[/quote]
,
such that 
is a divisor of 
.
,
and 
.
is prime, 
with 
and 
.
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
,
,
.
is parallel to 
,
is the midpoint of segment 
and 
are such that 
is a multiple of 
and 
is a multiple of 
and
meaning 
is the least common multiple of the numbers 
.
such that
for every positive integer 
be positive integers with 
.
,
,
,
,
,
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of 
be an acute triangle with circumcenter 
.
,
on side 
.
passing through 
at point 
and 
is mundane if there exist positive integers 
such that 
is a positive integer. Find, with proof, all prime numbers that are not mundane.
that has the following property: Any 
real numbers, not necessarily distinct, that lie in the interval ![$[100,101]$](http://latex.artofproblemsolving.com/8/e/3/8e3fe24e3b899bbf2128eebe08756e1ff963aaa0.png)
,
and 
such that
.
, 
and 
.
.
-
board.
board consisting of 
unit segments. A configuration of several 
Y by
Points 
lie on the sides 
of a square 
, respectively, such that the area of 
is equal to one half of the area of . Prove that some diagonal of is parallel to some side of .
Proposed by Josef Tkadlec - Czech Republic
lie on the sides 
of a square 
, respectively, such that the area of 
is equal to one half of the area of . Prove that some diagonal of is parallel to some side of .Proposed by Josef Tkadlec - Czech Republic
This post has been edited 1 time. Last edited by parmenides51, Jan 25, 2022, 1:54 PM
Let ![$[P]$](http://latex.artofproblemsolving.com/8/2/c/82c8af2036036acc93984fa2d3e7fac5cb926624.png)
denote the area of a polygon 
. Suppose that 
and let 
be the point on 
such that 
Note that ![$[KLMN'] = \frac12 [ABCD].$](http://latex.artofproblemsolving.com/4/a/b/4abc3884aacb3a221d73a2436fa862d4c96e423d.png)
Thus ![$[KLMN] = [KLMN']$](http://latex.artofproblemsolving.com/f/1/0/f10e8c02da11febfc27a4d8eb6703d5828bef33b.png)
,![$[KMN] = [KMN']$](http://latex.artofproblemsolving.com/5/7/1/5719500dcf2cad6e13860c6d5e0246e7e5ad6d72.png)
implying that 
Denote the length as on the figure and let the side of 
The sum of the areas of the right triangles in the corners is 
and so 
which can be modifier into 
and so 
or 
which gives the desired conclusion.
![$[DMN]+[MCL]+[LBK]+[KAN]=\frac{a^2}{2}$](http://latex.artofproblemsolving.com/2/b/a/2ba2c394a764edf75440bc672904de81c9f00777.png)
or 
and 
which makes 
a parallelogram and hence 
lie on line 
such that 
and 
so 
is parallelogram.
