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.
satisfying![\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]](http://latex.artofproblemsolving.com/8/0/f/80fee0440e9fc4031fa4dcff96b9605011ef0e68.png)
for all real numbers 
and 
.
, in which each unit cell is colored blue or red. There are exactly 
rows in which the majority of cells are blue, and exactly columns in which the majority of cells are red.
be an equilateral triangle with circumcenter 
. Let 
and 
be two points on segments 
and 
, respectively, such that 
. If 
is the reflection of with respect to line 
, prove that lines 
and 
are parallel.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter 
. be an acute triangle, its circumcircle and its circumcenter. The altitude from intersects in a point 
and the segment 
intersects the circumcircle of 
in a point 
. Finally, let 
be the midpoint of . Show that 
is parallel to 
.
be the set of all nonnegative integers. Find all the functions 
satisfying the relation![\[ f(f(f(n))) = f(n+1 ) +1 \]](http://latex.artofproblemsolving.com/8/b/9/8b953428105df2c2344963cbaf3eb81d8201e0df.png)
.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
, such that for any 
, 
very convex if![\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|
\]](http://latex.artofproblemsolving.com/3/b/a/3ba7733caf51cc70a090920ca21f8b47aad9b1ec.png)
and 
. Prove that no very convex function exists.
satisfying
for all integers 
.
, and for any positive integer 
, 
means applied times to , and 
means 
applied times to .
be a set of distinct positive even numbers and 
be a set of distinct positive odd numbers such that![\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]](http://latex.artofproblemsolving.com/5/d/f/5df36843d80d7f6a11401c2f6162ad10a00befb3.png)
Prove that![\[5m+12n\le 581.\]](http://latex.artofproblemsolving.com/8/8/b/88be1683ea815b49af99607e446dd7751130f527.png)
.
.
. that satisfy the above equation lie inside the circle 
are 
(approx). which lie inside the circle but also satisfy 
.
,
,(since (m,n) must lie inside the circle)
,
,
for integer 
.
, which gives that 
to be any of 
. For this, just find a general solution and use the fact that 
. Also, needs to be odd as otherwise the whole LHS would end up being even. and we proved that it cant be any of 
, we must have that 
since all are distinct , now it implies 
, now 
is not possible.
, clearly has to be odd and 
which really doesn't give any such and to be working.
which also doesn't give any such and 
which also doesn't works.
. 
so
Now show 
as above. 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
are distinct continuously.Sort 
and 
ascendingly.Then its easy to see 
and 
.
.
and 
adding up we get 
.
Then 
