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.
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)
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.
such that 
. Show that 
. When will equality hold?
.
has 
obtuse. Let , , be the feet of the altitudes from 
, 
, 
respectively. Let 
, 
, 
be arbitrary points on 
, , 
respectively. The circles with diameter 
, 
, 
are drawn. Show that the lengths of the tangents from the orthocentre of to these circles are equal.
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that Prove that
Equality holds iff 
or 
and any cyclic permutations.
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be positive real variables with 
Prove that
Equality holds iff 
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
be non-negative real variables with Prove that
Equality holds iff or 
where 
such that 
.
.
and 
.
,
and 
be non-negative numbers such that 
.
and set 
The inequality becomes
ưhere
which ends the proofs 