CombinatoricsDifficulty 5.2AIME, harderFind the answer
## Task 5 - 020515
Seven students are sitting at a table. One is named Fred, one is named Willi, four are named Lutz, and one is named Christian.
We also know that among them, there are two brothers with the last name Scheibner, one with the last name Franke, and four students with the last name Schulz.
a) Of whom can we be absolutely certain about both the first and last names?
b) Why must he be named that way?
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
Solution
a) One of the students must be named Lutz Schulz.
b) Four of the seven students have the first name Lutz. Similarly, four of the seven students have the last name Schulz.
Only three students do not have the last name Schulz. However, since four boys are named Lutz, at least one must have the last name Schulz.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.
Source: NuminaMath-1.5,
licensed Apache-2.0.
Statement and solution reproduced as published; topic and difficulty added by this site.