AoPS Academy
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
be a positive integer. We say that a positive integer 
is -good if 
is divisible by 
for all positive integers 
with 
. Suppose is a positive integer such that is -good, but 
is not -good. Prove that 
is prime.
be real numbers. For any set 
let 
. Assume that the function 
takes on at least 
values where runs over all 
subsets of 
. Prove that the number of sets 
for which 
does not exceed 
.
satisfying the following two conditions for all 
:![\[ f(x+y)=f(x)+f(y), \;\;\; f \left( \max \{x, y \} \right) = \max \left\{ f(x),f(y) \right\}. \]](http://latex.artofproblemsolving.com/3/9/1/391164166941b57fe54bde43b1f3ef3b49494180.png)
be a closed map with the property that the inverse image of each point in 
is a compact subset of 
. Prove that 
is compact whenever 
is compact in .
and 
.
.
-sum of type 
is a finite sequence of positive integers 
, such that:
;
, and 
. For example, there are five 
-sums of type , namely: ; 
; 
; 
; 
. An -sum of type 
is a finite sequence of positive integers 
such that: 
; 
;
each 
is in the sequence 
defined by 
, 
, 
; and
if 
, then 
is a subsequence. For example, there are five -sums of type , namely: ; 
; 
; 
; 
. Prove that for 
the number of type and type -sums is the same.
and 
.
such that
Prove that
derivable on 
with 
satisfied
for all 
and all 
.
be defined by 
(a) Show that 
has exactly one point of discontinuity. at its point of discontinuity.
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
just pick 
, otherwise 
is coprime to 
. By Chinese Remainder Theorem there exists an such that 
and 
, pick this , we see that both 
and leave remainder modulo , so we are done.
,
.
