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fresh_42

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As (almost) always: have a look on previous challenge threads, too. E.g. in https://www.physicsforums.com/threads/math-challenge-march-2019.967174/ are still problems to solve, and some of them easy, which I find, and in any case useful to know or at least useful to have seen.

##1.## (solved by @SpinFlop ; see also here. ) There are ##12## prisoners who are given the chance to be freed if they pass this test: the warden will put a hat on the head of each prisoner which will be either black or white. The prisoners will get informed that there will be at least one hat of each color. They will be able to see everyone else's hat except their own. Also, there will be no sort of communication among prisoners. The prisoners will be lined up every five minutes starting at ##10:00 \space\space a.m.## and ending at ##10:50 \space\space a.m.## In order to pass the test, all the prisoners with a black hat and only them, will have to step forward during the same line-up. If they succeed, all prisoners will be freed otherwise they will all be executed. Is there a way for the prisoners to pass the test?

##2.## (solved by @lpetrich ) For which real numbers ##\alpha## does the improper integral ##\int_0^1\int_{-\infty}^\infty\frac{1}{(x^2+y^2)^{\alpha}} dx dy## converge.

##3.## Let ##a\in\mathbb{C}## and define ##s=t^3+3t^2+at\in\mathbb{C}(t)##. For which ##a## is ##\mathbb{C}(t)/\mathbb{C}(s)## a Galois extension?

##4.## (solved by @wrobel ) Let ##f## be a continuous element of ##L^1((0,\infty))##. Suppose that ##x=x(t)## solves the differential equation ##\frac{dx}{dt}=-px+f(t)##. Prove that ##\lim_{t\to\infty}x(t)=0##.

##5.## Let ##f:S^5\to S^3\times S^2## be a smooth [edit: and surjective] function. Show that ##f## takes critical values in any set of the form ##S^3\times\{p\}## for ##p\in S^2##.

##6.## (solved by @lpetrich ) Find the area ##A## enclosed by the asteroid ##(x,y)=(\cos^3 t,\sin^3 t)## for ##0\leq t \leq 2\pi\,.##

##7.## (solved by @lpetrich ) Two surface ships on maneuvers are trying to determine a submarine's course and speed to prepare for an aircraft intercept. Ship ##A## is located at ##(4, 0, 0)##, whereas ship ##B## is located at ##(0, 5, 0)##. All coordinates are given in thousands of feet. Ship ##A## locates the submarine in the direction of the vector ##2\mathbf{i} + 3\mathbf{j} - (1/3)\mathbf{k}##, and ship ##B## locates it in the direction of the vector ##18\mathbf{i} - 6\mathbf{j} - \mathbf{k}##. Four minutes ago, the submarine was located at ##A=(2, -1, -1/3)##. The aircraft is due in ##20## minutes. Assuming that the submarine moves in a straight line at a constant speed, to what position should the surface ships direct the aircraft?

##8.## Calculate the following:

##a.)## (solved by @fbs7 ) ##\displaystyle{\int} \dfrac{\sqrt{(x^2-1)^3}}{x}\,dx ##

##b.)## (solved by @lpetrich ) The arc length ##L## of ##y=-\dfrac{x^2}{8}+\log x## for ##1\leq x \leq 2##

##9.## (solved by @lpetrich ) Find the similarity transformations to diagonalize the following matrices:

##a.)## ##A=\begin{pmatrix}1&-\sqrt{2}&1\\ \sqrt{2}&0&-\sqrt{2}\\ 1&\sqrt{2}&1\end{pmatrix}##

##b.)## ##B=\begin{pmatrix}\cos \varphi &-\sin \varphi \\ \sin \varphi& \cos \varphi \end{pmatrix}##

##10.## (solved by @Math_QED ) Suppose that ##\mathbb{F}## is a finite field with say ##|\mathbb{F}|= p^m = q## and that ##V## is a vector space of finite dimension ##n## over ##\mathbb{F}\,.## Find the order of ##\operatorname{GL}(V)\,.##

##1##. The king in chess can move to any neighboring square in three different ways: horizontally, vertically or diagonally. Now, we assume that we have an infinite chessboard and the king starts on some square.

##a.)## (solved by @fbs7 ) In how many different squares can it be after ##n## moves?

##b.)## (solved by @Master1022 ) Provide an optimal approach i.e. the minimum number of moves required to reach a destination.

##c.)## (solved by @Master1022 ) Answer ##a.)## if there is a constraint of no diagonal moves.

##2##. The squares of an ##8 \times 8## chessboard are mistakenly colored in blocks of two colors, such that beginning from top left corner of the chessboard and moving in rows of ##2 \times 2## squares, always from left to right, the colors of ##2 \times 2## squares are ##W - B - W - B## for the odd-numbered rows and ##B - W - B - W## for the even numbered rows - we begin counting of rows with number ##1##, where ##W## denotes White and ##B## denotes Black. This board needs to be cut in such a way so that a standard chessboard can be reassembled from the pieces obtained. What is the minimum number of pieces that you need to cut the board and in which way should they be reassembled?

##3##. An algorithm works like this: it starts with a single equilateral triangle and on each (subsequent) iteration, it adds new equilateral triangles all around the outside. When the algorithm finishes its ##n##-th iteration how many small triangles will be there?

##4.## (solved by @YoungPhysicist ) Can the numbers ##1, 2, 3, \ldots, 16## be arranged in a row so that each two adjacent numbers add up to a square number?

Example: ##2, 7, 9, 16, \ldots## would be a possibility for the first four numbers ##2 + 7 = 9, 7 + 9 = 16, 9 + 16 = 25##; but then we are stuck.

##5.## (solved by @ShreyJ ) We are looking for a ten-digit number ##N##, where the first digit indicates how many zeros occur in ##N##, the second digit, how many ones appear in ##N##, the third digit, how many doubles occur in ##N##, ... and the tenth digit, how many nines appear in ##N##.

*As a general note due to a given cause: Please do not edit your posts after they have been answered!*

This is in general - not only here - not acceptable, as it makes the threads unreadable and worst case change the entire meaning and thus disrespects the ones who answered! At most an additional commentary marked as "edit (1):" etc. could be added, but nothing within the text, except spell or comma mistakes.This is in general - not only here - not acceptable, as it makes the threads unreadable and worst case change the entire meaning and thus disrespects the ones who answered! At most an additional commentary marked as "edit (1):" etc. could be added, but nothing within the text, except spell or comma mistakes.

**Questions**##1.## (solved by @SpinFlop ; see also here. ) There are ##12## prisoners who are given the chance to be freed if they pass this test: the warden will put a hat on the head of each prisoner which will be either black or white. The prisoners will get informed that there will be at least one hat of each color. They will be able to see everyone else's hat except their own. Also, there will be no sort of communication among prisoners. The prisoners will be lined up every five minutes starting at ##10:00 \space\space a.m.## and ending at ##10:50 \space\space a.m.## In order to pass the test, all the prisoners with a black hat and only them, will have to step forward during the same line-up. If they succeed, all prisoners will be freed otherwise they will all be executed. Is there a way for the prisoners to pass the test?

##2.## (solved by @lpetrich ) For which real numbers ##\alpha## does the improper integral ##\int_0^1\int_{-\infty}^\infty\frac{1}{(x^2+y^2)^{\alpha}} dx dy## converge.

##3.## Let ##a\in\mathbb{C}## and define ##s=t^3+3t^2+at\in\mathbb{C}(t)##. For which ##a## is ##\mathbb{C}(t)/\mathbb{C}(s)## a Galois extension?

##4.## (solved by @wrobel ) Let ##f## be a continuous element of ##L^1((0,\infty))##. Suppose that ##x=x(t)## solves the differential equation ##\frac{dx}{dt}=-px+f(t)##. Prove that ##\lim_{t\to\infty}x(t)=0##.

##5.## Let ##f:S^5\to S^3\times S^2## be a smooth [edit: and surjective] function. Show that ##f## takes critical values in any set of the form ##S^3\times\{p\}## for ##p\in S^2##.

##6.## (solved by @lpetrich ) Find the area ##A## enclosed by the asteroid ##(x,y)=(\cos^3 t,\sin^3 t)## for ##0\leq t \leq 2\pi\,.##

##7.## (solved by @lpetrich ) Two surface ships on maneuvers are trying to determine a submarine's course and speed to prepare for an aircraft intercept. Ship ##A## is located at ##(4, 0, 0)##, whereas ship ##B## is located at ##(0, 5, 0)##. All coordinates are given in thousands of feet. Ship ##A## locates the submarine in the direction of the vector ##2\mathbf{i} + 3\mathbf{j} - (1/3)\mathbf{k}##, and ship ##B## locates it in the direction of the vector ##18\mathbf{i} - 6\mathbf{j} - \mathbf{k}##. Four minutes ago, the submarine was located at ##A=(2, -1, -1/3)##. The aircraft is due in ##20## minutes. Assuming that the submarine moves in a straight line at a constant speed, to what position should the surface ships direct the aircraft?

##8.## Calculate the following:

##a.)## (solved by @fbs7 ) ##\displaystyle{\int} \dfrac{\sqrt{(x^2-1)^3}}{x}\,dx ##

##b.)## (solved by @lpetrich ) The arc length ##L## of ##y=-\dfrac{x^2}{8}+\log x## for ##1\leq x \leq 2##

##9.## (solved by @lpetrich ) Find the similarity transformations to diagonalize the following matrices:

##a.)## ##A=\begin{pmatrix}1&-\sqrt{2}&1\\ \sqrt{2}&0&-\sqrt{2}\\ 1&\sqrt{2}&1\end{pmatrix}##

##b.)## ##B=\begin{pmatrix}\cos \varphi &-\sin \varphi \\ \sin \varphi& \cos \varphi \end{pmatrix}##

##10.## (solved by @Math_QED ) Suppose that ##\mathbb{F}## is a finite field with say ##|\mathbb{F}|= p^m = q## and that ##V## is a vector space of finite dimension ##n## over ##\mathbb{F}\,.## Find the order of ##\operatorname{GL}(V)\,.##

##1##. The king in chess can move to any neighboring square in three different ways: horizontally, vertically or diagonally. Now, we assume that we have an infinite chessboard and the king starts on some square.

##a.)## (solved by @fbs7 ) In how many different squares can it be after ##n## moves?

##b.)## (solved by @Master1022 ) Provide an optimal approach i.e. the minimum number of moves required to reach a destination.

##c.)## (solved by @Master1022 ) Answer ##a.)## if there is a constraint of no diagonal moves.

##2##. The squares of an ##8 \times 8## chessboard are mistakenly colored in blocks of two colors, such that beginning from top left corner of the chessboard and moving in rows of ##2 \times 2## squares, always from left to right, the colors of ##2 \times 2## squares are ##W - B - W - B## for the odd-numbered rows and ##B - W - B - W## for the even numbered rows - we begin counting of rows with number ##1##, where ##W## denotes White and ##B## denotes Black. This board needs to be cut in such a way so that a standard chessboard can be reassembled from the pieces obtained. What is the minimum number of pieces that you need to cut the board and in which way should they be reassembled?

##3##. An algorithm works like this: it starts with a single equilateral triangle and on each (subsequent) iteration, it adds new equilateral triangles all around the outside. When the algorithm finishes its ##n##-th iteration how many small triangles will be there?

##4.## (solved by @YoungPhysicist ) Can the numbers ##1, 2, 3, \ldots, 16## be arranged in a row so that each two adjacent numbers add up to a square number?

Example: ##2, 7, 9, 16, \ldots## would be a possibility for the first four numbers ##2 + 7 = 9, 7 + 9 = 16, 9 + 16 = 25##; but then we are stuck.

##5.## (solved by @ShreyJ ) We are looking for a ten-digit number ##N##, where the first digit indicates how many zeros occur in ##N##, the second digit, how many ones appear in ##N##, the third digit, how many doubles occur in ##N##, ... and the tenth digit, how many nines appear in ##N##.

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