MathDB

5

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(7)

2017 Team #5: Collinearity with circumcenter

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2/19/2017
Let ABCABC be an acute triangle. The altitudes BEBE and CFCF intersect at the orthocenter HH, and point OO denotes the circumcenter. Point PP is chosen so that ∠APH=∠OPE=90∘\angle APH = \angle OPE = 90^{\circ}, and point QQ is chosen so that ∠AQH=∠OQF=90∘\angle AQH = \angle OQF = 90^{\circ}. Lines EPEP and FQFQ meet at point TT. Prove that points AA, TT, OO are collinear.
geometrycircumcircle
2017 Algebra/NT #5: Weird Sum

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2/19/2017
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples (a,b,c)(a, b, c) of positive integers such that abc=2310abc=2310 on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute ∑a,b,c∈Nabc=2310(a+b+c),\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c), where N\mathbb{N} denotes the positive integers.
2017 Geometry #5: Circumradii of Small Triangles

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2/19/2017
Let ABCDABCD be a quadrilateral with an inscribed circle ω\omega and let PP be the intersection of its diagonals ACAC and BDBD. Let R1R_1, R2R_2, R3R_3, R4R_4 be the circumradii of triangles APBAPB, BPCBPC, CPDCPD, DPADPA respectively. If R1=31R_1=31 and R2=24R_2=24 and R3=12R_3=12, find R4R_4.
geometry
2017 Combinatorics #5: Kelvin the Frog's Favorite Numbers

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2/19/2017
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if did_i denotes the iith digit, then di≤di+1d_i\le d_{i+1} for at most one value of ii. For example, Kelvin likes the numbers 4321043210, 132132, and 33, but not the numbers 13371337 and 123123. How many 55-digit numbers does Kelvin like?
2017 Theme #5

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5/8/2018
Each of the integers 1,2,...,7291,2,...,729 is written in its base-33 representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: 12101112202122...12101112202122... How many times in this string does the substring 012012 appear?
combinatorics
2017 General #5

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5/8/2018
Given that a,b,ca,b,c are integers with abc=60abc = 60, and that complex number ω≠1\omega \neq 1 satisfies ω3=1\omega^3=1, find the minimum possible value of | a + b\omega + c\omega^2|.
algebra
2017 Guts #5: Diophantine equation II

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2/21/2017
Find the number of ordered triples of positive integers (a,b,c)(a, b, c) such that 6a+10b+15c=3000.6a + 10b + 15c = 3000.
number theoryDiophantine equation