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fresh_42

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## Summary:

- A lot of calculus questions this month, but a bit (linear and abstract) algebra, too. The functional analysis question is a bit tricky.

## Main Question or Discussion Point

**Questions**

1.(solved by @archaic ) Determine ##\lim_{n\to \infty}\cos\left(t/\sqrt{n}\right)^n## for ##t\in \mathbb{R}##.

1.

**2.**(solved by @Antarres ) Let ##a_0,\ldots,a_n## be distinct real numbers. Show that for any ##b_0,\ldots,b_n\in\mathbb{R}##, there exists a unique polynomial ##p## of degree at most ##n## such that ##p(a_i)=b_i## for each ##i##.

**3.**(solved by @Antarres ) Let ##r(t)=(x(t),y(t))## be an arc-length parameterization of a smooth closed curve ##C## with length ##2\pi##. Let ##A## be the area enclosed by ##C##

**a.)**Using Green's formula, show ##A^2\leq\left(\int_0^{2\pi}x(t)^2dt \right)\ \left(\int_0^{2\pi}y'(t)^2 dt\right)##

**b.)**By translating ##C##, we may assume without loss of generality that ##\int_0^{2\pi}x(t) dt=0##. Use this assumption together with the Wirtinger inequality (question 1 of the January challenge) to show that ##A\leq\pi.##

**c.)**By examining the equality condition for the Wirtinger inequality (Problem No. 1 in 01/2020), show that ##A=\pi## can only happen if ##C## is a circle.

**4.**(solved by @Antarres ) Let ##f:[0,\infty)\to\mathbb{R}## be a continuous function such that the sequence of functions ##f_n(x)=f(x+n)## converges uniformly to a function ##g:[0,\infty)\to\mathbb{R}##. Show that ##f## and ##g## are uniformly continuous.

**5.**

a.)(solved by @etotheipi ) Let ##A## = ##

a.)

\begin{bmatrix}

-1 & 0 \\

2 & 1

\end{bmatrix}

## Find ##A^{99}##, ##A^{2n}## and ##A^{2n + 1}##, ##n \in \mathbb{N}##

**b.)**(open) How would we conclude in case of any commutative ring using ideals? (##\operatorname{char} \neq 2##)

**6.**(solved by @Antarres )

**a.)**If ##a##,##b##,##k## are integers with ##b \neq 0## show that ##(a + kb, b) = (a,b)##

**b.)**We take the

*Fibonacci*sequence ##(F_n)## (starting from the third term): ##1, 2, 3, 5, 8, 13, \dots##, where each term after the second one, is the sum of the two previous terms. If ##F_n## is the ##n-##th term of the sequence, show that every two consecutive Fibonacci numbers are coprime.

**7.**

a.)(solved by @archaic ) If ##f: \mathbb{R} \to \mathbb{R}## is an integrable function in every closed interval of ##\mathbb{R}## show using calculus that

a.)

$$\int_{a}^{b} f(x)\,dx = \int_{a + d}^{b + d} f(x - d)\,dx$$

with ##a \leq b## and ##d \geq 0##.

**b.)**(open) The function ##\varphi\, : \,\mathbb{R}^2 \longrightarrow \mathbb{R}^2## with ##\varphi(t,r)=(t(r+2),t^2-r)## is injective on ##U:=(0,1)\times (-1,1)\,.## Show that ##\varphi\, : \,U\longrightarrow V:= \varphi(U)## is a diffeomorphism.

Next let ##f\, : \,\mathbb{R}^2\longrightarrow \mathbb{R}## be integrable over ##V##. Write

$$

\int_V f\,d\lambda = \int_{\ldots}^{\ldots} \int_{\ldots}^{\ldots} \ldots f(\ldots\, , \,\ldots)\,dr\,dt

$$

and calculate the area of ##V##.

**8.**(solved by @julian ) Calculate ##\displaystyle{\sum_{k=1}^\infty} \dfrac{1}{\binom{2k}{k}}##

**9.**(solved by @Antarres ) Let ##a## be an integer and ##p## an odd prime which does not divide ##a##. The left multiplication

$$

\lambda_{a,p}\, : \,\mathbb{Z}_p^\times \longrightarrow \mathbb{Z}_p^\times \, ; \,x \longmapsto ax \operatorname{mod} p

$$

is then a permutation on ##\{\,1,\ldots,p-1\,\}\,.## Prove

$$

\left(\dfrac{a}{p}\right) = \operatorname{sgn}\left(\lambda_{a,p}\right)

$$

**10.**(solved by @suremarc and @Antarres ) Let ##\mathcal{H}## be a real Hilbert space and ##\beta## a continuous bilinear form, ##\mathcal{H}^*## its dual space of continuous functionals on ##\mathcal{H}##, and ##\beta(f,f)\geq C\|f\|^2>0.##

Prove that for any given continuous functional ##F \in \mathcal{H}^*## there is a unique vector ##f^\dagger \in \mathcal{H}## such that

$$

F(g)=\beta(f^\dagger,g)\quad \forall\,\,g\in \mathcal{H}

$$

**High Schoolers only**

11.Answer the following questions:

11.

**a.)**(solved by @mmaismma ) How many knights can you place on a ##n\times m## chessboard such that no two attack each other?

**b.)**(solved by @Not anonymous ) In how many different ways can eight queens be placed on a chessboard, such that no queen threatens another? Two solutions are not different, if they can be achieved by a rotation or by mirroring of the board. (For this part there is no proof required.)

**12.**(solved by @Not anonymous ) Determine (with justification, but without explicit calculation) which of

**a.)**##1000^{1001}## and ##{1002}^{1000}##

**b.)**(solved by @etotheipi ) ##e^{0.000009}-e^{0.000007}+e^{0.000002}-e^{0.000001}## and ##e^{0.000008}-e^{0.000005}##

is larger.

**13.**(solved by @etotheipi , @Not anonymous)

**a.)**Let ##A=(-2,0)\, , \,B=(0,4)## and ##M=(1,3)\,.## What is ##\alpha = \sphericalangle (AMB)##?

**b.)**Let ##C=(-1,2+\sqrt{5})\, , \,D=(-1,2-\sqrt{5})## and ##M=(1,3)\,.## What is ##\beta = \sphericalangle (CMD)##?

**14.**(solved by @Not anonymous ) Find the exact formula of the function ##f(x) =\sup_{t \in \mathbb{R}}(2tx - t^2)##, ##x \in \mathbb{R}##.

**15.**(solved by @etotheipi ) In what way does the oscillation period of a pendulum change, if the suspension pivot moves

**a.)**vertically up with acceleration ##a##,

**b.)**vertically down with acceleration ##a < g##,

**c.)**horizontally with acceleration ##a##?

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