MathDB

9

Part of 2019 BMT Spring

Problems(5)

Product of products is a root of unity (BMT 2019 Algebra #9)

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5/6/2019
Let an a_n be the product of the complex roots of x2n=1 x^{2n} = 1 that are in the first quadrant of the complex plane. That is, roots of the form a+bi a + bi where a,b>0 a, b > 0 . Let r=a1⋯a2⋅…⋅a10 r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} . Find the smallest integer k k such that r r is a root of xk=1 x^k = 1 .
The incircle-generating tetrahedron (BMT 2019 Geo #9)

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5/18/2019
Let ABCD ABCD be a tetrahedron with ∠ABC=∠ABD=∠CBD=90∘ \angle ABC = \angle ABD = \angle CBD = 90^\circ and AB=BC AB = BC . Let E,F,G E, F, G be points on AD‾ \overline{AD} , BD BD , and CD‾ \overline{CD} , respectively, such that each of the quadrilaterals AEFB AEFB , BFGC BFGC , and CGEA CGEA have an inscribed circle. Let r r be the smallest real number such that [△EFG][△ABC]≤r \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r for all such configurations A,B,C,D,E,F,G A, B, C, D, E, F, G . If r r can be expressed as a−bcd \dfrac{\sqrt{a - b\sqrt{c}}}{d} where a,b,c,d a, b, c, d are positive integers with gcd⁡(a,b) \gcd(a, b) squarefree and c c squarefree, find a+b+c+d a + b + c + d .
Note: Here, [P] [P] denotes the area of polygon P P . (This wasn't in the original test; instead they used the notation area(P) \text{area}(P) , which is clear but frankly cumbersome. :P)
geometry3D geometrytetrahedron
Mathlete's log: this is an awfully complex problem! (BMT 2019 Discrete #9)

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5/26/2019
Let z=12(2+i2) z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) . The sum ∑k=01311−zek⋅iπ7 \sum_{k = 0}^{13} \dfrac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} can be written in the form a−bi a - bi . Find a+b a + b .
2019 BMT Team 9

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1/7/2022
You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices. Each edge is adjacent to both of its vertices. What is the minimum number of colors required to do this?
combinatorics
2019 BMT Individual 9

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1/9/2022
Define an almost-palindrome as a string of letters that is not a palindrome but can become a palindrome if one of its letters is changed. For example, TRUSTTRUST is an almost-palindrome because the RR can be changed to an SS to produce a palindrome, but TRIVIALTRIVIAL is not an almost-palindrome because it cannot be changed into a palindrome by swapping out only one letter (both the AA and the LL are out of place). How many almost-palindromes contain fewer than 44 letters.
combinatorics