QUESTION
3.
In the sequence of fractions
1
CO 380 Spring 2012
Assignment #3 Due **Thursday** May 24th, 2012, 10:00 a.m.
1.
Observe that
1 = 1
2 + 3 + 4 = 3
3 + 4 + 5 + 6 + 7 = 5
4 + 5 + 6 + 7 + 8 + 9 + 10 = 7
State, and prove, a generalization suggested by these equations.
2.
The numbers 1 to 9 are to be placed in the circles in such a way
that the sum of the four numbers along each side of the triangle
has the same value, S.
(a)
Prove that 17 S 23.
(b)
Find a suitable arrangement of the numbers when S = 23.
(c)
Show that when S = 20 there are at most 8 di erent choices
for the collection of three numbers which should be placed at
the vertices of the triangle.
1
;
2
1
;
1
2
;
3
1
;
2
2
;
1
3
;
4
1
;
3
2
;
2
3
;
1
4
;
5
1
;
4
2
;
3
3
;
2
2
2
2
2
4
;
1
5
;
6
1
; : : :
fractions equivalent to any given fraction occur many times. For example, fractions equivalent
to
1
2
occur for the rst two times in positions 3 and 14. In which position is the fth occurrence
of a fraction equivalent to
3
7
?
4.
Each of 50 people knows a di erent piece of information. They are allowed to give the
information they know by a phone call between themselves and one other person. During any
call, just one person is permitted to speak and tells the other person all of the information
that they know. With justi cation, determine the minimum number of calls required to enable
each person to know all of the information, and demonstrate how all people can come to learn
all of the information in this minimum number of calls.
5. xyz is a positive three digit integer, with x 6 = 0, z 6 = 0, whose digits are not necessarily
distinct. How many possible xyz exist such that both xyz and zyx are divisible by 4.
6.
Find all quadruples of real numbers (x; y; u; v) satisfying the system of equations
x
2
+ y
2
+ u
2
+ v
2
= 4
xu + yv + xv + yu = 0
xyu + yuv + uvx + vxy = 2
xyuv = 1:
SOLUTION
4) First person convey his information in 49 ways.
Now, Second person will have to make 48 phone calls to convey his information to remaining person.
Similarly,3rd person will have to make 47 phone calls to convey his information.
….
…
Hence, 50th person need not to call any one to convey his information.
Hence ,Required Number of calls = 49 +48+47……2+1+0 = 49(49+1)/2
= 49* 25 =
= 1225 phone calls Answer
(2) (1)There are 18 ways to solve this.
1 Here ,we see sum of values of each
5 7 sides lies between 17 to 23.
9 6
2 4 8 3
1
5 8
9 3
4 2 6 7
1 Note: Here gray color
6 9 triangle for part 2
8 4 and yellow color triangle
2 5 7 3 for part 3.
1
6 9
8 2
4 3 5 7
Here Gray color triangle show sum of each sides equal 20 .
1
6 7
8 3
5 2 4 9
2
4 7
9 3
5 1 6 8
2
6 9
7 1
5 3 4 8
2
6 9
8 1
3 4 5 7
3
2 6
9 4
7 1 5 8
3
4 9
8 1
5 2 6 7
3
4 7
8 2
6 1 5 9
3
5 9
6 1
7 2 4 8
3
5 8
7 1
6 2 4 9
4
2 7
9 3
5 1 8 6
4
3 9
8 1
5 2 7 6
(2) Here Triangles values in Yellow is required arrangement.
7
2 4
6 3
8 1 5 9
7
3 6
5 1
8 2 4 9
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