Posts Tagged ‘Analysis’

Putnam 2017 A3 – Solution

December 4, 2017 Leave a comment

Problem A3. Denote {\phi = f-g}. Then {\phi} is continuous and {\int_a^b \phi = 0}. We can see that

\displaystyle I_{n+1}-I_n = \int_a^b (f/g)^n \phi = \int_{\phi\geq 0} (f/g)^n \phi+ \int_{\phi<0} (f/g)^n \phi

Now note that on {\{ \phi>=0\}} we have {f/g \geq 1} so {(f/g)^n \phi \geq \phi}. Furthermore, on {\{\phi<0\}} we have {(f/g)^n <1} so multiplying with {\phi<0} we get {(f/g)^n \phi \geq \phi}. Therefore

\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} \phi + \int_{\phi<0} \phi = \int \phi = 0.

To prove that {I_n} goes to {+\infty} we can still work with {I_{n+1}-I_n}. Note that the negative part cannot get too big:

\displaystyle \left|\int_{ \phi <0 } (f/g)^n \phi \right| \leq \int_{\phi<0} |\phi| \leq \int_a^b |f-g|.

As for the positive part, taking {0<\varepsilon< \max_{[a,b]} \phi} we have

\displaystyle \int_{\phi\geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon}(f/g)^n \varepsilon.

Next, note that on {\{ \phi \geq \varepsilon\}}

\displaystyle \frac{f}{g} = \frac{g+\phi}{g} \geq \frac{g+ \varepsilon}{g}.

We would need that the last term be larger than {1+\delta}. This is equivalent to {g\delta <\varepsilon}. Since {g} is continuous on {[a,b]}, it is bounded above, so some delta small enough exists in order for this to work.

Grouping all of the above we get that

\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon} \varepsilon (1+\delta)^n.

Since {|\phi>\varepsilon|>0} we get that {I_{n+1}-I_n} goes to {+\infty}.


Some of the easy Putnam 2016 Problems

December 11, 2016 Leave a comment

Here are a few of the problems of the Putnam 2016 contest. I choose to only list problems which I managed to solve. Most of them are pretty straightforward, so maybe the solutions posted here may be very similar to the official ones. You can find a complete list of the problems on other sites, for example here.

A1. Find the smallest integer {j} such that for every polynomial {p} with integer coefficients and every integer {k}, the number

\displaystyle p^{(j)}(k),

that is the {j}-th derivative of {p} evaluated at {k}, is divisible by {2016}.

Hints. Successive derivatives give rise to terms containing products of consecutive numbers. The product of {j} consecutive numbers is divisible by {j!}. Find the smallest number such that {2016 | j!}. Prove that {j-1} does not work by choosing {p = x^{j-1}}. Prove that {j} works by working only on monomials…

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IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

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IMC 2016 – Day 1 – Problem 1

July 27, 2016 Leave a comment

Problem 1. Let {f:[a,b] \rightarrow \Bbb{R}} be continuous on {[a,b]} and differentiable on {(a,b)}. Suppose that {f} has infinitely many zeros, but there is no {x \in (a,b)} with {f(x)=f'(x) = 0}.

  • (a) Prove that {f(a)f(b)=0}.
  • (b) Give an example of such a function.

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SEEMOUS 2016 Problem 4 – Solution

March 6, 2016 3 comments

Problem 4. Let {n \geq 1} be an integer and set

\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.

Prove that

a) {\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}

b) {\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}.

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SEEMOUS 2016 – Problems

March 5, 2016 3 comments

Problem 1. Let {f} be a continuous and decreasing real valued function defined on {[0,\pi/2]}. Prove that

\displaystyle \int_{\pi/2-1}^{\pi/2} f(x)dx \leq \int_0^{\pi/2} f(x)\cos x dx \leq \int_0^1 f(x) dx.

When do we have equality?

Problem 2. a) Prove that for every matrix {X \in \mathcal{M}_2(\Bbb{C})} there exists a matrix {Y \in \mathcal{M}_2(\Bbb{C})} such that {Y^3 = X^2}.

b) Prove that there exists a matrix {A \in \mathcal{M}_3(\Bbb{C})} such that {Z^3 \neq A^2} for all {Z \in \mathcal{M}_3(\Bbb{C})}.

Problem 3. Let {A_1,A_2,...,A_k} be idempotent matrices ({A_i^2 = A_i}) in {\mathcal{M}_n(\Bbb{R})}. Prove that

\displaystyle \sum_{i=1}^k N(A_i) \geq \text{rank} \left(I-\prod_{i=1}^k A_i\right),

where {N(A_i) = n-\text{rank}(A_i)} and {\mathcal{M}_n(\Bbb{R})} is the set of {n \times n} matrices with real entries.

Problem 4. Let {n \geq 1} be an integer and set

\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.

Prove that

a) {\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}

b) {\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}.

Some hints follow.

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Vojtech Jarnik Competition 2015 – Problems Category 2

March 27, 2015 Leave a comment

Problem 1. Let {A} and {B} be two {3 \times 3} matrices with real entries. Prove that

\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs {(n,m)} of positive integers satisfying the equation

\displaystyle 5^n = 6m^2+1.

Problem 3. Determine the set of real values {x} for which the following series converges, and find its sum:

\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).

Problem 4. Find all continuously differentiable functions {f : \Bbb{R} \rightarrow \Bbb{R}}, such that for every {a \geq 0} the following relation holds:

\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),

where {D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}

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