18.S34 PROBLEMS #1
Fall 2007
Problems are marked by the following diculty ratings.
[1] Easy. Most students should be able to solve it.
[2] Somewhat dicult or tricky. Many students should be able to solve it.
[3] Dicult. Only a few students should be able
/Problema 2
#include <stdio.h>
#include <stdlib.h>
void menor(int n)cfw_
int dist=0, lim=0;/cuida que no todos los numeros sean iguales y que no se
pasen del limite
int a, b, i;
printf("Introduce el numero 1 a comparar: ");
scanf("0", &a);
if(a<1|a>1000)
18.S34 PROBLEMS #7
Fall 2007
74. [1] (a) What is the least number of weights necessary to weigh any
integral number of pounds from 1 lb. to 63 lb. inclusive, if the
weights must be placed on only one of the scale-pans of a balance?
Generalize to any numb
18.S34 PROBLEMS #5
Fall 2007
50. [1] A person buys a 30-year $100, 000 mortgage at an annual rate of
8%. What is his or her monthly payment?
51. (a) [1] Person A chooses an integer b etween 0 and 211 1, inclusive.
Person B tries to guess As number by ask
18.S34 PROBLEMS #4
FALL 2007
39. [1] Three students A, B , C compete in a series of tests. For coming
in rst in a test, a student is awarded x p oints; for coming second,
y p oints; for coming third, z p oints. Here x, y , z are p ositive integers
with x
18.S34 (FALL 2006)
PROBLEMS ON INEQUALITIES
1. Let a be a real number and n a p ositive integer, with a > 1. Show that
n+1
n1
an 1 n a 2 a 2 .
2. Let xi > 0 for i = 1, 2, . . . , n. Show that
1
1
1
(x1 + x2 + + xn )
+
+
n2 .
x1 x2
xn
3. If xi > 0, qi >
18.S34 (FALL 2007)
PROBLEMS ON ROOTS OF POLYNOMIALS
Note. The terms root and zero of a polynomial are synonyms.
Those problems which appeared on the Putnam Exam are stated as they
appeared verbatim (except for one minor correction and one clarication).
1.
18.S34 PROBLEMS #6
Fall 2007
62. [1] Find the missing term:
?
63. [1] (a) A drugstore received a shipment of ten b ottles of a certain
drug. Each b ottle contains one thousand pills. The drugstore
received a telegram from the drug company saying that the
18.S34 PROBLEMS #3
Fall 2007
28. [1] Let x, y > 0. The harmonic mean of x and y is dened to be
2xy /(x + y ). The geometric mean is xy . The arithmetic mean (or
average ) is (x + y )/2. Show that
2xy
x+y
xy
,
x+y
2
with equality if and only if x = y .
18.S34 (FALL 2007)
PROBLEMS ON CONGRUENCES AND
DIVISIBILITY
1. (55P) Do there exist 1, 000, 000 consecutive integers each of which con
tains a repeated prime factor?
2. A p ositive integer n is powerful if for every prime p dividing n, we have
that p2 div
18.S34 (FALL, 2007)
PIGEONHOLE PROBLEMS
Note: Notation such as (78P) means a problem from the 1978 Putnam
Exam.
1. (78P) Let A be any set of 20 distinct integers chosen from the arith
metic progression 1, 4, 7, . . . , 100. Prove that there must be two di
18.S34 (FALL, 2007)
PROBLEMS ON PROBABILITY
1. Three closed b oxes lie on a table. One b ox (you dont know which)
contains a $1000 bill. The others are empty. After paying an entry fee,
you play the following game with the owner of the b oxes: you p oint
18.S34 PROBLEMS #2
Fall 2007
14. [1] Here is a square divided into four congruent pieces:
Can a square be divided into ve congruent pieces?
15. [1] Choose any 1000 p oints in the plane. Does there always exist a
straight line which divides the p oints ex
18.S34 (FALL, 2007)
PROBLEMS ON GENERATING FUNCTIONS
Note. All the problems b elow can be done using generating functions.
Many of them can also be done by other methods. However, you should hand
in only solutions which use generating functions. No credit