-
Solve the equation
-
The function
has
derivatives. Show that if
and
then we can find
such that
.
-
Let
.
For, let
where the sum ranges over all pairs
of positive integer satisfying the indicated inequalities. Evaluate:
.
- This problem deals to elementary functional analysis and is taken from very old paper of Putnam Competitions.
has continuous second derivative and
,
for all
.
are integers such that
are all integers. Put
and
.
•Prove that
.
• Show that for, there are atmost
such indices
.
• Show that there are atmostlattice points on the curve
.
![{(\dfrac{1}{10})}^{\log_{\frac{x}{4}} {\sqrt [4] {x} -1}} -4^{\log_{10} {\sqrt [4] {x} +5}} =6, \forall x \ge 1](http://s0.wp.com/latex.php?latex=%7B%28%5Cdfrac%7B1%7D%7B10%7D%29%7D%5E%7B%5Clog_%7B%5Cfrac%7Bx%7D%7B4%7D%7D+%7B%5Csqrt+%5B4%5D+%7Bx%7D+-1%7D%7D+-4%5E%7B%5Clog_%7B10%7D+%7B%5Csqrt+%5B4%5D+%7Bx%7D+%2B5%7D%7D+%3D6%2C+%5Cforall+x+%5Cge+1&bg=ffffff&fg=333333&s=0 "{(\dfrac{1}{10})}^{\log_{\frac{x}{4}} {\sqrt [4] {x} -1}} -4^{\log_{10} {\sqrt [4] {x} +5}} =6, \forall x \ge 1")
has 
derivatives. Show that if 
and ![\log [f(b)+f'(b)+f"(b)+ \ldots +f^n(b)] - \log [f(a)+f'(a)+f"(a)+ \ldots +f^n(a)] =b-a](http://s0.wp.com/latex.php?latex=%5Clog+%5Bf%28b%29%2Bf%27%28b%29%2Bf%22%28b%29%2B+%5Cldots+%2Bf%5En%28b%29%5D+-+%5Clog+%5Bf%28a%29%2Bf%27%28a%29%2Bf%22%28a%29%2B+%5Cldots+%2Bf%5En%28a%29%5D+%3Db-a&bg=ffffff&fg=333333&s=0 "\log [f(b)+f'(b)+f\"(b)+ \ldots +f^n(b)] - \log [f(a)+f'(a)+f\"(a)+ \ldots +f^n(a)] =b-a")
then we can find 
such that 
.
.
, let 
where the sum ranges over all pairs 
of positive integer satisfying the indicated inequalities. Evaluate:
.
![f: [0, \mathbf{N}] \to \mathbf{R}](http://s0.wp.com/latex.php?latex=f%3A+%5B0%2C+%5Cmathbf%7BN%7D%5D+%5Cto+%5Cmathbf%7BR%7D&bg=ffffff&fg=333333&s=1 "f: [0, \mathbf{N}] \to \mathbf{R}")
has continuous second derivative and 
, 
for all 
.
are integers such that 
are all integers. Put 
and 
.
.
, there are atmost 
such indices 
.
lattice points on the curve 
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