and 
.
Let 
.
Let 
.
Let 
.
does the following system of equations has a solution other than 
in the set of real numbers? 
be an acute triangle, and let 
and 
-
-
and 
are selected on 
such that 
and 
.
and 
are selected on 
such that 
and 
.
and 
meet at 
.
and 
are perpendicular.
plot 
equally spaced points around a circle. Label one of them 
,
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at 
for all 
.
Y by
p1. The sequence of numbers 
is obtained by writing the positive integers in order, one after the other. What position is 
in the first time does 
appear in succession?
p2. Consider a square in the plane with vertices
where 
, 
are integer numbers for each 
. Suppose the area of the square is a power of 
. Prove that its sides are parallel to the axes.
p3. Prove that for every integer
, it is true that 
p4.
is a triangle of area 
with circumcenter 
and 
is the midpoint of 
. We choose the points 
on the sides 
and 
respectively such that is at 
and segments 
and are parallel. Suppose the area of the triangle 
is 
. Calculate the angle 
.
is obtained by writing the positive integers in order, one after the other. What position is 
in the first time does 
appear in succession?p2. Consider a square in the plane with vertices
where 
, 
are integer numbers for each 
. Suppose the area of the square is a power of 
. Prove that its sides are parallel to the axes.p3. Prove that for every integer
, it is true that 
p4.
is a triangle of area 
with circumcenter 
and 
is the midpoint of 
. We choose the points 
on the sides 
and 
respectively such that is at 
and segments 
and are parallel. Suppose the area of the triangle 
is 
. Calculate the angle 
.This post has been edited 3 times. Last edited by parmenides51, Dec 21, 2022, 11:13 PM
similar to ![$[ABC]=4[APQ]$](http://latex.artofproblemsolving.com/e/7/4/e7418edc234f799d3e824bba5b40a828a9f3f065.png)
,
and 
.
.
