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.
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
be a function satisfying the following conditions:
, if 
and 
, then 
;
is surjective.
.
be a triangle with 
and 
. Let 
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
denote the number of ways this can be done. For example, clearly 
because we can have either a red or a blue tile. Also 
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
for all 
.
for all 
.![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](http://latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
. such that the following statement holds: Suppose real numbers 
, 
, 
, 
, 
, 
, , 
satisfy 
for all 
. Then there exists 
, 
, , 
, each of which is either 
or 
, such that![\[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]](http://latex.artofproblemsolving.com/6/5/8/658ddc5e8626e7d5ef29809c8c3ac641e3a87757.png)
. The center 
of the semi-circle is the midpoint of segment 
. We have 
and 
..
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are vertices 
such that if the numbers 
are assigned to them respectively, then 
and is a parallelogram.
be the roots of the equation 
. What is the value of the expression
?
such that
Proposed by Ngo Van Trang, Vietnam
such that 
. Find the value of 
and 
.
and 
such that 
is 
, then find the value of
.
be the set of integers. Determine all functions 
such that for all integer :
of a triangle are positive integers. Let
for any positive integer .
and 
, determine all possible perimeters of the triangle .
has two roots, where one is twice of the other, find all possible values of .
Find the value of 
such that 
is an integer.
is 
in triangle , find the ratio of the side length 
to the side length 
.
and the points 
.
, then 
with 
and 
.
intersects the line 
in the point 
.
intersects the circle in the point 
.
intersects the line 
in the point 
.
is the midpoint of ![$[DE]$](http://latex.artofproblemsolving.com/4/f/5/4f55b2be1d3d9963afec61b4973bfecc6141b1ff.png)
.
