Can you solve these counting problems?

There are 100 players in a single elimination tournament. How many matches need to be played? Three sets have their union as the integers from 1 to 10, and their intersection is the empty set. How many such sets are possible? Special thanks this month to: Mike Robertson, Daniel Lewis, Kyle, Lee Redden. Thanks to all supporters on Patreon! http://www.patreon.com/mindyourdecisions

0:00 problems
2:25 solution 1
5:18 solution 2

Problem 1
The College Panda
https://thecollegepanda.com/12-tricky-sat-math-questions/
Brainly
https://brainly.com/question/13082487
AoPS 2023 AMC 10B, P15
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_15
Problem 2
1985 Putnam A1
https://prase.cz/kalva/putnam/psoln/psol851.html
Twitter
https://twitter.com/BannedIran/status/1416592272699842564/photo/1
1985 Putnam
https://kskedlaya.org/putnam-archive/1985.pdf
Math StackExchange
https://math.stackexchange.com/questions/1190662/1985-putnam-a1-solution
1985 Putnam solutions
https://math.berkeley.edu/~giventh/19109_85s.pdf

More Putnam videos
--------------------------------------
1968 Putnam exam A1
An integral that implies pi is less than 22/7
https://youtu.be/MFnOaaXwOJk

2004 Putnam A1
If your free throw percentage is below 75%, and later is above 75%, is there necessarily a point where it is exactly 75%?
https://youtu.be/vqpWwgpDs58

1978 Putnam B1
A convex octagon inscribed in a circle has four consecutive sides of length 3 and four consecutive sides of length 2. Find the area of the octagon.
https://youtu.be/3Ze-59CxFbE

1994 Putnam A2
Finding the area of a sector of an ellipse
https://youtu.be/fNlr1z_kNlc

1989 Putnam A4
For an irrational number &alpha between 0 and 1, is there a finite game with an unbiased coin such that the probability of one player winning the game is α?
https://youtu.be/lORQ_wt2MZY

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