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 set consisting of non-negative integers such that:
,
, both 
and 
..
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
and 
such that![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](http://latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
. Points 
and 
lie on segments 
and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point 
. The circumcircles of triangles and 
intersect at a second point 
. Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
| 
with 
are positive integers and 
is prime (
), we get: 
, both greater than 1, such that the following holds: of degree 
with integer coefficients and for any prime 
, there exists a set 
of at most 
integers, such that![\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]](http://latex.artofproblemsolving.com/f/9/1/f919f2391121c2b0551d29d4f09ea6cde631a8ba.png)
contains a complete residue system modulo (i.e., intersects with every residue class modulo ).
such that there exists a prime number , such that 
is a power of 
. is a number of the form 
where is a positive integer.
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
+1 w
and 
such that 
.
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly 
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than 
and no more than 
games.
be positive integers such that is a permutation of 
, and
is a permutation of 
.
.
2021 IMOC Problems
, 
(where 
). For any given integer 
, consider a graph 
with 
vertices that contains a Hamiltonian cycle 
and has more than 
edges. For which lengths 
must the graph necessarily contain a cycle of length 
?
.