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.
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](http://latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to 
and the line segments 
and 
are concurrent.
that satisfies: for all different 
, we have 
?
be a triangle with circumcenter 
and internal bisector 
. Let cuts 
again at 
and 
cuts again at 
. Point 
lie on such that 
. The line pass through and perpendicular to intersects 
at 
respectively. Let circumcircle of 
cuts at 
. Prove that 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist. and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
be the number of spanning trees in a graph 
, where may have multiple edges and loops.
and 
of , let 
, 
, and 
denote the graphs obtained by contracting the edges , , and both and in 
, respectively.![\[
k(G/\{e,f\}) \cdot k(G) \leq k(G/e) \cdot k(G/f)
\]](http://latex.artofproblemsolving.com/7/8/2/78296088b860654fe0310b75c3aaeaf980789cef.png)
over the circle in some order so that among any 
successive numbers at least 
numbers are multiplies of 
or 
(or both and ) ?
where a and b are relatively prime integers, what is b − a?
be the set of real numbers. Determine all functions 
such that 
and for any real numbers 
and 
,
Let 
and 
is the sum of 
square numbers. Prove that: 
is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
be equal to divided by the largest power of 
that divides . Legendre's three-square theorem says that is expressible as the sum of three squares iff is not mod .
is never mod , so even always works. If is odd, if 
is mod , then 
is even and thus 
is even. Thus 
is odd. Observe that if is odd then 
. Thus if is mod then 
is mod . However, any odd number to an odd power is congruent to itself mod , so this means 
, contradiction.
, which would make it trivial.
satisfying
Prove
the following equation :
