[/quote]
,
such that the number 
is a perfect square for all positive integers 
.
and 
be two similar right-angled triangles (in the same plane) with 
such that 
and 
lie on opposite sides of the line 
.
,
.
.
2/ Let 
![$f:[0,1]\rightarrow [0,1].$](http://latex.artofproblemsolving.com/d/0/7/d076b5242cc57c3c4af7971c9cb104023effca65.png)
I'm interested in what solutions there are other than constant functions.
for which there exists an integer 
such that 
is a divisor of 
.
where they are integers and 
,
are in a gp.
?
. Prove that
board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a 
square to another midpoint of the vertical segment, according to the rules:
When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from 
to 
)
When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from 
to 
When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from 
)
board.
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
is a triangle where 
is a point on the side 
such that 
and 
,
.
is extended to 
such that 
be an acute triangle with 
,
of side 
,
)
.
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
![\[ a_{m^2 + m + n} = a_{m^2} + a_m + a_n\]](http://latex.artofproblemsolving.com/7/4/4/7447c19e23f40b093bb4cd0cb85a931398570210.png)
for all positive integers 
Y by
Given a finite set 
, and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to 
inequalities, these given comparisons satisfy the following three constraints:
1. The sum of elements of any non-empty subset is greater than 0.
2. For any two subsets, removing or adding the same elements does not change their comparison of the sums of elements.
3. For any two disjoint subsets
and 
, if the sums of elements of and are greater than those of subsets 
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
The question is: Does there necessarily exist a positive solution
that satisfies all these conditions?
, and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to 
inequalities, these given comparisons satisfy the following three constraints:1. The sum of elements of any non-empty subset is greater than 0.
2. For any two subsets, removing or adding the same elements does not change their comparison of the sums of elements.
3. For any two disjoint subsets
and 
, if the sums of elements of and are greater than those of subsets 
and 
respectively, then the sum of elements of the union 
is greater than that of 
.The question is: Does there necessarily exist a positive solution
that satisfies all these conditions?This post has been edited 2 times. Last edited by bobaboby1, Mar 12, 2025, 3:28 PM
Reason: Wrong typing.
Reason: Wrong typing.
?
