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.
such that there exists a sequence of positive integers 
, 
,
, 
satisfying: ![\[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\]](http://latex.artofproblemsolving.com/7/b/4/7b4ab849cdc409906514e2ebb688802f17df41ea.png)
for every 
with 
.
, there exists the positive integers 
, such that for 
, With 
may be formed as a triangle side length , and the area of the triangle is a positive integer.
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and be two fivedigit integers with the same permutation sum.
.
students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)
in it, and every other time, Henrik picks a triangle where he writes a 
. If one of the players gets four in a row in some direction (see figure), he wins the game.
on the diagonal 
of a given rectangle 
, such that 
. The point 
is the intersection point between 
and the parallel line to 
that contains 
. Prove that the triangle 
is equilateral if and only if is a square.
satisfies 
. Find the value of 
.
be a convex quadrilateral, such that
and 
. is a kite.
be a triangle with orthocentre 
. The tangent lines from 
to the circle with diameter 
touch this circle at 
and 
. Prove that 
and are collinear.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer . The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice. is one of the first prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
be a set of distinct positive even numbers and 
be a set of distinct positive odd numbers such that![\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]](http://latex.artofproblemsolving.com/5/d/f/5df36843d80d7f6a11401c2f6162ad10a00befb3.png)
Prove that![\[5m+12n\le 581.\]](http://latex.artofproblemsolving.com/8/8/b/88be1683ea815b49af99607e446dd7751130f527.png)
.
.
.
that satisfy the above equation lie inside the circle 
are 
(approx). which lie inside the circle but also satisfy 
.
,
,(since (m,n) must lie inside the circle)
,
,
for integer 
.
, which gives that 
to be any of 
. For this, just find a general solution and use the fact that 
. Also, needs to be odd as otherwise the whole LHS would end up being even. and we proved that it cant be any of 
, we must have that 
since all are distinct , now it implies 
, now 
is not possible.
, clearly has to be odd and 
which really doesn't give any such and to be working.
which also doesn't give any such and 
which also doesn't works.
. 
so
Now show 
as above. 
,
prove that 
Prove that
are distinct continuously.Sort 
and 
ascendingly.Then its easy to see 
and 
.
.
and 
adding up we get 
.
Then 
