AoPS Academy
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)
,such that the inequality 
holds for all positive reals 
and for every positive real 
, there exist positive reals 
, such that the equality holds.
, let 
be the real root of equation 
. Prove that
;
.
be an integer, 
a finite set of integers (not necessarily positive) and 
subsets of . Suppose that, for every 
, the sum of the elements of 
is 
. Prove that contains at least 
elements.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
, respectively. Let 
be the midpoint of 
. Let 
be the intersection point of a perpendicular line passing through to 
. Line 
intersects the circumcircle of 
at 
. The circumcircle of 
intersects line 
at 
. Prove that quadrilateral 
is cyclic.
for which there exists a sequence of distinct positive integers 
, 
, 
, 
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
be a positive integer, and let be a subset of 
. An -partition of into 
parts is a representation of n as a sum 
, where the parts 
belong to and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set 
.-partition of into parts is optimal if there is no -partition of into 
parts with 
. Prove that any optimal -partition of contains at most ![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
be a triangle inscribed in circle 
is a point on its Euler line. 
is circumcevian triangle of 
and 
are midpoints of 
and 
respectively. Prove that 
and 
are concurrent.
be a triangle inscribed in circle is a point on its Euler line. is circumcevian triangle of 
and are the reflections of 
and 
through midpoints of 
and 
respectively. Prove that 
and 
are concurrent. be a triangle inscribed in circle is a point such that on its isogonal conjugate lies on Euler line of 
is circumcevian triangle of and are the reflections of and through and respectively. Prove that and are concurrent.
be a triangle inscribed in circle is a point on its Euler line. is circumcevian triangle of and are the reflections of 
and 
through midpoints of and respectively. Prove that 
and 
are concurrent.
and any point . Let 
is the pedal triangle of and 
is center of 
. Let 
and 
is circumcevian triangle of 
wrt . Then 
are perspective.
be a triangle inscribed in circle is a point on its Euler line. is circumcevian triangle of and are the reflections of and through midpoints of and respectively. Prove that and are concurrent.
be orthocenter of 
; 
, 
intersect (
) again at , , , draw lines which parallel with 
, , intersect () again at 
, 
, 
, 
, , are collinear; , , are collinear, , , , , , we have: , 
, 
are collinear, , , , 
, , we have: 
, 
, are collinear, , are collinear then: , , , , are collinear intersect at a point lie on Euler line of intersect 
at a point lie on Euler line of 
, , concurrent at point lie on Euler line of
be centroid of ; 
, 
, 
, 
, 
are images of 
, , through the homothety with centers , , ratio 2 , , are also images of , , through the homothety with center ratio 4 with 2 point , and 
, 
, 
intersect () again at , , ; 
, 
, 
intersect () again at , , ; we have: the intersections of these pairs of lines (
, 
), (
, 
), (
, 
) are collinear, , concurrent or , , concurrent
