MathDB

2

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #2: Arithmetic and Geometric Series

Source:

2/16/2013
Let {an}n≥1\{a_n\}_{n\geq 1} be an arithmetic sequence and {gn}n≥1\{g_n\}_{n\geq 1} be a geometric sequence such that the first four terms of {an+gn}\{a_n+g_n\} are 00, 00, 11, and 00, in that order. What is the 1010th term of {an+gn}\{a_n+g_n\}?
HMMTarithmetic sequencegeometric sequence
2013 Team #2: Cafe Tables and Seats

Source:

2/18/2013
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, MM groups consisting of a total of NN people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of M+NM + N?
2013 HMMT Guts #2: Squares Forming Arithmetic Progression

Source:

3/26/2013
The real numbers xx, yy, zz, satisfy 0≤x≤y≤z≤40\leq x \leq y \leq z \leq 4. If their squares form an arithmetic progression with common difference 22, determine the minimum possible value of ∣x−y∣+∣y−z∣|x-y|+|y-z|.
HMMTarithmetic sequence
2013 HMMT Geometry # 2

Source:

3/3/2024
Let ABCDABCD be an isosceles trapezoid such that AD=BCAD = BC, AB=3AB = 3, and CD=8CD = 8. Let EE be a point in the plane such that BC=ECBC = EC and AE⊥ECAE \perp EC. Compute AEAE.
geometry