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.
and 
. Prove that:
be an integer and 
be a real number. Feeling discouraged while taking the mock IMO, Ma Pu Kywae looks outside the window and observes 
snails labelled 
climbing up a fence of height 
units. Initially, all the snails start from the ground, and each snail 
steadily climbs up at the speed of 
units per minute until it reaches the top. At the end of each minute, one of the snails whose height is strictly less than drops onto the ground, resetting its journey up the wall. so that no matter how the snails drop, at least one will eventually make it to the top.
and have answers
. Define set![\[V_n:=\{(a_1,\cdots ,a_n)\in\mathbb R^n\mid a_1,\cdots ,a_n\ge 0\text{ and }a_1+\cdots +a_n=1\}.\]](http://latex.artofproblemsolving.com/f/c/1/fc1a6e5a65337f92ad93929e2f47c2195fc467f9.png)
For 
, 
, define 
, such that for all 
, there exists 
such that 
.
. We say a real sequence 
satisfies a linear recurrence of order 
if is the least positive integer for which there exist real constants 
such that![\[
u_{k+d} \;=\; c_1\,u_{k+d-1} + c_2\,u_{k+d-2} + \cdots + c_d\,u_k
\quad\text{for all }k\ge0.
\]](http://latex.artofproblemsolving.com/8/c/1/8c17b119134a2e8ee1efb1b77d319e732997e882.png)
In this case we call![\[
m_u(x)\;=\;x^d \;-\;c_1x^{d-1} \;-\;\cdots-\;c_d
\]](http://latex.artofproblemsolving.com/0/b/9/0b96010e4cfa56b389a6aea0e0678acd983636d5.png)
the minimal polynomial of the sequence, of degree .
satisfies such a recurrence of order , then 
. and 
be two sequences whose minimal polynomials 
and 
have no common root. Prove that the sequence 
satisfies a recurrence whose minimal polynomial is![\[
m_{u+v}(x) \;=\; m_u(x)\,m_v(x),
\]](http://latex.artofproblemsolving.com/0/0/1/0018a78bbfe0132cf7cb526cd3a2868947884e63.png)
in particular 
.
whose minimal polynomial has degree exactly .
be a triangle. The points 
and 
lie on the segments 
and 
respectively, such that the lines 
and 
intersect in a common point. Prove that it is possible to choose two of the triangles 
and 
whose inradii sum up to at least the inradius of the triangle .
and 
such that
where 
.
is a friend of 
, then is a friend of .) designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a 
-best friend. More generally, if 
is a positive integer, then a user is an -best friend provided that they have been designated the best friend of someone who is an 
-best friend. Someone who is a -best friend for every positive integer is called popular.
, is it possible to obtain the number 
by repeating this step finitely many times?
be an acute triangle with 
. Let 
be the reflection of point over the midpoint of segment 
. Points 
and 
lie on the same side of 
as and satisfy 
and 
.
and 
are tangent. Prove that 
.
+1 w
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
be the circumcenter and 
the orthocenter of an acute triangle . Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
