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.
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.
+1 w
(use in a problem for junior) to practice. Can anyone share to me?
be a non-isosceles triangle with orthocenter 
, circumcenter 
. Let 
be midpoints of 
and 
be reflections of across 
. Prove that triangle 
have the same centroid.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
be circumscribed about circle 
, and let be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at . Prove that 
is tangent to .
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be the maximal number of the vertices of a convex lattice polygon with exactly lattice points in the interior. Show that:
for 
for some constant 
.
be a point lies inside the parallelogram such that 
.
are concyclic.
with scale 
. You'll see that circumcenter-formed quadrilateral is cyclic iff lines perpendicular to 
passing through , where 
form a cyclic quadrilateral. It's equivalent with 
Let 
be possibly other point than 
lying on line 
such that 
. Then 
so points 
are concyclic.
such that 
- that's just a translation of triangle 
to side . We have 
and 
so points 
form an isosceles trapezoid. Thus they are concyclic.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
