MathDB

9

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

2023 geometry #9

Source:

2/28/2024
Point YY lies on line segment XZXZ such that XY=5XY = 5 and YZ=3Y Z = 3. Point GG lies on line XZXZ such that there exists a triangle ABCABC with centroid GG such that XX lies on line BCBC, YY lies on line ACAC, and ZZ lies on line ABAB. Compute the largest possible value of XGXG.
geometry
HMMT Feb 2023 team p9

Source:

2/20/2023
Let ABCABC be a triangle with AB<ACAB < AC. The incircle of triangle ABCABC is tangent to side BCBC at DD and intersects the perpendicular bisector of segment BCBC at distinct points XX and YY. Lines AXAX and AYAY intersect line BCBC at PP and QQ, respectively. Prove that, if DP⋅DQ=(AC−AB)2DP \cdot DQ = (AC-AB)^2 then AB+AC=3BC.AB + AC = 3BC.
2023 Combinatorics #9

Source:

2/28/2024
There are 100100 people standing in a line from left to right. Half of them are randomly chosen to face right (with all (10050){100 \choose 50} possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.
combinatorics
2023 Algebra NT #9 s_{20}(n) - s_{23}(n)

Source:

2/28/2024
For any positive integers aa and bb with b>1b > 1, let sb(a)s_b(a) be the sum of the digits of aa when it is written in base bb. Suppose nn is a positive integer such that ∑i=1⌊log⁡23n⌋s20(⌊n23i⌋)=103   and   ∑i=1⌊log⁡20n⌋s23(⌊n20i⌋)=115\sum^{\lfloor \log_{23} n\rfloor}_{i=1} s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\, \sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115 Compute s20(n)−s23(n)s_{20}(n) - s_{23}(n).
number theory