1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a 6. (10) 2008 circular coins, possibly of different diameters, are placed on the surface of a flat table in such a way that no coin is on top of another coin. What is the largest number of points at which two of the coins could be touching? 7. (9) Let G be a group of finite order and identity e. Suppose that is an automorphism of G onto itself with the following properties: (1) (x) = x if and only if x = e; (2) ((x)) = x for each element x of G. (a) Give an example of a group and automorphism for which these conditions are satisfied. (b) Prove that G is commutative (i.e., xy = yx for each pair x, y of elements in G). 8. (11) Let b≥2 be an integer base of numeration and let 1 < r
1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a 6. (10) 2008 circular coins, possibly of different diameters, are placed on the surface of a flat table in such a way that no coin is on top of another coin. What is the largest number of points at which two of the coins could be touching? 7. (9) Let G be a group of finite order and identity e. Suppose that is an automorphism of G onto itself with the following properties: (1) (x) = x if and only if x = e; (2) ((x)) = x for each element x of G. (a) Give an example of a group and automorphism for which these conditions are satisfied. (b) Prove that G is commutative (i.e., xy = yx for each pair x, y of elements in G). 8. (11) Let b≥2 be an integer base of numeration and let 1 < r
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...
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![1. (10) An m x n rectangular array of distinct real numbers has the
property that the numbers in each row increase from left to right.
The entries in each column, individually, are rearranged so that the
numbers in each column increase from top to bottom. Prove that
in the final array, the numbers in each row will increase from left to
right.
2. (11) Determine distinct positive integers a, b, c, d, e such that the
five numbers a, b², c³, d4, e5 constitute an arithmetic progression.
(The difference between adjacent pairs is the same.)
3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub-
sets with the same sum if and only if k divides 1½n(n + 1) and
n≥ 2k - 1.
4. (6) Suppose that f(x) is a continuous real-valued function defined
on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies
(i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1.
(a) Prove that there exists a number a for which 0 < a < 1 and
f'(a) < f(1);
(b) Prove that there exists a unique number b for which a <b<1
and f'(a) = f(b)/b.
=
5. (6) For x 1 and x 0, let
-
f(x)=
−8[1 − (1 − x) 1/2]3
-
=
x2
(a) Prove that limo f(x) exists. Take this as the value of f(0).
(b) Determine the smallest closed interval that contains all values
assumed by f(x) on its domain.
(c) Prove that f(f(f(x))) = f(x) for all x ≤ 1.
6. (4) Let h(n) denote the number of finite sequences {a1, a2,.
of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak
ak}
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Transcribed Image Text:1. (10) An m x n rectangular array of distinct real numbers has the
property that the numbers in each row increase from left to right.
The entries in each column, individually, are rearranged so that the
numbers in each column increase from top to bottom. Prove that
in the final array, the numbers in each row will increase from left to
right.
2. (11) Determine distinct positive integers a, b, c, d, e such that the
five numbers a, b², c³, d4, e5 constitute an arithmetic progression.
(The difference between adjacent pairs is the same.)
3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub-
sets with the same sum if and only if k divides 1½n(n + 1) and
n≥ 2k - 1.
4. (6) Suppose that f(x) is a continuous real-valued function defined
on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies
(i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1.
(a) Prove that there exists a number a for which 0 < a < 1 and
f'(a) < f(1);
(b) Prove that there exists a unique number b for which a <b<1
and f'(a) = f(b)/b.
=
5. (6) For x 1 and x 0, let
-
f(x)=
−8[1 − (1 − x) 1/2]3
-
=
x2
(a) Prove that limo f(x) exists. Take this as the value of f(0).
(b) Determine the smallest closed interval that contains all values
assumed by f(x) on its domain.
(c) Prove that f(f(f(x))) = f(x) for all x ≤ 1.
6. (4) Let h(n) denote the number of finite sequences {a1, a2,.
of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak
ak}
>
Transcribed Image Text:6. (10) 2008 circular coins, possibly of different diameters, are placed
on the surface of a flat table in such a way that no coin is on top of
another coin. What is the largest number of points at which two of
the coins could be touching?
7. (9) Let G be a group of finite order and identity e. Suppose that
is an automorphism of G onto itself with the following properties:
(1) (x) = x if and only if x = e; (2) ((x)) = x for each element
x of G.
(a) Give an example of a group and automorphism for which these
conditions are satisfied.
(b) Prove that G is commutative (i.e., xy = yx for each pair x, y of
elements in G).
8. (11) Let b≥2 be an integer base of numeration and let 1 < r <b−1.
Determine the sum of all r-digit numbers of the form
Ar−1ɑr—2 . . . A2A1a0 = ar−1b"−1 + ar-26"-2+
· + air + ao
whose digits increase strictly from left to right: 1≤ar-1 < ar−2 <
· · · < a1 < ao ≤b-1.
9. (2) For each positive integer n, let
Sn =
n 2k
k2.
k=1
Prove that Sn+1/S is not a rational function of n.
10. (8) A point is chosen at random (with the uniform distribution) on
each side of a unit square. What is the probability that the four
points are the vertices of a quadrilateral with area exceeding ?
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