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LSU MATH 2020 - MATH 2020 Midterm Exam

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Math2020 Fall 2005 Midterm examDue Friday October 21 2005This is a take home exam. You are allowed to discuss questions with other people, but you mustmake sure that the final work is your own. You should understand the answers well enoughto write out solutions on your own without looking at or referring to any other material, e.g.,without copying from other students or from text books. You should be prepared to answerany questions about how you arrived at your answers. If there is any doubt about any studentscopying without understanding work, then this exam will be given as a class time exam.For each question, explain how you arrive at your answer.Proof by induction questions1) Prove the following formula by induction13+ 33+ 53+ · · · + (2n − 1)3= n2(2n2− 1).2) For this question, you may assume the triangle inequality, which says that for a, b realnumbers,|a + b| ≤ |a| + |b|.Assuming the triangle inequality, prove by induction that if x1, x2, · · · xnare n real numbers,then|x1+ x2+ · · · + xn| ≤ |x1| + |x2| + · · · + |xn|3)I have three types of tiles, as follows:There are as many of each type as I need, and each tile can be rotated or turned over.a) Show how to fill each of the following 3 × 3 grids, with one square removed, with the giventiles (tiles must not overlap).b) Prove by induction that for all positive integers n, (3n)2− 1 is divisible by 4.(For full credit, this must be a proof by induction, though other proofs of this result are possible.)c)Show how to fill the following regions with the three above types of tiles:1d)Making use of the above results, prove by induction that if I give you any 3n× 3nboard, withany one square of my choice removed, you can fill in the remainder with the above 3 kinds oftiles. (Tiles may be rotated and turned over, and as many as required may be used, but theymay not overlap).e)Below, on the left is an example of a possible tiling when n = 2. Fill in the diagram on theright, but leaving the black square uncovered, all other squares covered, with the allowed tiles.Counting problems4)I have 6 segments of a ring, with colors as listed below, which can be put together to form acomplete ring.3 pieces are red2 pieces are blue1 piece is yellow                   The pieces:One possible ring:a) How many ways can I put these together to form different rings? Note that a ring can berotated or turned over, which does not change it into a different ring.b) Draw all the possible rings. (Don’t draw rotations or flips of any ring, just draw each oneonce, in some position).c) Now I have three large boxes, one contains lots of red pieces, one contains lots of blue, andone contains lots of yellow pieces (pieces all as above). How many different ways can I choose6 pieces, each of which must be red, yellow, or blue. E.g., I could take (Y, Y, Y, B, B, B), whichis counted as the same choice as (Y, B, Y, B, Y, B).d) How many possible rings are there of 6 of the above pieces, from red, yellow and blue pieces?(Now the order does matter, but rings can still be rotated and turned over).5)Starting at a point (0, 0, 0) in a three dimensional lattice, with x, y, z coordinates, how manyways are there to get to a point (m, n, p)? Give a formula in terms of m, n, p.Here n, m, p are positive integers, and each possible move is a step of 1 unit in the positive x,y, or z direction. I.e., from a point (r, s, t), you can move to one of (r + 1, s, t), (r, s + 1, t), or(r, s, t + 1). E.g., to get from (0, 0, 0) to (1, 1, 1), one possible sequence of steps is(0, 0, 0) → (1, 0, 0) → (1, 0, 1) → (1, 1,


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LSU MATH 2020 - MATH 2020 Midterm Exam

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