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The 62nd William Lowell Putnam Mathematical Competition Saturday, December 1, 2001

The document is the description for the 62nd William Lowell Putnam Mathematical Competition from December 1, 2001. It contains 6 problems labeled A-1 through A-6 and 6 additional problems labeled B-1 through B-6. The problems cover a range of mathematical topics including sets, probability, polynomials, geometry, number theory, and real analysis.

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Victor Manuel Grijalva Altamirano
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86 views1 page

The 62nd William Lowell Putnam Mathematical Competition Saturday, December 1, 2001

The document is the description for the 62nd William Lowell Putnam Mathematical Competition from December 1, 2001. It contains 6 problems labeled A-1 through A-6 and 6 additional problems labeled B-1 through B-6. The problems cover a range of mathematical topics including sets, probability, polynomials, geometry, number theory, and real analysis.

Uploaded by

Victor Manuel Grijalva Altamirano
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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The 62nd William Lowell Putnam Mathematical Competition

Saturday, December 1, 2001

A-1 Consider a set and a binary operation  , i.e., for each B-2 Find all pairs of real numbers 6/ QE
satisfying the sys-
 ,    . Assume   
   for all tem of equations
 . Prove that     
  for all  .
" "
A-2 You have coins      . For each , ! is bi- ( 6 /  (SR Q 
&RE/  ( Q 

ased so that, when tossed, it has probability "#$ &%'()"

/ % Q
" "
of fallings heads. If the * coins are tossed, what is the 2 Q %$ Q 1 2T/ 1
U
probability that the number of heads is odd? Express / %
the answers as a rational function of * .
A-3 For each integer + , consider the polynomial B-3 For any positive integer * , let VW*YX denote the closest in-
,.- teger to Z * . Evaluate
&/
0 /$12) &%'+3(54
/  ( 6+725%
 
,8- [
For what values of + is &/
the product of two non- \ %E_ ` (M%aY_ ^` 
constant polynomials with integer coefficients?
^]8 % 
A-4 Triangle 9:; has an area 1. Points < => lie, respec-
tively, on sides :; , ?9 , 9?: such that 9?< bisects : =
at point @ , : = bisects  > at point , and  > bisects
B-4 Let
b denote the set of rational
hgi numbers
P different from
9< at point A . Find the area of the triangle @ A . 2;"  D  "dc . Define e7f by e. 6/ /j23"#'/ .
Prove or disprove that
A-5 Prove that there are unique positive integers  , * such [
that 
B8 2)  (C"
 %ED'DF" . k
eml ^n
o )p$
A-6 Can an arc of a parabola inside a circle of radius 1 have ]8
a length greater than 4?
B-1 Let * be an even positive integer. Write the numbers where e l n denotes e composed with itself * times.
"  % G *  in the squares of an *IHJ* grid so that the
-th row, from left to right, is B-5 Let  and  be real numbers in the interval 6D  "#E%
,
and let q be a continuous real-valued function such
6K2L"
&*
()"  6;2M"
*N(M% O 6;2M"
*P(5*  that qY rqs 6/
O
t u qs 6/
(  / for all real / . Prove that
Color the squares of the grid so that half of the squares qs 6/
 Cv / for some constant v .
in each row and in each column are red and the other
half are black (a checkerboard coloring is one possi- B-6 Assume that  
$w. is an increasing sequence of pos-
bility). Prove that for each coloring, the sum of the itive real numbers such that xryrz  F#'* D . Must
numbers on the red squares is equal to the sum of the there exist infinitely many positive integers * such that
numbers on the black squares.   (  ^B %   for } "  % G *K2)" ?
a{ {|

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