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 
integers which satisfy![\[ a_{n+5}+a_n>a_{n+2}+a_{n+3}\]](http://latex.artofproblemsolving.com/8/8/b/88bbb2f304525a3261851571d34863bf4d4a1e51.png)
for all integers 
. Find the minimum possible value of the difference between the maximum value and the minimum value among .
be a cyclic quadrilateral inscribed in circle 
whose diagonals meet at 
. Lines 
and 
meet at 
. Segment 
intersects at 
. Lines 
and meet at 
, and lines 
and meet at 
. Prove that 
and 
concur with the tangent to at .
, point 
lies on side and point 
lies on side 
. Let 
and 
be points on segments 
and 
, respectively, such that 
is parallel to . Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.
be the circumcircle of a triangle 
. A circle passing through points 
and 
meets the sides 
and 
at 
and 
, respectively. The lines 
and 
meet again at 
and 
, respectively. The tangent lines of at and meet the line 
at 
and 
, respectively. Prove that the lines 
and 
meet at .
is the orthocenter of an acute triangle , and let be the midpoint of . Suppose 
meets and at 
respectively. 
meets 
at , and the line through perpendicular to meets 
at . Prove that 
are collinear.
prime number, there exists only one positive number 
that makes the equation 
a perfect square of a positive integer
be a real-valued function defined on the set of real numbers that satisfies![\[f(x + y) \leq yf(x) + f(f(x))\]](http://latex.artofproblemsolving.com/c/9/4/c94075d26e09712f5393fb66ecbfdc2573c239b6.png)
and 
. Prove that 
for all 
.
, 
. The internal angle bisector of 
and the external angle bisector of intersect the ray at points 
and , respectively. Given that 
, prove that 
.
be an acute triangle, and let 
and be its circumcenter and orthocenter, respectively. For 
, points 
and 
lie on lines 
and 
(where 
), respectively, such that 
is a parallelogram. Prove that![\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]](http://latex.artofproblemsolving.com/b/a/5/ba563336236b6a04fc9ef661760fa1251bfb07d6.png)
in this order. Prove that![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](http://latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
boxes 
, 
, 
,..,
. The -th box 
contains balls.
greater than 
and choose 
a multiple of .
, 
, 
and move the 
balls to the box 
.?
be nonnegative real numbers such that 
. Prove that![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
be denoted by 
respectively. The condition 
implies that 
is the bisector of 
Therefore, 
and 
Using the law of cosines in 
and 
we see that 
and 
are the solutions of the quadratic 
and therefore 
However, the area of the quadrilateral is given by 
.
,
,
,