[/quote]
,
be an acute scalene triangle 
be the orthogonal projection of 
on 
,
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
,
,
are positive integers that satisfy
and 
.
such that
be real positive numbers such that: 
.
:
such that:
for all x,y E 
,
,
,
be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, 
.
for all 
.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
be a trapezoid inscribed in circle 
,
.
intersects 
.
is orthocenter of triangle 
take a point 
so that 
.
coincide
be integer numbers 
satisfy that
.
is a perfect square number
can’t be any power 
of a positive integer number
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
is given where 
with the following property:
number 
.
and 
and 
Find the minimum 
satisfying![\[f(x+f(y))=x+f(f(y))\]](http://latex.artofproblemsolving.com/8/d/a/8daa6bcb3155768ee1ba4bd22beb53a85a6e3923.png)
and 
,
.
Y by Mango247
The desert is divided into square cells, infinite on all sides one of which is a lion. The hunter sets up fences, each of which divides 
neighboring cells of the desert, and the lion after setting each subsequent fence runs to one of the neighboring cells (ie with a common side, while the lion can not jump over the fence). In order to hunt a lion, the hunter must completely enclose some cell of the desert, which is a lion. Can he do this with restriction to install 
fences?
(Bryman W.)
neighboring cells of the desert, and the lion after setting each subsequent fence runs to one of the neighboring cells (ie with a common side, while the lion can not jump over the fence). In order to hunt a lion, the hunter must completely enclose some cell of the desert, which is a lion. Can he do this with restriction to install 
fences?(Bryman W.)
This post has been edited 4 times. Last edited by parmenides51, Apr 26, 2021, 11:47 AM
Reason: removed original wording
Reason: removed original wording
fences. To do this consider a 
square with the lion at the center. In teh first move the hunter sets a fence in one of the two borders of the four corners, so that in every border cell there is only one way out of the square. When the fourth turn is completed the lion is at most in a border cell (since he can perché at most 4 cells). So now the hunter can always block the exit from the border cell the lion is in, and if it is already blocked, he can advantage himself and block some other cell. Eventually, the hunter will complete the perimeter 
fences, and the lion will be inside.
,
,
,
,
,
,
and finally 
(in the worst cases when the lion goes in the bigger part), by always splitting the next rectangle in two rectangle as evenly as possible. In this way we use at most 
fences.
