MathDB

4

Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #4

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2/12/2018
Distinct prime numbers p,q,rp,q,r satisfy the equation 2pqr+50pq=7pqr+55pr=8pqr+12qr=A2pqr+50pq=7pqr+55pr=8pqr+12qr=A for some positive integer A.A. What is AA?
number theoryprime numbers
2018 Combinatorics #4

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2/12/2018
How many ways are there for Nick to travel from (0,0)(0,0) to (16,16)(16,16) in the coordinate plane by moving one unit in the positive xx or yy direction at a time, such that Nick changes direction an odd number of times?
2018 Geometry #4

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2/12/2018
A paper equilateral triangle of side length 22 on a table has vertices labeled A,B,C.A,B,C. Let MM be the point on the sheet of paper halfway between AA and C.C. Over time, point MM is lifted upwards, folding the triangle along segment BM,BM, while A,B,A,B, and CC on the table. This continues until AA and CC touch. Find the maximum volume of tetrahedron ABCMABCM at any time during this process.
geometry3D geometrytetrahedron
2018 Team #4

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2/12/2018
In acute triangle ABCABC, let DD, EE, and FF be the feet of the altitudes from AA, BB, and CC respectively, and let LL, MM, and NN be the midpoints of BCBC, CACA and ABAB, respectively. Lines DEDE and NLNL intersect at XX, lines DFDF and LMLM intersect at YY, and lines XYXY and BCBC intersect at ZZ. Find ZBZC\frac{ZB}{ZC} in terms of ABAB, ACAC, and BCBC.
2018 General #4

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11/12/2018
Find the number of eight-digit positive integers that are multiples of 99 and have all distinct digits.
number theory
2018 Theme #4

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11/13/2018
I have two cents and Bill has nn cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all nn of his cents on some combination of pencil packages to get PP pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all n+2n+2 of his cents on some combination of pencil packages to get fewer than PP pencils. What is the smallest value of nn for which this is possible?
Note: Both times Bill must spend all of his cents on pencil packages, i.e. have zero cents after either purchase.
2018 November Team #4

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11/13/2018
Let aa and bb be real numbers greater than 1 such that ab=100ab=100. The maximum possible value of a(log⁡10b)2a^{(\log_{10}b)^2} can be written in the form 10x10^x for some real number xx. Find xx.