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.
, are there infinitely many positive integers 
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
satisfying
of injective functions from the set of real numbers to itself satisfying
and 
. (We say a function 
is injective if 
for any distinct real numbers 
and 
.)
, find the smallest integer 
such that any graph with vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
{
} consisting 
distinct positive integers is 
if for every 
the number 
is divisible by the product of all numbers in except 
. Find all integers such that exists a set consisting of 
positive integers.
with 
and 
.
?
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
, such that for 
? be a cyclic quadrilateral for which 
. Construct two squares 
and 
externally on the sides 
and 
respectively.
meets 
at 
, 
meets 
at 
and 
denotes circumcenter of . Prove that 
and 
meets again on 
.
. Then, 
if and only if 
. Since 
, we have 
, so if 
where 
are primes, then 
. Suppose that not all of the are equal. Then, 
can only appear in the prime factorization less than times, so if it appears times, then 
, so since 
for 
, this means that 
, which is a contradiction. Therefore, appears times in the prime factorization of , so no other prime can appear. Therefore, if and only if 
for some prime 
. The smallest such number at least 
is 
.
to get![\[
g(N) = \sum_{1\le i\le k}\frac{\alpha_i}{p_i},\quad\text{where}\quad N = \prod_{1\le i\le k}p_i^{\alpha_i}
\]](http://latex.artofproblemsolving.com/7/a/1/7a17a49f1cb88c8b74defff78650ed5cbb8f4c2a.png)
for primes 
. Now, 
. Let be as above. Multiplying both sides by 
where 
, we get 
. Hence 
. From here, 
for some prime . From here, we immediately conclude 
to be the smallest such number.
![\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]](http://latex.artofproblemsolving.com/9/5/9/9591f7c842d1b6afcba11b15b3ea3f9e01bc937a.png)
![\[
2^a + 3^b + 5^c = n!
\]](http://latex.artofproblemsolving.com/b/c/6/bc6b0754dd874f646fcab124c91583b620db273c.png)
be a function such that
for every prime 
for every pair of positive integers 
such that 
