Download Linear algebra 1. Let A and B be n × n matrices such that AB + A + B

Linear algebra
1. Let A and B be n × n matrices such that AB + A + B = 0. Prove that AB = BA.
2. Let A be a square integer matrix such that the sum of each column is divisible by n. Show that det A
is divisible by n too.
3. Prove that there exists a sequence of An 0 − 1 matrices such that their determinant is exponentially
large, that is, An is an n × n matrix whose elements are only 0’s and 1’s, and det An > cn + d for some
c > 1, d constants.
4. * Let a1 , a2 , . . . an be distinct integers. Show that the polynomial
(x − a1 )(x − a2 ) · · · (x − an ) − 1
is irreducible over the integers.
5. Show that if the real polynomial f (x) is nonnegative for every real x, then f (x) = (a(x))2 + (b(x))2
for some a(x), b(x) real polynomials.
6. Show that the polynomial
f (x) = 1 +
x2
xn
x
+
+ ··· +
1!
2!
n!
has no multiple roots.
7. Prove that if A2 = I for the n × n matrix A, then
rank(I − A) + rank(I + A) = n.
8. Let p(x) = x5 + x and q(x) = x5 + x2 . Find all complex pairs (w, z) for which w 6= z and p(w) = p(z)
and q(w) = q(z).
9. Let A be a complex n × n matrix such that A 6= λI for every λ ∈ C. Prove that A is similar to a
matrix with at most one nonzero element in its diagonal.
10. Compute the inverse of the following matrix:
1
ε
ε2
1
ε2
ε4
···
···
···
ε
ε2n−2
εn−1
ε2n−2
···
ε(n−1)

1
1

1

 ..
.
1

1
n−1
2






2π
where ε = cos( 2π
n ) + i sin( n ).
11. * Let C, X, Y be n × n complex matrices. Show that the equation
XY − Y X = C
has a solution iff trC = 0.
12. Compute the value of the following determinant:

1
b1

b1

b1

 ..
.
1
a1
b2
b2
1
a1
a2
b3
... 1
. . . a1
. . . a2
. . . a3
..
.
b1
b2
b3
...
bn

1
a1 

a2 

a3 



an
13. Let a1 , . . . , a9 be positive integers whose prime divisors are all smaller than 20. Show that the product
of some (at least 1) of them is a square number.
14. Let A be an n × n matrix such that every element in its diagonal is 0 and every other element is either
1 or −1. Show that det A 6= 0.
15. Prove that if A is a complex n×n matrix for which |aik | ≤ 1 ∀i, k = 1, . . . , n holds, then | det A| ≤ nn/2 .
Find also a matrix for which equality hold.
16. Let A, B be n × n matrices.
(a) Show that tr(AB) = tr(BA).
(b) Show matrices A, B, C such that tr(ABC) 6= tr(ACB).
17. * Suppose that every root of the polynomial xn + an−1 xn−1 + · · · + a0 is real. Show that the interval
an−1
n−1
−
±
n
n
r
a2n−1 −
2n
an−2
n−1
contains all of the roots.
18. Let A be an m × n matrix, m ≥ n. Show that the eigenvalues of AAT are the same as those of AT A
and m − n times 0.
19. Let A and B be real n × n matrices, AB = I. Show that if the elements of A are all nonnegative
numbers, then B has at least n positive and at least n negative elements.
20. Let B and C matrices such that they have the same number of rows, and let A denote the matrix we
obtain by writing the columns of C after the columns of B. Show that
det(AT A) ≤ (det(B T B))(det(C T C)).
21. * Prove that for any a element of any A generating set of the group hQ, +i, A − {a} is also a generating
set.
22. Prove that every real polynomial p(x) has a (nonzero) multiple p(x)q(x) such that it consists of prime
powers of x only.
23. * Let p(x) be a polynomial with N nonzero coefficients, that is, of the form
p(x) = a1 xn1 + · · · + aN xnN .
Show that every nonzero root of p has multiplicity less than N .
24. * Let 0 ≤ α1 < α2 < · · · < αn integers and 0 < x1 < · · · < xn real numbers. Show that the
determinant of the following matrix is positive.
1
xα
1
α
x2 1

 ..
 .

1
xα
n
2
xα
1
α2
x2
2
xα
n
n
. . . xα
1
αn
. . . x2
..
.
n
. . . xα
n





25. Let v1 , v2 , . . . , vd+2 be points in Rd . Show that they can be partitioned into two classes such that the
convex hulls of the two classes intersect. (This is Radon’s Theorem.)
Number theory
26. * Let p be prime and a, b, c, d ≥ 0 integers, b, d < p. Prove that
pa + b
pc + d
a
b
≡
(mod p).
c
d
27. Let f be a nonconstant polynomial with integer coefficients. Prove that there are infinitely many
primes which divide f (n) for some n positive integer.
28. Let pn denote the n-th prime. Show that for n ≥ 2, pn + pn+1 is the product of at least 3 (not
necessarily distinct) primes.
29. * Compute the determinant of the n × n matrix ((i, j))i,j , where (i, j) denotes the greatest common
divisor of i and j.
30. (a) Prove that for every k positive integer there exist k consecutive composite numbers.
(b) Prove that from every (strictly increasing) arithmetic progression of positive integers, for every
k there exist k consecutive elements of the sequence which are all composite.
31. * The sequence bn is defined by
b1 = 0,
b2 = 2,
b3 = 3,
bn+1 = bn−1 + bn−2
(n ≥ 3)
Show that for p prime p|bp .
32. Let p, q > 1 be integers, (p, q) = 1. We say that the number m can be built (from p and q) if there
exist x, y ≥ 0 integers such that px + qy = m.
(a) Show that only finitely many positive integers can not be built.
(b) Which is the largest number which can not be built?
(c) How many numbers can not be built?
33. Show a polynomialf with integer coefficients such that it has no rational roots, but f (x) = 0 (mod m)
has a solution for every m positive integer.
34. * Let n be a positive integer. Prove that there exists an N positive integer such that whenever p > N
is prime, then xn + y n = z n (mod p) has a nontrivial solution. (A solution is trivial if at least one of
x, y, z is 0.)
35. Find all integer k’s for which the equation
x2 + y 2 + z 2 = kxyz
has a positive integer solution.
36. Show that if n is an odd positive integer, then n|2n! − 1.
37. * Let an be the coefficient of wn in (1 + w + w2 )n . Show that if p > 3 is prime then
ap ≡ 1
(mod p)
38. Let a, b be integers, d 6= 0. Show that from the sequence
a, a + d, a + 2d, . . . , a + nd, . . .
has infinitely many distinct elements such that they have the same prime divisors.
39. Let f (x) be a nonconstant polynomial with integer coefficients. Prove that there are infinitely many
primes which divide f (x) for some integer x.
40. Show infinitely many distinct pair of positive integers m, n such that m and n have the same prime
divisors and m + 1 and n + 1 also have the same prime divisors.
41. Show that every positive integer has a multiple with only 0 and 1 digits.
42. (a) Show infinitely many distinct pair of positive integers a, b such that a|b2 + b + 1 and b|a2 + a + 1.
(b) Determine all such pairs.
43. Show that σ(n) (that is, the sum of positive divisors of the positive integer n) is even iff n is a square
number or the double of a square number.
44. Determine all the integers from the interval [1, n] which are relative primes (coprimes) to all the others.
45. Let f (x) = xy (mod x + y), and for some x0 , x1 positive integers let us define the sequence xk the
following way:
xk = f (xk−1 , xk−2 ).
Show that for every positive integer n there exist x0 and x1 such that the resulting sequence is a cycle
of length n.
46. Let n, k be positive integers. Show that the sum
1
1
1
+
+ ··· +
n n+1
n+k
can not be an integer.
47. * Let s0 = 0, s1 integer and, for n ≥ 0, sn+2 = A0 sn + A1 sn+1 for some fixed A0 , A1 . Show that for
every positive integers n, k
(sn , sk ) = s(n,k)
where (a, b) denotes the greatest common divisor of a and b.
2
48. Show that if n ≥ 5 integer, then b n3 c is composite.
49. Which elements of the following sequence are primes?
101, 10101, 1010101, . . .
50. Show that from among 10 consecutive porisitve integers there is always at least one which is relatively
prime to all the others.
x
51. Show that for every m ∈ Z+ there exists xZ+ such that each of x + 1, xx + 1, xx + 1, . . . is divisible
by m.
52. Show that for every positive integer n there is a multiple of 2n which contains no 0 digit.
Series, sums, limits
53. Show that if a1 , a2 , . . . are distinct positive integers for which the decimal fraction
0, (a1 )(a2 ) . . .
is rational, then the sum
P∞
1
i=1 ai
converges.
54. * Let a1 , a2 , . . . an infinite sequence of which every element is either 2 or 3. Prove that the number
1
1
1
+
+
+ ···
a1
a1 a2
a1 a2 a3
is rational iff ai is periodic after a certain index.
55. Show that the only accumulation point of the sequence an = n!e − [n!e] is 0.
56.
(n2 + 1)(n2 + 2) . . . (n2 + n)
=?
n→∞ (n2 − 1)(n2 − 2) . . . (n2 − n)
lim
57.
√
2·
√
4
4·
√
8
8···
√
2n
2n · · · =?
58. Drop all terms from the sum
1 1 1
+ + + ...
1 2 3
which contain 6 as a digit. Show that the remaining sum converges.
59. Let Π : Z+ → Z+ be a bijection. Find C < ∞ such that
∞
X
1
< C.
nΠ(n)
n=1
60. Show a sequence of numbers p1 , p2 , . . . such that pn ≥ 0 ∀n,
bijection
∞
X
P∞
n=1
pn = 1 and for every S : Z+ → Z+
pn S(n) = ∞
n=1
holds. (That is, if we have a probability distribution on Z+ determined by the probabilities (pn ), it
has an infinite expectation no matter how it is actually distributed among the numbers).
P∞
P∞
61. Is there a sequence an of positive integers such that n=1 ann2 and n=1 a1n both converge?
62. We call a number a whole power if it is of the form ab , where a, b > 1 integers.
(a) Show that for every n ∈ Z+ there is an arithmetic sequence of length n whose elements are all
whole powers.
(b) Show that there is no infinite arithmetic sequence of whole powers.
63. Compute the following sum.
∞
X
1
n(n
+ 1)
n=1
64. Compute the following sum.
∞
X
1
(2n
−
1)2n
n=1
65. *
Y p prime
1
1− 2
p
=?
21. Let α > 1 be irrational, and β is the solution of 1/α + 1/β = 1. Prove that among the numbers
bnαc, bnβc each positive integer appears exactly once.
Analysis
66. * The real sequence an is subadditive if
am+n ≤ am + an
Prove that if an is subadditive, then
an
n
∀m, n ∈ N
converges either to a real number or −∞.
67. Is there a 3 times differentiable real function f such that f 6= f 0 , f 6= f 00 and f = f (3) ?
68. The function f is positive on [0, 1] and is Riemann-integrable. Does it follow that
R1
0
f (x)dx is positive?
69. Let f (x) = 2x(1 − x) and
n
z }| {
fn = f ◦ · · · ◦ f .
(a) limn→∞
R1
fn (x)dx =?
R1
(b) Compute 0 fn (x)dx for n = 1, 2, . . .
0
70. Compute the following.
Z
2x
lim
x→0+
x
sinm t
dt
tn
(m, n ∈ N).
P∞
71. The series n=1 an is divergent,
not going either to +∞ or −∞, and an → 0. Prove that the set of
Pn
accumulation points of i=1 an is an interval.
72. * Compute the following limit.
lim
√
t→1−0
2
1 − t (1 + t + t4 + t9 + · · · + tn + . . . )
73. * Let a and d be positive numbers, and let An denote the arithmetic mean and Gn denote the geometric
mean of the numbers a, a + d, . . . , a + (n − 1)d. Show that
lim
n→∞
Gn
2
= .
An
e
74. * Let the function f (x, y) be positive and continuous on the rectangle
a ≤ x ≤ a0 ,
Show that
Z
lim
n→∞
a
a0
Z
b0
n
[f (x, y)] dy
b ≤ y ≤ b0
−1
−1/n
dx
= min max f (x, y).
x
b
y
75. The real function f satisfies the following for every x and y:
f (x) ≤ x,
f (x + y) ≤ f (x) + f (y).
Show that f (x) = x for every real x.
76. The pnk ≥ 0 real numbers are given for every 0 ≤ k ≤ n such that pn0 + pn1 + · · · + pnn = 1.
For a sequence sn , let tn = pn0 s0 + pn1 s1 + · · · + pnn sn for every 0 ≤ n integer. We say that this
transformation (of sn into tn ) is regular if sn and tn are equiconvergent and if they are convergent,
their limit is the same. Prove that the transformation is regular iff 0 = limn→∞ pnk for every k.
77. Compute the following limit:
an =
1
1+
n
n
1
·
e
n
.
78. Let z1 , . . . , zk , . . . be complex numbers in the half-plane 0 ≤ Re(z). Prove that if z1 + · · · + zk + . . .
and z12 + · · · + zk2 + . . . are both convergent, then so is |z1 |2 + · · · + |zk |2 + . . . .
79. Can the set of irrational numbers be written as a countable union of closed sets?
80. Is there a sequence of continuous functions fn : R → R such that limn→∞ fn (x) exists and is finite for
every irrational x, but for each rational x . . .
(a) . . . limn→∞ f (x) = +∞?
(b) . . . fn (x) is a bounded divergent sequence?
81. Prove that if for any x and given a 6= 0
f (x + a) =
1 + f (x)
1 − f (x)
holds, then f (x) is periodic.
82. Let f be a continuous real function on the interval (a, b), f 6≡ 0. Prove that if
b
Z
0=
b
Z
b
xf (x) dx = · · · =
f (x) dx =
a
Z
a
xn−1 f (x) dx
a
then f changes sign at least n times.
83. Calculate the volume of the n-dimensional sphere with radius R.
84. Prove that if p(z) = z n + an−1 z n−1 + · · · + a1 z + a0 , then max|z|=1 (|f (z)|) ≥ 1.
85. Suppose that the polynomial p(z) = z + a2 z 2 + a3 z 3 + · · · + an z n maps the |z| < 1 unit disc injectively.
Prove that from this it follows that n|an | ≤ 1.
86. Prove that if a complex polynomial has main coefficient 1, and all other coefficients have absolute
value at most 1, then its roots have absolute values less than 2.
87. Suppose that a 6= 0 is the unique root of the polynomial f with minimal absolute value. Let
−
f 0 (z)
= c0 + c1 z + c2 z 2 + · · · + ck z k + . . .
f (z)
Show that
a = lim
n→∞
cn−1
.
cn
88. Let z1 , z2 , . . . , zn be complex numbers such that the |z1 | + |z2 | + · · · + |zn | = 1. Show that there is a
subset of them such that their sum has absolute value at least 61 .
Combinatorics and Graph Theory
89. Let n ≥ 3. Count the positive integer solutions of x + y + z = n for which the following hold:
x ≤ y + z,
y ≤ z + x,
z ≤ x + y.
90. Prove that each positive integer can be written as the sum of different positive integers in exactly as
many ways as it can be written as the sum of odd positive integers. For example, 6 as the sum of
different integers:
6,
1 + 5,
2 + 4,
1 + 2 + 3;
and as the sum of odd integers:
1 + 5,
3 + 3,
1 + 1 + 1 + 3,
1 + 1 + 1 + 1 + 1 + 1.
91. n couples live on an island; every couple consists of a hunter and a farmer. The island is divided into n
hunting grounds of the same area and, independently, n farms. Show that the areas can be distributed
among the hunters and farmers in such a way that for every couple, the hunting and farming lands
intersect (in a positive area).
92. Weights (nonnegative real numbers) are placed in the vertices of a simple graph such that the total
weight is 1. On each edge, we write the product of the weights of its end vertices. For what distribution
of the weights will the sum of the numbers written on the edges be maximal and what is this maximum?
93. Show that if a brick-shaped box can be filled with bricks of size 1 × 2 × 4, then it can be filled in such
a way too where the equal sides of the small bricks are parallel.
94. Determine the number of trees on the vertices {v1 , . . . , vn , w1 , . . . , wm } with the property that each
edge runs between vi and wj for some i, j.
95. Prove that each connected graph has a vertex such that the graph remains connected if the vertex is
deleted.
Geometry
96. (a) Finitely many points are given on the 2-dimensional plane. Is there a line which contains exactly
2 of the points?
(b) Finitely many points are given in the 3-dimensional space. Is there a plane which contains exactly
3 of the points?
97. Cover a circle of diameter 10 with the minimum number of strips of width 1.
98. n blue and n red points are given on the plane such that no 3 are collinear. Prove that they can
be connected by n segments such that the endpoints of each segment are of different colors and the
segments do not intersect.
√
99. Show that if n points are given on the plane, there can be no more than cn n unit-distances between
them.
100. Consider a triangle ABC. Let OA be the center of the square with side BC drawn outwards. OB and
OC are defined similarly. Show that COC and OA OB are equal and perpendicular.
Probability
101. k is uniform on the integers 1, 2, . . . , n. Where goes the probability of the event n mod k ≤ k/2 as
n → ∞?
102. Alice thinks up a number between 1 and n. Bob is guessing at the number in the following way: for
each question, he generates a random subset of the numbers uniformly, and asks whether the number
is among them. What is the expected number of questions needed? How does this compare to the
optimal (deterministic) strategy?
Riddles
103. Determine the convex domain in the plane with the smallest area for which the following holds: a
worm of unit length can be covered by the domain no matter how the worm bends.
104. Tom is standing at the edge of a square-shaped swimming pool and wants to catch Jerry, who is
swimming inside. Tom can not swim, but he runs exactly four times as fast as Jerry can swim. Jerry
runs faster than Tom. Can Jerry escape?
105. An army has attacked the king’s castle. The battle goes as follows: in the beginning, there is a finite
number of soldiers standing on each of the steps leading to the gates (there is a finite number of steps
as well). The leader of the attacking army divides his army into two groups in any way he wants, then
the king may send home one of the two groups, and finally every soldier in the other group advances
one step. After this, they repeat this process. The attackers win if a soldier can reach the gate, and
the king wins if he can send home the whole army. In what starting configures can the king defend
his castle?
106. It takes 100 days to cross the desert. I want to cross it, but I can carry water enough only for one
day. I may hire locals to help; each local can carry water enough for one person for one day. I may
hire any number of locals, but I want all of them to survive (they must reach either side of the desert,
having enough water at all times). Is it possible to cross the desert?
107. 5 people are held captive. On each of their heads, a black or white cap is placed, so that everybody
sees everybody else’s cap, but not their own. Everybody has to make a guess at their own colour. If
at least 1 person manages to guess the colour of his own cap, all of them are released. The guesses
are made at the same time, so no one can make his guess based on information from others’ guesses.
However, before the guessing, they can discuss a strategy how to guess. Can they escape?
108. An infinite number of people are held captive. On each of their heads, a black or white cap is placed,
so that everybody sees everybody else’s cap, but not their own. Everybody has to make a guess at
their own colour. If only finitely many of the guesses miss, the people are released. The guesses are
made at the same time, so no one can make his guess based on information from others’ guesses.
However, before the guessing, they can discuss a strategy how to guess. Can they escape?
109. Determine the shortest line in a regular triangle which cuts it into two parts of the same area.
110. Is there an uncountable chain of subsets of N? (A set of sets forms a chain if it is completely ordered
to the subset relation.)
111. You have a coin which is loaded; it comes up heads with a probability p (0 < p < 1); however, p is
unknown. Produce an event with probability 1/2 using the coin.
112. Is there a real multi-variable polynomial which has no zero value but has values arbitrarily close to
zero?
113. Is there a real function which is everywhere continuous but nowhere differentiable?
114. Is there a real function which is strictly monotonic and takes only irrational values?