note that 
which means that 
are colinear.
which does in fact show that 
is isosceles and therefore 
,
,
where 
(clearly then 
is A-excenter of 
)
which means that 
is cyclic but by summing arcs we end up realising 
is diameter and in fact now this means 
is 
be the miquelpoint of 
then 
colinear and then Reim's twice gives 
and from double Reim's once again we have that 
and this is excellent news because now we can note that 
which shows that 
are tangent at 
and 
.
and 
be the projections of 
be the tangency point of 
is tangent to 
is a parallelogram, by projecting onto 
.
Simplifying yields 
,
But 
With Thales' Theorem, 
so 
thus getting 
and the conclusion follows.
are in centres, 
meets 
in 
and tha
.
