MathDB

8

Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #8

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2/12/2018
For how many pairs of sequences of nonnegative integers (b1,b2,…,b2018)(b_1,b_2,\ldots, b_{2018}) and (c1,c2,…,c2018)(c_1,c_2,\ldots, c_{2018}) does there exist a sequence of nonnegative integers (a0,…,a2018)(a_0,\ldots, a_{2018}) with the following properties:
[*] For 0≤i≤2018,0\leq i\leq 2018, ai<22018.a_i<2^{2018}. [*] For 1≤i≤2018,bi=ai−1+ai1\leq i \leq 2018, b_i=a_{i-1}+a_i and ci=ai−1∣aic_i=a_{i-1}|a_i;
where ∣| denotes the bitwise or operation?
2018 Combinatorics #8

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2/12/2018
A permutation of {1,2,…,7}\{1, 2, \dots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation (3,4,2,1,6,5,7)(3, 4, 2, 1, 6, 5, 7) can be partitioned correctly into the blocks [3,4,2,1][3, 4, 2, 1] and [6,5,7][6, 5, 7], since when these blocks are sorted, the permutation becomes (1,2,3,4,5,6,7)(1, 2, 3, 4, 5, 6, 7). Find the expected value of the maximum number of blocks into which the permutation can be partioned correctly.
2018 Geometry #8

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2/12/2018
Let ABCABC be an equilateral triangle with side length 8.8. Let XX be on side ABAB so that AX=5AX=5 and YY be on side ACAC so that AY=3.AY=3. Let ZZ be on side BCBC so that AZ,BY,CXAZ,BY,CX are concurrent. Let ZX,ZYZX,ZY intersect the circumcircle of AXYAXY again at P,QP,Q respectively. Let XQXQ and YPYP intersect at K.K. Compute KX⋅KQ.KX\cdot KQ.
geometrycircumcircle
2018 Team #8

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2/12/2018
Allen plays a game on a tree with 2n2n vertices, each of whose vertices can be red or blue. Initially, all of the vertices of the tree are colored red. In one move, Allen is allowed to take two vertices of the same color which are connected by an edge and change both of them to the opposite color. He wins if at any time, all of the verices of the tree are colored blue.
(a) Show that Allen can win if and only if the vertices can be split up into two groups V1V_1 and V2V_2 to size nn, such that each edge in the tree has one endpoint in V1V_1 and one endpoint in V2V_2.
(b) Let V1={a1,…,an}V_1 = \left\{ a_1, \ldots, a_n \right\} and V2={b1,…,bn}V_2 = \left\{ b_1, \ldots, b_n \right\} from part (a). Let MM be the minimum over all permutations σ\sigma of {1,…,n}\left\{ 1, \ldots, n \right\} of the quantity ∑i=1nd(ai,bσ(i)), \sum\limits_{i = 1}^{n} d(a_i, b_{\sigma(i)}), where d(v,w)d(v, w) denotes the number of edges along the shortest path between vertices vv and ww in the tree. Show that if Allen can win, then the minimum number of moves that it can take for Allen to win is equal to MM.
2018 General #8

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11/12/2018
Equilateral triangle ABCABC has circumcircle Ω\Omega. Points DD and EE are chosen on minor arcs ABAB and ACAC of Ω\Omega respectively such that BC=DEBC=DE. Given that triangle ABEABE has area 33 and triangle ACDACD has area 44, find the area of triangle ABCABC.
geometrycircumcircle
2018 Theme #8

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11/13/2018
Crisp All, a basketball player, is dropping dimes and nickels on a number line. Crisp drops a dime on every positive multiple of 1010, and a nickel on every multiple of 55 that is not a multiple of 1010. Crisp then starts at 00. Every second, he has a 23\frac{2}{3} chance of jumping from his current location xx to x+3x+3, and a 13\frac{1}{3} chance of jumping from his current location xx to x+7x+7. When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp stops on a dime?
probability
2018 November Team #8

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11/13/2018
Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 2\sqrt2 away from the vertex.When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?
geometry3D geometry