AoPS Academy
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, 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 sequence of positive integers with 
, 
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
. Show that if 
is a prime less than 
for some positive integer 
, then there exists 
such that 
.
be a triangle with circumcircle 
, incentre 
. The second intersections of 
, 
, and 
with are 
, 
, and 
, respectively. Lines and 
intersect at 
and lines 
and 
intersect at 
. Suppose 
and 
intersect again at 
. Lines 
and 
intersect the circumcircle of triangle 
again at 
and 
, respectively.
lies on 
.
for which
(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of less than is the difference between two numbers on the sheet, and no other positive number is.
be a square with 
side length and with center 
. A rhombus 
is drawn where point 
lies on line 
. What is the area of ?
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist square that is colored in chess order.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
, such that each of its sides contains a vertex of the polygon. Then there are two diagonals 
such that 
and 
, so that 
, double the area of the rectangle, at least as large as double the area of the polygon.
, this value appears only once in the set (no repetition of equal values) of the lengths of the segments joining the vertices of the polygon, so more work is needed.
be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of 