PUTNAM 2012 WEEK 4. INEQUALITIES.
Easier Problems
1. In each case, which number is larger (or largest): (a) 19 + 99 or
20 + 98? (b) 2000! or 10002000 ? (c) 19991999 , 20001998 , or 19982000 ?
2. Prove
PUTNAM 2011 WEEK 8
DERIVATIVES AND INTEGRALS
In the interest of drama, this worksheet is intended to be even harder than
usual. Fasten your seat belts! Buy your brain a sugary drink!
Easier Problems
1
PUTNAM 2011 WEEK 3:
GEOMETRY, VECTORS, COMPLEX NUMBERS.
It is unfortunate that American kids learn less high school geometry than
their peers in other countries because geometry is a lot of fun! The t
PUTNAM 2010 WEEK 1
INDUCTION AND PIGEONHOLE PRINCIPLE
Work in groups. Try small cases. Do examples. Look for patterns.
Use lots of paper. Talk it over. Choose effective notation. Try the
problem with
PUTNAM 2011 WEEK 2. GAMES AND INVARIANTS.
Easier Problems
1. Beavis and Butthead play double chess: all the rules are like in
usual chess but every player makes two moves every time. Prove
that the wh
PUTNAM 2010 WEEK 3. INEQUALITIES.
Easier Problems
1. Which is larger, (a) 19 + 99 or 20 + 98? (b) 2000! or 10002000 ?
(c) 19991999 , 20001998 , or 19982000 ?
2. Prove that ab (a + b)/2 with equality i
PUTNAM 2011 WEEK 6: GRAPH THEORY.
A graph is a collection of vertices and a collection of edges connecting
some vertices. We can visualize a graph by drawing a picture where vertices are dots and edge
PUTNAM 2011 WEEK 10: PROBABILITY.
This is the last worksheet. The Putnam exam will take place on Saturday,
December 3, and will take the whole day. I will post the exam location on
the website as soon
PUTNAM 2017 WEEK 7
SEQUENCES AND SERIES
Easier Problems
1. Prove that any two consecutive Fibonacci numbers are coprime:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
2. Find lines in the Pascal triangle th
PUTNAM 2011 WEEK 1. PROOF BY INDUCTION.
General but surprisingly effective advice. Work in groups. Try small
cases. Do examples. Look for patterns. Use lots of paper. Talk it over.
Choose effective no
PUTNAM 2011 WEEK 5: NUMBER THEORY.
Basic number theory concepts to remember and use: modular arithmetic,
unique factorization, greatest common divisor (a,b) can be written as a linear combination ax+b