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.
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.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
+1 w
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides 
and 
, respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by 
the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex 
is denoted by 
. Prove that 
.
![\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 
.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
+1 w
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 ) ?
and 
are same then obviously your answer is wrong. Actually the answer would be 
I guess. Trying to prove.....
case is trivial which gives us 
....and the rest as follows
doesn't necessarily divide that part....
be a prime such that 
for 
. Using![\[
a^3+b^3+1 - 3ab = (a+b+1)(a^2+b^2+1-ab-a-b)
\]](http://latex.artofproblemsolving.com/e/9/7/e978baa1bbf867ec90243096ff0c1edf3725f438.png)
we get 
. Suppose first that 
(the case 
is symmetric). Then, 
forces 
which together with 
forces 
. So, 
, where 
are arbitrary are both solution families.
. Then 
and 
. Since 
, we must have 
, giving the family 
where 
are arbitrary.
satisfying
Prove
the following equation :
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
.
,
,
,
,
.
divides 
and 
,
,
for all 
for some non-negative integers 
.
.
is odd then one of this numbers has to be even and one odd and then this pair works in this problem 
)
.
.
and