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It's time for a basic math challenge! For more advanced problems you can check our other intermediate level math challenge thread!

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

a) (solved by @lpetrich ) Find a minimal polynomial to determine the factor ring which is isomorphic to ## \mathbb{F} _8## .

b) (resolved in post #83) From there determine a basis of ## \mathbb{F}_8## over ## \mathbb{F}_2## and write down its multiplication and addition laws.

c) (solved by @lpetrich ) Why does the algebraic closure of a finite field have to be infinite?##\space##

##2.## (solved by @nuuskur ) If ##a## is an odd integer show that ##a^{2^n} \equiv 1(\mod{2^{n+2}})## for all ##n \in \mathbb{N} - \{0\}## ##\space##

##3.## (solved by @Zafa Pi ) A carnival has a

Heads: ##+1## as a payoff

Tails: ##+3## as a payoff

Other: game over, lose all accrued winnings.

Otherwise the player may stop at any time and keep the accrued winnings.

How much should a risk neutral player be willing to pay in order to play this game?

(For avoidance of doubt, this refers to playing one 'full' game, which is complete upon termination, and termination occurs on the first occurrence of (a) result of coin toss equals ##\text{Other}## or (b) the player elects to stop.)

How many rounds would it take on average for the game to terminate? (You may assume a

Now suppose the player doesn't care about the score and just loves flipping coins -- how long will the game take to terminate, on average, in this case? ##\space##

##4.## (solved by @Hiero ) Calculate the volume ##\mu(A)## of

## A =\{(x,y,z)\in \mathbb{R}^3\,: \,x,y,z \ge 0\; , \;x+y+z \leq \sqrt{2}\; , \;x^2+y^2 \leq 1\,\}## ##\space##

##5.## (solved by @nuuskur ) Determine the open balls with radius ##3## around ## (2,1) \in \mathbb{R}^2## w.r.t.

a) the French Railway metric with Paris at the origin ##P## and Reims at ##R=(2,1)## .

b) the Manhattan metric.

c) the maximum metric. ##\space##

##6.##(solved by @Zafa Pi ) Two urns contain the same total number of balls, some black and some white in each. From each urn are drawn ## \big(n \geq 3\big)## balls with replacement. Find the number of drawings and the composition of the two urns so that the probability that all white balls are drawn from the first urn is equal to the probability that the drawing from the second is either all white or all black. ##\space##

##7.## (solved by @lpetrich ) Show that if for some complex number ##z##, ##{\lvert \sin z\rvert}^2 + {\lvert \cos z\rvert}^2 = 1## then ##z \in \mathbb{R}## ##\space##

##8.## (solved by @dRic2 ) Show that of all triangles with given base and given area, the one with the least perimeter is isosceles ##\space##

##9.## (solved by @lpetrich ) Solve ##\int_{\Gamma} \omega## with the curve ##\Gamma = \gamma([0,1])## given by

##\gamma : \mathbb{R} \longrightarrow \mathbb{R}^3\; , \;\gamma(t)=(t^2,2t,1)\text{ and }\omega = z^2 dx +2ydy+xzdz##

and compute the exterior derivative ## \nu = d\omega## . As such, the result is an exact ##2-## form. Is it also closed? Show this by direct calculation. ##\space##

##10.## (solved by @lpetrich ) Prove that ##\lim_{n\to\infty} (n!e - \lfloor n!e \rfloor) = 0## ##\space##

**RULES:**1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

**5) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted by**@fresh_42##1.## Finite Field ##\mathbb{F}_8##a) (solved by @lpetrich ) Find a minimal polynomial to determine the factor ring which is isomorphic to ## \mathbb{F} _8## .

b) (resolved in post #83) From there determine a basis of ## \mathbb{F}_8## over ## \mathbb{F}_2## and write down its multiplication and addition laws.

c) (solved by @lpetrich ) Why does the algebraic closure of a finite field have to be infinite?##\space##

**(by**@fresh_42**)**##2.## (solved by @nuuskur ) If ##a## is an odd integer show that ##a^{2^n} \equiv 1(\mod{2^{n+2}})## for all ##n \in \mathbb{N} - \{0\}## ##\space##

**(by**@QuantumQuest**)**##3.## (solved by @Zafa Pi ) A carnival has a

*3 sided coin*with outcomes ## \big \{\text{Heads, Tails, Other}\big\}## and respective probabilities of ## \big\{\frac{2}{5}, \frac{2}{5}, \frac{1}{5}\big\}##.*Rules of play:*Heads: ##+1## as a payoff

Tails: ##+3## as a payoff

Other: game over, lose all accrued winnings.

Otherwise the player may stop at any time and keep the accrued winnings.

*question:*How much should a risk neutral player be willing to pay in order to play this game?

(For avoidance of doubt, this refers to playing one 'full' game, which is complete upon termination, and termination occurs on the first occurrence of (a) result of coin toss equals ##\text{Other}## or (b) the player elects to stop.)

*optional:*How many rounds would it take on average for the game to terminate? (You may assume a

*mild*preference for shorter vs longer games in the event of any tie breaking concerns.)Now suppose the player doesn't care about the score and just loves flipping coins -- how long will the game take to terminate, on average, in this case? ##\space##

**(by**@StoneTemplePython**)**

##4.## (solved by @Hiero ) Calculate the volume ##\mu(A)## of

## A =\{(x,y,z)\in \mathbb{R}^3\,: \,x,y,z \ge 0\; , \;x+y+z \leq \sqrt{2}\; , \;x^2+y^2 \leq 1\,\}## ##\space##

**(by**@fresh_42**)**##5.## (solved by @nuuskur ) Determine the open balls with radius ##3## around ## (2,1) \in \mathbb{R}^2## w.r.t.

a) the French Railway metric with Paris at the origin ##P## and Reims at ##R=(2,1)## .

b) the Manhattan metric.

c) the maximum metric. ##\space##

**(by**@fresh_42**)**##6.##(solved by @Zafa Pi ) Two urns contain the same total number of balls, some black and some white in each. From each urn are drawn ## \big(n \geq 3\big)## balls with replacement. Find the number of drawings and the composition of the two urns so that the probability that all white balls are drawn from the first urn is equal to the probability that the drawing from the second is either all white or all black. ##\space##

**(by**@StoneTemplePython**)**##7.## (solved by @lpetrich ) Show that if for some complex number ##z##, ##{\lvert \sin z\rvert}^2 + {\lvert \cos z\rvert}^2 = 1## then ##z \in \mathbb{R}## ##\space##

**(by**@QuantumQuest**)**##8.## (solved by @dRic2 ) Show that of all triangles with given base and given area, the one with the least perimeter is isosceles ##\space##

**(by**@QuantumQuest**)**##9.## (solved by @lpetrich ) Solve ##\int_{\Gamma} \omega## with the curve ##\Gamma = \gamma([0,1])## given by

##\gamma : \mathbb{R} \longrightarrow \mathbb{R}^3\; , \;\gamma(t)=(t^2,2t,1)\text{ and }\omega = z^2 dx +2ydy+xzdz##

and compute the exterior derivative ## \nu = d\omega## . As such, the result is an exact ##2-## form. Is it also closed? Show this by direct calculation. ##\space##

**(by**@fresh_42**)**##10.## (solved by @lpetrich ) Prove that ##\lim_{n\to\infty} (n!e - \lfloor n!e \rfloor) = 0## ##\space##

**(by**@QuantumQuest**)**

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