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.
with incenter 
,
is a point on 
,the bisector of 
meet 
at 
.Prove that 
such that
for all positive real numbers 
and 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
real number on the board such that each number is more than 
.
on the board and write 
.
and 
.
?
be triangle and point be 
altitude of . be a point on 
and 
such that 
and 
. Point 
is the intersection of 
and 
. Point 
be intersection of 
and . Line 
intersect circumcircle of again at 
.
.
bisect .
, define a sequence of integers 
by letting 
and 
for 
. Let 
. Determine the largest possible 
such that, for some positive integer , the numbers 
are all prime.
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 
. The only restriction on this function is that 
for all 
(that is, 
is idempotent). such that 
. Then is equivalent to saying that every point must map to a fixed point. There are no other restrictions, so if there are fixed points, then there are 
ways to choose the fixed points and 
ways to decide where to map the other points. Therefore,![\[N=\sum_{k=0}^7\binom7k\cdot k^{7-k}=1\cdot0+7\cdot1+21\cdot32+35\cdot81+35\cdot64+21\cdot25+7\cdot6+1\cdot1=6\boxed{322}.\]](http://latex.artofproblemsolving.com/1/1/d/11dd3cc7ff71c8ecb2af3867942ec831bf8e9656.png)
case is trivial, my stupid brain also thought the 
case was trivial and submitted 321. fml
such that 
. Then for all , so there must exist at least one such that . If there are values such that , there are 
ways to pick them, and each of the other 
values must map to one of these values, so we get 
. Thus the answer is![\[\binom711^6+\binom722^5+\binom733^4+\binom744^3+\binom755^2+\binom766^1+\binom777^0\equiv \boxed{322}\pmod{1000}.\]](http://latex.artofproblemsolving.com/f/8/2/f82b67dbc954d3f277f01886529b18884d522313.png)
, then for this value of then 
. This can be shown by noting that if there exists a , then 
. So, .
. If there are 
different values, then we select the values which satisfy . For the remaining 
values, there are values for each. So, the sum is equivalent to
after manual calculation.
.
a
.
and 
Prove that
and 
Prove that
denote the number of 
-
such that for each 
,![\[ 1 \leq a_i \leq 7 \quad\text{and}\quad a_{a_i} = a_i. \]](http://latex.artofproblemsolving.com/5/1/7/51741f4725c64c1ad29b14949ad890f131b89687.png)
Compute the remainder when 
.
![\[N = \sum_{k=1}^{7}\binom{7}{k}k^{7-k} = 6322\]](http://latex.artofproblemsolving.com/4/1/3/4130662083552b6e8cfd2175b87e1e4e50e99718.png)
to itself such that 
for all 
It is not difficult to see that every 
so the total number of desired functions is
Hence, the answer is 
