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The Math challenge threads have returned!

Rules:

1. You may use google to look for anything except the actual problems themselves (or very close relatives).

2. Do not cite theorems that trivialize the problem you're solving.

3. Do not solve problems that are way below your level. Some problems may be intended for high school or early university level students. If a problem looks way too easy for you, leave it for someone else :)

4. Have fun!

a) Both sides' pieces are scrambled in the same way, so each piece is still opposite an enemy piece of the same type, along the file.

b) The king is between the two rooks.

c) The two bishops are opposite colors.

Verify that there are 960 possible starting positions for Fischer random, explaining the variant's other common name: "chess 960". It might seem that when you reflect the board, you get a position with equivalent game play, so there are effectively only 480 inequivalent starting positions. What is wrong with this logic?

a) Explain why we may assume without loss of generality that ##M=M'##

b) Consider the path of top forms ##\omega_t=(1-t)\omega+t\omega'## with ##0\leq t\leq 1.## Explain why each ##\omega_t## is a volume form on ##M.##

c) We will construction the diffeomorphism as the flow of a 1-parameter family of vector fields. Show that if ##X_t## is a 1-parameter family of vector fields and ##\phi_t:M\to M## is its flow map, then ##\frac{d}{dt}\phi_t^*\omega_t= \phi_t^*\left(d (\iota_{X_t}\omega_t)+(\omega'-\omega)\right),## where ##\iota## means interior multiplication. You will need Cartan's formula for the Lie derivative of a differential form, found on the same page.

d) Show that you may smoothly choose the vector fields ##X_t## such that ##\frac{d}{dt}\phi_t^*\omega_t=0## and conclude that ##\phi_1:M\to M## is a diffeomorphism satisfying ##\phi_1^*\omega'=\omega.##

Rules:

1. You may use google to look for anything except the actual problems themselves (or very close relatives).

2. Do not cite theorems that trivialize the problem you're solving.

3. Do not solve problems that are way below your level. Some problems may be intended for high school or early university level students. If a problem looks way too easy for you, leave it for someone else :)

4. Have fun!

**1.**(solved by @PeroK and @mfb) Fischer random is a variant of chess with all of the same rules, except that in the starting position, the positions of the pieces on the back ranks are scrambled, according to the following rules:a) Both sides' pieces are scrambled in the same way, so each piece is still opposite an enemy piece of the same type, along the file.

b) The king is between the two rooks.

c) The two bishops are opposite colors.

Verify that there are 960 possible starting positions for Fischer random, explaining the variant's other common name: "chess 960". It might seem that when you reflect the board, you get a position with equivalent game play, so there are effectively only 480 inequivalent starting positions. What is wrong with this logic?

**2.**solved by @martinbn) Let ##v## be a known nonzero vector in ##\mathbb{R}^3.## If you know ##v\cdot w## and ##v\times w,## can you determine what the vector ##w## must be? If so, give a formula for ##w## in terms of ##v## and ##v\cdot w## and ##v\times w##. If not, give an example where there are multiple possible solutions for ##w##.**3.**solved by @martinbn and @anuttarasammyak)Evaluate the sum ##\sum_{n=1}^\infty\frac{1}{n(n+1)(n+2)}.##**4.**(solved by @pasmith and @anuttarasammyak)Evaluate ##\int_0^1\left(\frac{x^2-4x+2}{2-x}\right)^{100} dx.##**5.**(solved by @julian) Show that ##\sum_{n=1}^\infty\frac{\sin(n)}{n}=\frac{\pi-1}{2}.##**6.**(solved by @martinbn) Let ##A## be the ##n\times n## matrix whose diagonal entries are 1 and whose off-diagonal entries are 2. Find all eigenvalues of ##A## and their multiplicities.**7.**(solved by @martinbn) Let ##S_5## be the group of permutations on a 5 element set. Let ##X=\{(\sigma,\pi)\in S_5\times S_5: \sigma\pi=\pi\sigma\}.## What is ##|X|?##**8.**(solved by @martinbn) Let ##\alpha## be a complex number on the unit circle. Suppose that ##f\in\mathbb{Q}[x]## is irreducible and has ##\alpha## as a root. Show that if ##\beta## is any root of ##f##, then ##1/\beta## is also a root.**9.**(solved by @mathwonk)Let ##M_g## be the closed orientable surface of genus ##g\geq 0.## Let ##\vee_{i=1}^n S^1## be the wedge product of ##n## circles. Show that if there exists a continuous map ##r:M_g\to\vee_{i=1}^n S^1## with a right homotopy inverse, then ##n\leq g##. ##r## having a right homotopy inverse means that there exists ##\iota:\vee_{i=1}^n S^1\to M_g## with ##r\circ \iota## homotopic to the identity map on ##\vee_{i=1}^n S^1##. An example of a map with a right homotopy inverse would be a retraction onto a subspace.**10.**The point of this problem is to show that if two (connected and closed) diffeomorphic manifolds have the same volume, then there is a diffeomorphism between them that preserves volume. More precisely, let ##M## and ##M'## be diffeomorphic manifolds, which are closed and connected. If ##\omega## and ##\omega'## are volume forms on ##M## and ##M'##, respectively, with ##\int_M \omega=\int_{M'}\omega'##, then there is a diffeomorphism ##f:M\to M'## such that ##f^*\omega'=\omega.##a) Explain why we may assume without loss of generality that ##M=M'##

b) Consider the path of top forms ##\omega_t=(1-t)\omega+t\omega'## with ##0\leq t\leq 1.## Explain why each ##\omega_t## is a volume form on ##M.##

c) We will construction the diffeomorphism as the flow of a 1-parameter family of vector fields. Show that if ##X_t## is a 1-parameter family of vector fields and ##\phi_t:M\to M## is its flow map, then ##\frac{d}{dt}\phi_t^*\omega_t= \phi_t^*\left(d (\iota_{X_t}\omega_t)+(\omega'-\omega)\right),## where ##\iota## means interior multiplication. You will need Cartan's formula for the Lie derivative of a differential form, found on the same page.

d) Show that you may smoothly choose the vector fields ##X_t## such that ##\frac{d}{dt}\phi_t^*\omega_t=0## and conclude that ##\phi_1:M\to M## is a diffeomorphism satisfying ##\phi_1^*\omega'=\omega.##

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