[/quote]
,
doesn't have any solution on integers.
is strictly increasing : 
such that 
for all 
be a polynomial with real coefficients such that 
for all 
.
for some real constant 
and 
.
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
.
chords, one of which has length 2025. Find the minimum possible value of 
.
order, and if the constituents of its first, second, ..., 
figurate numbers of the first, second, ..., 
for which 
is a perfect square.
be a hyperboloid of one sheet and let 
be another quadric that intersects the hyperboloid at the curve 
.
be a point on 
be a line through 
,
to be the line through 
.
then this is true for all initial choices of 
be positive integers such that 
is a square. Prove that 
is not prime.
be a real polynomial of degree 
that has only simple roots. It is possible to determine a positive quantity 
so that for every pair of positive real numbers 
,
with 
,
has exactly 
or 
sign variations.
such that 
Y by Adventure10
Let 
be real numbers. Consider the following curves on 
plane.
![\[ y=e^{|x|}\ \ \ \ \ \ \ \ \ \ [1]\]](//latex.artofproblemsolving.com/2/6/9/269e966da23aef955a98054744956624fce0f0ca.png)
![\[ \ \ \ \ \ \ y=ax+b\ \ \ \ \ \ \ [2]\]](//latex.artofproblemsolving.com/1/4/a/14ab8d9a27c07c7b155d586708553b16d4365f01.png)
(1) Suppose those curves have a common tangent line, express
interms of 
, sketch the graph on 
plane.
(2) Denote
the graph of (1). For a constant 
, find the maximum value of 
and the value of giving the maximum value.
be real numbers. Consider the following curves on 
plane.![\[ y=e^{|x|}\ \ \ \ \ \ \ \ \ \ [1]\]](http://latex.artofproblemsolving.com/2/6/9/269e966da23aef955a98054744956624fce0f0ca.png)
![\[ \ \ \ \ \ \ y=ax+b\ \ \ \ \ \ \ [2]\]](http://latex.artofproblemsolving.com/1/4/a/14ab8d9a27c07c7b155d586708553b16d4365f01.png)
(1) Suppose those curves have a common tangent line, express
interms of 
, sketch the graph on 
plane.(2) Denote
the graph of (1). For a constant 
, find the maximum value of 
and the value of giving the maximum value.
yields 
,
be tangent to it, then we have:
(for 
)
be the tangency point, then:![$y-a = a[x-\ln(a)] \Rightarrow y=ax+a[1-\ln(a)] \Rightarrow \textcolor{red}{b = a[1-\ln(a)]}$](http://latex.artofproblemsolving.com/c/5/6/c56ca8aee271d3b18fc52ba0c9e6eaa5c6839dde.png)
.![$g(a) = pa + a[1-\ln(a)]$](http://latex.artofproblemsolving.com/b/9/9/b99baad0278f99449dbb5ec26a019bdc2d0e1285.png)
for some real constant 
equal to zero yields:
;
(a maximum).
.
