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.
is a positive divisor of other than itself.
consists of positive integers, each of which has at least three proper divisors. For each 
, the integer 
is the sum of the three largest proper divisors of 
.
.
+2 w
+1 w
grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
positive integers into 2 groups. Are there a 2 distinct number in the same group whose sum is a cube? Does the statement still hold true for 
.
of positive integers such that for any pair of positive integers 
, the arithmetic and geometric means![\[ \frac{a_m + a_{m+1} + \cdots + a_n}{n-m+1}\quad\text{and}\quad (a_ma_{m+1}\cdots a_n)^{\frac{1}{n-m+1}}\]](http://latex.artofproblemsolving.com/2/0/b/20b6f3097b2a8fa8e3fbed54af1f7f892f4ca2fc.png)
are both integers.
and 
be circles with centres 
and , respectively, such that the radius of is less than the radius of . Suppose and intersect at two distinct points 
and 
. Line 
intersects at 
and at 
, so that 
lie on in that order. Let 
be the circumcentre of triangle 
. Line 
meets again at 
and meets again at 
. Let 
be the orthocentre of triangle 
. parallel to is tangent to the circumcircle of triangle 
.
+2 w
which is known to both players. On the 
th turn of the game (starting with 
) the following happens: is odd, Alice chooses a nonnegative real number 
such that![\[
x_1 + x_2 + \cdots + x_n \le \lambda n.
\]](http://latex.artofproblemsolving.com/f/5/e/f5e4f751946952d021736c6eafa0b7ec02d02055.png)
[*]If is even, Bazza chooses a nonnegative real number such that![\[
x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
\]](http://latex.artofproblemsolving.com/f/f/4/ff40463336f4c5dc7381b14e71abe7d45b78a1c0.png)
[/list], the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players. for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
.
be a positive integer. There are 
knights sitting at a round table. They consist of pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.
where 
, 
are coprime positive integers. Show that for any integer 
, there exists 
positive integers , 
, 
, satisfying![\[\sum_{1\le i<j\le n}\frac 1{a_ia_j}=r.\]](http://latex.artofproblemsolving.com/c/1/4/c143afc6fa3bfadb257ad8ef8e339ac4648bd1ee.png)
Created by Xianbang Wang, High School Affiliated to Renmin University of China
where 
is