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I'll present my choice of challenging problems here.

Most of the problems are from math analysis so I put thread here.

I've chosen them from various sources like college competitons,journals

or textbooks.Some of the problems are well known,and some are

new and original.Level of difficulty vary among them,but no problem

is trivial.I expect that really really good math students could solve about

50% of the problems.And if you are math analysis begginer you will hardly

solve more than 2 problems.I showed the problems to one assistant professor

and he told me that even a college math professors,with the help of

reference materials ,and knowledge, would be hardly able to solve them all.

If you're mathematician or professor ,I'm looking forward to see your feedback: how do you rate each of these problems (from easiest to the hardest)?

Finally ,if you want, feel free to post your solution (if you think you have an elegant one).

Problems

I.

Let the function of real parameter u be defined by the improper integral:

[tex]f(u)=\int_{0}^{\infty}\frac{dx}{e^{x^{u}}}[/tex]

a)Calculate [itex]\lim_{u\to\infty}f(u)[/itex]

b)Given is the series:

[tex]\sum_{n=1}^{\infty}\frac{(2n-1)!!}{f(1/n)}(\frac{x}{1+x^2})^2[/tex]

Find radius of convergence and sum of the series when it converges.

II.

Calculate the sum:

[tex]\sum_{k=0}^{\infty}\frac{1}{k^4+\frac{1}{64}}[/tex]

III.

Prove the identity:

[tex]\int_{0}^{1}x^x dx=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^n}[/tex]

IV.

Let B be a Borel-measurable subset of {z[itex]\in\mathbb{C}[/itex]:0<|z|<1}.

Prove that

[tex]-1\leq\int_{B}\frac{z}{|z|}dz\leq 1[/tex]

V.

Given is the function of complex variable:

[tex]f(z)=1+z+z^2+z^4+z^8+z^{16}+...[/tex]

Show that it can't be analyticaly extended to area outside the unit circle |z|=1.

VI.

Given are two real polynomials:

[tex]p(x)=(1+x^2-x^3)^{2000},q(x)=(1-x^2+x^3)^{2000}[/tex]

Which one has the larger coefficient [itex]a_{200}[/itex] in term [itex]a_{200}x^{200}[/itex],in canonical representation?Explain answer.

VII

Find minimum and maximum of real function:

[tex]f(x)=Asin^2 x + Bsin(x)cos(x) + Ccos^2 x[/tex]

VIII.

Let

[tex]f(x)=\sum_{n=1}^{100}cos(n^{3/2}x)[/tex]

Prove that real function f(x) has at least 40 zeroes in interval [0,1000]

IX.

Evaluate integrals:

a)

[tex]\int_{0}^{\frac{\pi}{2}}ln(sin^2(x)+a^2cos^2(x))dx[/tex]

b)

[tex]\int_{D}\int \frac{arctg(x+y)}{(x^2+y^2)^2} dxdy[/tex]

[tex]D[/tex] is the area of integration defined by {x>0.y>0,x+y>1}

X.

Given the condition [tex]z(x,y)=f(x)f(y)[/tex] solve PDE:

[tex]\frac{\partial^2 z}{\partial x^2}=\frac{\partial z}{\partial y}[/tex]

What do you think?

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# Some Challenging problems

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