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.
and 
be elements of the set 
such that 
, 
is a divisor of 
, and 
is a divisor of 
. Determine the largest value that 
can take.
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
have the property that 
is a perfect square for infinitely many positive integers 
?
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
. Points 
are chosen on , such that 
, and point 
is the foot of the altitude from 
to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
that is not isosceles at 
, let 
be its incircle, which is tangent to 
at 
, respectively. The lines 
and 
intersect the line passing through and parallel to 
at 
and 
, respectively. The lines passing through 
and perpendicular to 
intersect 
and 
at 
and 
, respectively.
are collinear and that 
are concurrent.
positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining 
numbers that are not hidden. What is the maximum number of the written numbers that can be perfect squares?
is a prime number and 
are positive integers. Prove that there exist infinitely such that 
are divisible by 
, let 
if is not squarefree and 
if is a product of 
primes, and let 
be the sum of the divisors of . Prove that for all we have![\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]](http://latex.artofproblemsolving.com/8/e/b/8eb88de7cd7ea8e2853dbc80cae24bdc6b77f9be.png)
be a sequance of natural integers. Proof that, any number in the sequance can be written only as or 
for any 
and not necessarily distinct primes 
for 
. And and 
are same in some case.
has no integer solutions
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and be two fivedigit integers with the same permutation sum.
.
(where are the digits and ) we define the \textit{permutation sum} as the value For example the permutation sum of 20253 is Let and be two fivedigit integers with the same permutation sum..
![\[
S(n) = 1111a_{n} + 10111b_{n} + 11011c_{n} + 11101d_{n} + 11110e_{n}
\]](http://latex.artofproblemsolving.com/d/c/6/dc68086c322208f7443e2192f623d3e1e06efe09.png)
![\[
S(m) = 1111a_{m} + 10111b_{m} + 11011c_{m} + 11101d_{m} + 11110e_{m}
\]](http://latex.artofproblemsolving.com/0/a/f/0af218635db948bbca114f174d600bb3945d9c76.png)
Also,![\[
S(n) + n = 11111(a_{n} + b_{n} + c_{n} + d_{n} + e_{n})
\]](http://latex.artofproblemsolving.com/1/0/3/10363f5e9a09ae68291dc24dee8347d9bae6685e.png)
![\[
S(m) + n = 11111(a_{m} + b_{m} + c_{m} + d_{m} + e_{m})
\]](http://latex.artofproblemsolving.com/8/8/c/88ca334b50162649bad05531dd2f89588f8b713f.png)
Since,![\[
S(n) = S(m)
\]](http://latex.artofproblemsolving.com/b/9/4/b94b10e0be06bdf17e09a93a9ae40e51ad0b459f.png)
![\[
\implies 11111(a_{n} + b_{n} + c_{n} + d_{n} + e_{n}) - n = 11111(a_{m} + b_{m} + c_{m} + d_{m} + e_{m}) - m
\]](http://latex.artofproblemsolving.com/f/e/d/fed32efe41634ce352403707b77dea0a3aa119e8.png)
![\[
m-n = 11111((a_{m} + b_{m} + c_{m} + d_{m} + e_{m})-(a_{n} + b_{n} + c_{n} + d_{n} + e_{n}))
\]](http://latex.artofproblemsolving.com/a/a/2/aa2f615fc83b016c8ee6dac955e0005dd048fcc6.png)
![\[
\implies m \equiv n \pmod{11111}
\]](http://latex.artofproblemsolving.com/7/f/9/7f95c981e2e2d4e4191b8c7ae8ea9f0711da5b06.png)
And,![\[
S(n) \equiv 4n \pmod{9}
\]](http://latex.artofproblemsolving.com/0/7/5/075a303409c5dbfaf66d9ac3afee9c632b43dc8a.png)
![\[
S(m) \equiv 4m \pmod{9}
\]](http://latex.artofproblemsolving.com/9/0/1/90109fa1fc52df75576af6150ad79937a27b01a7.png)
So,![\[
4m \equiv 4n \pmod{9}
\]](http://latex.artofproblemsolving.com/c/5/8/c5803068a7342d178221c049fde5f2a05550e7f1.png)
![\[
m \equiv n \pmod{9}
\]](http://latex.artofproblemsolving.com/4/b/4/4b4e83800ad64d3169f5a412d80a139ccf56d0d9.png)
Now, Using Chinese Remainder Theorem,![\[
m \equiv n \pmod{99999}
\]](http://latex.artofproblemsolving.com/d/c/b/dcb4eb91378d92060f4f3498194d27b659397ba1.png)
![\[
\boxed{m=n}
\]](http://latex.artofproblemsolving.com/5/8/4/5841b978a06e6e866677dc956130dcafcd7108d3.png)
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
passing through 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
satisfying: 
.
![\[2^{xy}f(xy-1)+2^{x+y+1}f(x)f(y)=4xy-2,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/d/1/d/d1de7196fd3ba60732c14fa2b156ec880b4829d0.png)
denote the digital sum. Then 
for two different 
and 
if they have the same permutation sum. But observe that 
must be 
.
,
,
.
is considered unique. Thus, only one value can be given for