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 positive integer. A positive integer 
is called a -good number if it satisfies has exactly 
digits in decimal representation (it cannot have leading zeros). digits and the last digits of are considered as two separate -digit.
is a 
-good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
, determine 
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
be a fixed positive real number. Find the minimum possible value 
such that for any positive reals 
, 
, 
, 
, we have
and with no common divisor greater than 1 such that 
divides 
. (Professor Yongjin Song)
and 
. Prove that there always exists a positive integer 
such that 
. is a power of a prime number, but for arbitrary then well.....
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, , with , . Prove that there exists such that for every (For and , the notation represents . )
be the circumcircle of acute triangle 
. Points 
and 
are on segments 
and 
respectively such that 
. The perpendicular bisectors of 
and 
intersect minor arcs and of at points 
and 
respectively. Prove that lines 
and 
are either parallel or they are the same line., , with , . Prove that there exists such that for every (For and , the notation represents . )
is a positive real number, find the minimum of the following expression
such that 
divides 
for any integers 
and there exists a polynomial 
with integer coefficients, such that 
for all 
.
also divides .
such that 
but 
. Then, we can choose some 
such that 
. This implies 
, leading to 
, a contradiction.
must be a power of for all primes . Since 
for all 
, there exists some 
such that 
for infinitely many primes . It follows that 
for all , where is odd.
infinitely often along primes, let's justify the last step why 
for 
. Taking 
and keeping fixed, 
implies that 
. Using 
(the inverse exists as 
), we obtain![\[
p^k f(n) + 1\equiv (-1)^k \frac{f(n)}{n^k}+1\pmod{pn+1}\Rightarrow pn+1\mid \left|f(n)\pm n^k\right|.
\]](http://latex.artofproblemsolving.com/f/3/9/f392fe8bd086ec7abc433e8d35bb97f55e61f065.png)
By taking 
large, we are left with (as 
). A byproduct of this argument is that need be odd.
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
