[/quote]
,
,
,
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 
.
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
satisfying
for all integers 
.
,
,
means 
applied 
,
means 
applied 
satisfying 
.
such that 
(the number of positive divisors of 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers 
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
.
is tangent to 
,
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
other than 
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
Y by
In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set 
of cities is said beautiful if it contains at least three different cities and from each city 
in at least two buses depart, each one goes directly to a different city in and none of them is (if there is a direct bus from to a city 
in , there is not necessarily a direct bus from to ). Show that if there exists a beautiful set of cities , then there exists a beautiful 
subset of , such that for any two cities in , you can get from one to another by taking buses that only pass through cities in .
Note: A bus goes directly from one city to another if it does not pass through any other city.
of cities is said beautiful if it contains at least three different cities and from each city 
in at least two buses depart, each one goes directly to a different city in and none of them is (if there is a direct bus from to a city 
in , there is not necessarily a direct bus from to ). Show that if there exists a beautiful set of cities , then there exists a beautiful 
subset of , such that for any two cities in , you can get from one to another by taking buses that only pass through cities in .Note: A bus goes directly from one city to another if it does not pass through any other city.
This post has been edited 1 time. Last edited by parmenides51, Nov 13, 2021, 11:20 PM
Reason: typo: removed <em>
Reason: typo: removed <em>
that we cannot get to from a city 
in 
This implies that the city 
that leads to 
that leads to 
and any city that 
Since there is always another city leading to any city leading to 
and any city that 
It's also a beautiful set, since each city in it leads to another two cities in turn. So if a set is not satisfactory, there exists a smaller beautiful subset of the set. Since the smallest possible set (of 3 cities) is satisfactory, for any 
