MathDB

11

Part of 2017 AIME Problems

Problems(2)

Median-ception

Source: 2017 AIME I #11

3/8/2017
Consider arrangements of the 99 numbers 1,2,3,…,91, 2, 3, \dots, 9 in a 3×33 \times 3 array. For each such arrangement, let a1a_1, a2a_2, and a3a_3 be the medians of the numbers in rows 11, 22, and 33 respectively, and let mm be the median of {a1,a2,a3}\{a_1, a_2, a_3\}. Let QQ be the number of arrangements for which m=5m = 5. Find the remainder when QQ is divided by 10001000.
AMCAIMEAIME I2017 AIME Icombinatoricsgrids
The Importance of Being Earnestrongly connected graph

Source: 2017 AIME II #11

3/24/2017
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
AMCAIMEgraph theorycounting