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Time for the october challenge! This time a lot of people sent me suggestions for challenges. I wish to thank them a lot! If you think of a good challenge that could be included here, don't hesitate to send me!Ranking [and previous challenges] here: https://www.physicsforums.com/threads/micromass-big-challenge-ranking.879070/ 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.1.I use the notation from the Frenet-Serret frame for curves:a) Find the trajectory of an object moving in the ##x## direction with velocity ##v_0## at time ##t=0## that experiences a centripetal force (in the ##N## direction) that increases linearly with time (so that acceleration ##a=btN##).b) Does the speed of the object change?2.Definitions:The following functions from ##\mathbb N^k## to ##\mathbb N##, with ##k\ge 0##, are called basic functions:B1) The zero constant function ##Z:\mathbb N^0\to \mathbb N##, given by [tex]Z()=0[/tex](so ##Z## is actually the constant ##0##).B2) For all ##n\ge 0## and ##i## such that ##1\le i\le n##, the i:th n-ary projection function ##P_i^n:\mathbb N^k\to\mathbb N##, given by [tex]P_i^n(x_1, x_2,\dots, x_n)=x_i,[/tex] for all ##x_1,x_2,\dots,x_n\in \mathbb N##.B3) The successor function ##S:\mathbb N^1\to \mathbb N##, given by [tex]S(x_1)=x_1+1,[/tex] for all ##x_1\in \mathbb N##.We obtain new functions from old ones by applying the following operations:O1) Composition: For all ##m,n\ge 0## and all functions ##g:\mathbb N^m\to \mathbb N## and ##h_1,h_2,\dots, h_m:\mathbb N^n\to \mathbb N##, we obtain the function ##f:\mathbb N^n\to \mathbb N##, given by[tex]f(x_1,x_2,\dots, x_n)=g(h_1(x_1,x_2,\dots,x_n),h_2(x_1,x_2,\dots,x_n),\dots,h_m(x_1,x_2,\dots,x_n)),[/tex]for all ##x_1,x_2,\dots,x_n\in \mathbb N##.O2) Primitive recursion: For all ##n\ge 0## and all functions ##g:\mathbb N^n\to\mathbb N## and ##h:\mathbb N^{n+2}\to\mathbb N##, we obtain the (unique) function ##f:\mathbb N^{n+1}\to\mathbb N## which satisfies the following conditions:i)[tex]f(x_1,x_2,\dots,x_n,0)=g(x_1,x_2,\dots,x_n),[/tex]for all ##x_1,x_2,\dots,x_n\in\mathbb N##,ii)[tex]f(x_1,x_2,\dots,x_n,x_{n+1}+1)=h(x_1,x_2,\dots, x_n, x_{n+1},f(x_1,x_2,\dots,x_n,x_{n+1})),[/tex]for all ##x_1,x_2,\dots,x_n,x_{n+1}\in\mathbb N##.A function ##f:\mathbb N^n\to \mathbb N##, for any ##n\ge 0##, is called primitive recursive if it can be obtained from the basic functions B1-B3 by finitely many applications of the operations O1 and O2.Ackermann's function is the (unique) function ##A:\mathbb N^2\to \mathbb N## which satisfies the following conditions:a) ##A(0,n)=n+1##,b) ##A(m+1,0)=A(m,1)##,c) ##A(m+1,n+1)=A(m,A(m+1,n))##,for all ##m,n\in \mathbb N##.a) Prove that the factorial function ##n!## is primitive recursive.b) Prove that the "truncated subtraction" ##f(a,b) = \max\{a-b,0\}## is primitive recursive.c) Prove that to each primitive recursive function ##f:\mathbb N^n\to \mathbb N##, for any ##n\ge 0##, there is an ##m\in\mathbb N## such that[tex]f(x_1,x_2,\dots,x_n)\le A(m, \max_{1\le i\le n} x_i),[/tex]for all ##x_1,x_2,\dots, x_n\in\mathbb N##.d) Prove that ##A## is not primitive recursive.3) Let ##X## denote the set of all bounded real-valued sequences. As was shown in last challenge, a generalized limit exists for any such sequence. A generalized limit is any function function ##L:X\rightarrow \mathbb{R}## such that1) ##L((x_n + y_n)_n) = L((x_n)_n) + L((y_n)_n)##2) ##L((\alpha x_n)_n) = \alpha L((x_n)_n)##3) ##\liminf_n x_n \leq L((x_n)_n)\leq \limsup_n x_n##4) If ##x_n\geq 0## for all ##n##, then ##L((x_n)_n)\geq 0##.5) If ##y_n = x_{n+1}##, then ##L((x_n)_n) = L((y_n)_n)##6) If ##x_n\rightarrow x##, then ##L((x_n)_n) = x##.Find (in ZFC) the number (= cardinality) of all generalized limits.4) A pirate ship is at the origin of ##\mathbb{R}^2##. A merchant vessel is at ##(x_0,0)## with ##x_0

eq 0## as is sailing vertically upwards with constant speed ##v##. The pirate vessel is sailing with constant speed ##V## in such a way that at every instant it is always sailing directly towards the merchant vessel.a)Find the curve in which the pirate vessel moves.b)For ##V>v##, find the total distance traveled by the pirate ship until they capture the merchant vessel.c)In the case ##v=V## show that the pirate ship will lag behind the merchant vessel endlessly. Find the asymptotic distance between pirate ship and merchant vessel.5)Consider the recursion relation ##S_{n+2} = 2(S_{n+1} - S_n)## with ##S_1 = a## and ##S_2 = b##. Find all ##a## and ##b## such that ##S_n = 0## for infinitely many ##n##.6)Suppose ##X## and ##Y## are independent random variables distributed according to a uniform distribution on ##[0,1]##. Find the distribution of ##X^Y##.7)A laser is located ##b## units directly across from position ##a## on an infinite straight wall. An angle ##\theta## is chosen uniformly out of ##(-\pi/2,\pi/2)## and the laser is positioned such that it makes an angle ##\theta## when shining on the wall. Find the expected value and the variance of the random variable ##X## which is the (signed) distance from ##a##.8)On a table are ##2016## bells standing in a sequence. At every turn you have to pick ##2## bells, ring them and then exchange their place.For example, if there were only ##4## bells, they stand initially as ##A-B-C-D##. In turn ##1##, you pick bells ##A## and ##D##, ring them and exchange them to get ##D-B-C-A##. In turn ##2##, you pick bells ##D## and ##B##, ring them and exchange them to get ##B-D-C-A##.The goal of the bell ringer is to take ##n## turns after which the sequence of bells is reversed. For example an easy way to reverse the order in ##A-B-C-D## is first to ring ##A## and ##D## to get ##D-B-C-A## and then to ring ##B## and ##C## to get ##D-C-B-A##. We have reversed the bells in ##2## turns.Show that it is impossible to reverse ##2016## bells in an odd number of turns.9)Find all ##10##-digit numbers such thata) each digit ##\{0,1,2,3,4,5,6,7,8,9\}## is used exactly onceb) the first ##n## digits form a number divisible by ##n## (##n\in \{1,2,3,4,5,6,7,8,9,10\}##).For example, if my number would be ##1234567890##, then ##1## must be divisble by ##1##, ##12## must be divisible by ##2##, ##123## must be divisible by ##3## and so on.10)Find all 10-digit numbers where the first digit is how many zeros are in the number, the second digit is how many 1s are in the number etc. until the tenth digit which is how many 9s are in the number.1) Show that it is impossible to find four distinct squares as the subsequent elements in an arithmetic progression.The following sketch may be usedAssume ##\alpha##, ##\beta##, ##\gamma## and ##\delta## are four distinct squares that are consecutive elements of an arithmetic progression.Show that we may assume that they are relatively prime nonnegative integers.Show that ##\text{gcd}(\alpha,\beta) = \text{gcd}(\beta,\gamma) = \text{gcd}(\gamma,\delta) = 1## and that ##\alpha##, ##\beta##, ##\gamma##, ##\delta## are all odd.Show that ##\text{gcd}(\beta+\alpha, \beta - \alpha) = \text{gcd}(\gamma + \beta,\gamma - \beta) = \text{gcd}(\delta+\gamma,\delta - \gamma) = 2##.Define[tex]2a = \text{gcd}(\beta+\alpha,\gamma + \beta)[/tex][tex]2b = \text{gcd}(\beta + \alpha, \gamma - \beta)[/tex][tex]2c = \text{gcd}(\beta - \alpha, \gamma + \beta)[/tex][tex]2d = \text{gcd}(\beta-\alpha, \gamma-\beta)[/tex]Show[tex]\beta +\alpha = 2ab,~\beta-\alpha = 2cd[/tex][tex]\gamma + \beta = 2ac,~\gamma-\beta = 2bd[/tex][tex]\delta + \gamma = 2bc,~\delta - \gamma = 2ad[/tex]Show that ##(a+d)b = (a-d)c##and ##(c+d)a = (c-d)b##.Show that ##\text{gcd}(a+d, a-d)## and ##\text{gcd}(c+d,c-d)## are both either ##1## or ##2##.Show that[tex]a+d = mc,~a-d= mb[/tex][tex]c+d = nb,~c-d=na[/tex]with ##m,n\in \{1,2\}##.Derive a contradiction.2)Let ##p

eq 0## be a real number. Let ##x_1,...,x_n## be positive real numbers, we define the ##p##-mean as[tex]M_p(x_1,...,x_n) = \sqrt[p]{\frac{1}{n}\sum_{i=1}^n x_i^p}[/tex]Note that ##M_1(x_1,...,x_n)## is the usual mean.Prove that for all ##p,q\in \mathbb{R}\cup \{- \infty,+\infty\}## has that ##p\leq q## implies ##M_p(x_1,...,x_n)\leq M_q(x_1,...,x_n)##.Use Jensen's inequality with the function ##x^{p/q}##.3)Take a wire stretched between two posts, and have a large number of birds land on it at random. Take a bucket of yellow paint, and for each bird, paint the interval from it to its closest neighbour. The question is: what proportion of the wire will be painted. More strictly: as the number of birds goes to infinity, what is the limit of the expected value of the proportion of painted wire, assuming a uniform probability distribution of birds on the wire.1) Let ##A,B,C,D## be complex numbers with length ##1##. Prove that if ##A+B+C+D=0##, then these four numbers form a rectangle.2) On an arbitrary triangle, we produce on each side an equilateral triangle. Prove that the centroids of these three triangles forms an equilateral triangle.3)Find the total area of the red spot below:4)A man starts at the origin of ##\mathbb{R}^2##. He walks ##1## meter left and ##1## meter up. He walks then ##1/2## of the last distance right. He walks ##1/3## of the last distance down. He walks ##1/4## of the last distance left. Thus the man walks in a spiral. Where does he converge to?5).Ackermann's function is the (unique) function ##A:\mathbb N^2\to \mathbb N## which satisfies the following conditions:a) ##A(0,n)=n+1##,b) ##A(m+1,0)=A(m,1)##,c) ##A(m+1,n+1)=A(m,A(m+1,n))##,for all ##m,n\in \mathbb N##.a) Find (and prove) a closed form expression for ##A(1,n)##.b) Find (and prove) a closed form expression for ##A(2,n)##.c) Find (and prove) a closed form expression for ##A(3,n)##.6)Consider the integrals I and J.##I = \int\limits_0^{\frac{\pi}{2}}\frac{\sin x\cos x}{x+1}dx####J = \int\limits_0^{\pi}\frac{\cos x}{(x+2)^{2}}dx##What is I in function of J?7) Compute the integral ##\int \sqrt{\tan(x)}dx## (the final answer must be written with elementary functions only).8)Find all bases ##b>6## where ##5654## is a prime power.9)Let ##I_n = \int\limits_0^1 x^n\sqrt{1-x}dx##Show that ##I_n = \frac{2n}{2n+3}I_{n-1} \quad \forall n \in \mathbb{N}\backslash\{0\} ##10)Calculate:##\int\limits_{-1}^1\frac{1}{(e^x+1)(x^2 +1)}dx##11)Calculate:##\int \frac{\sqrt[6]{\frac{x}{x-3}} - \sqrt[4]{\frac{x}{x-3}}}{x^2 -3x}dx##12).Prove that every matrix ##A \in M_{n,n}(\mathbb{R})## can be written as the sum of a symmetric and an antisymmetrix matrix.1) Show that for ##0