MathDB

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Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/1: Affine geometry

Source:

2/17/2019
Let ABCDABCD be a parallelogram. Points XX and YY lie on segments ABAB and ADAD respectively, and ACAC intersects XYXY at point ZZ. Prove that ABAX+ADAY=ACAZ.\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.
HMMTgeometryanalytic geometry
HMMT Algebra/NT 2019/1: Not a sum of two palindromes

Source:

2/17/2019
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits are the same when read backwards.)
HMMTnumber theory
HMMT Combinatorics 2019/1: Palindromes

Source:

2/17/2019
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a continuous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)
HMMT
HMMT Geometry 2019/1: There were too many algebra in the previous round

Source:

2/17/2019
Let dd be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than dd degrees. What is the minimum possible value for dd?
HMMTalgebra