[/quote]
,
is chosen inside a convex quadrilateral 
.
be non-negative real numbers such that 
Prove that
consecutive positive integers were written on the board, where 
.
of the written numbers were erased, and it turned out that any two of the remaining numbers are coprime. Find the largest possible value of 
and 
are equal, and they intersect at 
.
and 
intersect at point 
,
and 
intersect at point 
lying inside triangle 
.
.
be a positive integer. Egor has 
cards with the number 
”
”
.
be a point on side 
be a point on the extension of 
such that 
.
be the circumcenter of 
,
satisfying 
.
children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
,
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:
rectangle, one of the numbers is the arithmetic mean of the other two.
is located in the middle cell of the table.
,
and 
.
.
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
Y by
Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$
One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
