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.
such that 
is a perfect square.
+1 w
and 
be non-zero integers such that 
Prove that ![\[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]](http://latex.artofproblemsolving.com/b/f/f/bffd77e507fc4ddc8368c8239c523a4df40448c4.png)
is a perfect square.
be positive real numbers.Prove that![$$\sqrt[3]{ab}+\sqrt[3]{cd}\leq\sqrt[3]{(a+b+c)(c+d+a)}.$$](http://latex.artofproblemsolving.com/b/7/9/b796d0f9a2d9430bf5efb83b901427b13386f478.png)
When equality holds?
be an integer. Ion draws a regular -gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the -gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.
with incentre 
and excentres 
and 
, opposite the vertices 
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than .
and 
, then find the maximum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
segments and 
are parallel. Angle bisectors of 
and 
meet at 
. Angle bisectors of 
and 
meet at 
. Prove that 
is cyclic
.
be positive real numbers such that 
and 
, prove that
. Points 
and 
lie on segments and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point . The circumcircles of triangles and 
intersect at a second point . Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
satisfy![\[
a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015}
\]](http://latex.artofproblemsolving.com/8/c/0/8c0b0cfdd569ea250e97590a3364794920d86e04.png)
and 
. Determine the value of![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
(in degrees).
that satisfy 
safisty: 
=
wich belongs to set A ?
.
The RHS becomes 
,
,
Thus, 
.
,
.
and 
.
,
.
,
.