Solutions to Problems for Math. H90
Issued 26 Oct. 2007
Problem 1: What is the smallest positive multiple of 11 whose decimal digits have an odd sum?
Solution 1: 209 = 1119 is the smallest. Here is why: Because 10N (1)N mod 11 , an
integer is divisible by
Solutions to Problems for Math. H90
Issued 21 Sep. 2007
Problem 1(a): Maple V used to say that the series S := 1 1 + 1 1 + 1 1 + converged to
1/2 . It was wrong because S converges to P/Q where P and Q are your favorite odd primes.
For instance, if P = 3
Solutions to Problems for Math. H90
Issued 27 Aug. 2007
Problem 1: Given is an ellipse E neither a circle nor degenerate (i.e. a straight line segment).
Let be the largest of the areas of triangles inscribed in E . How many inscribed triangles have
maxima
Solutions to Problems for Math. H90
Issued 7 Sep. 2007
Problem 1: Our Omnipotent and Merciful Emperor is resolved to punish his advisors. There
are two dozen of them and, according to our Mighty and Magnanimous Emperor, they all seem
obsessed with advisin
Solutions to Problems for Math. H90
Issued 14 Sep. 2007
Problem 1: Prove that no rectangular array of real numbers ij can satisfy the inequality
( i (j ij)2 ) j ( i ij2 ) unless ij = ij for some two arrays i and j 0 .
Proof 1: Let column vector bj have el
File: Putnam09
Some Solutions for the 2009 Putnam Exam
December 11, 2009 1:05 pm
The purpose of this document is to provide students with more examples of good mathematical
exposition, taking account of all necessary details, with clarity given priority o
Solutions to Problems for Math. H90
Issued 5 Oct. 2007
Problem 1: Calculus is bunk! Any silliness can be proved by calculus. Here is an example:
Choose any positive unteger K . Then K2 = K + K + K + + K + K + K with K repeated K
times on the right-hand si
Solutions to Problems for Math. H90
Issued 16 Nov. 2007
You dont have to solve them all.
Problem 0: A Wire-Frame Cube is constructed out of idealized wire of innitesimal thickness
run along the edges of the cube and fastened at its corners thus:
It can be
Solutions to Problems for Math. H90
Issued 2 Nov. 2007
Problem 1: Four ghostly galleons call them E, F, G and H, sail at night on a ghostly sea so
foggy that one side of a ship cannot be seen from the other. Each ship pursues its course steadily,
changing
Solutions to Problems for Math. H90
Issued 19 Oct. 2007
Problem 1: The Gregorian calendar puts 365 days in every ordinary year, 366 days in every
leap year. Leap years are those years evenly divisible by 4 except for century years (the years
divisible by
Solutions to Problems for Math. H90
Issued 12 Oct. 2007
Problem 1: Enumerating Ordered Pairs of Positive Integers
Supply fast arithmetical procedures to Enumerate provably all ordered pairs of positive integers.
These procedures must achieve a Bijection (
Prof. W. Kahan
Problems Solutions for Math. H90
September 21, 2000 3:35 am
1. Exhibit n and n positive integers k1, k2, , kn whose sum k1 + k2 + + kn = 174 and
whose product k1k2kn is as big as possible, and explain why.
Solution: n = 174/3 = 58 , and k1