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is a square.
such that there is a polynomial 
with integer coefficients for which 
and where 
is a prime number .
letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
defined on the positive reals such that 
for all 
.
such that 
is a perfect square, i.e., the square of an integer.
,
.
such that 
,
.
satisfying the system of equations:
,
.
is given by 
.
without satisfying 
.
Y by
In the center of the square of side 
shown in the figure is an ant. At one point the ant starts walking until it touches the left side 
, then continues walking until it reaches the bottom side 
, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of 
.
![[asy]
unitsize(2 cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$a$", (0,0.5), W);
label("$b$", (0.5,0), S);
dot((0.5,0.5));
[/asy]](//latex.artofproblemsolving.com/texer/g/guvxiblm.png)
shown in the figure is an ant. At one point the ant starts walking until it touches the left side 
, then continues walking until it reaches the bottom side 
, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of 
.![[asy]
unitsize(2 cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$a$", (0,0.5), W);
label("$b$", (0.5,0), S);
dot((0.5,0.5));
[/asy]](http://latex.artofproblemsolving.com/texer/g/guvxiblm.png)
This post has been edited 6 times. Last edited by parmenides51, Jun 1, 2024, 11:38 AM
