MATHEMATICAL ANALYSIS

PROBLEMS

1.

Let

1

:::;

a

<

f3

be real numbers. Prove that there are integers m,

n

>

1

such

that

a

<

ifiii

<

f3.

2.

Let

(an)n�O

and

(bn)n�o

be the sequences of integers defined by

1

+ =

an + bn n

E

N.

Find a recursive relation for each of the sequences

(an)n�O

and

(bn)n�o,

3.

Study the convergence of the sequence

(xn)n�O

satisfying the following prop­ erties:

1) Xn

>

1, n

=

0, 1,2, .. .

2) - Xn+l -2 1 (

--Xn+l Xn - 1 1

)

=

--Xn

+

1 , n = 0,1,2, .. .

4.

Study the convergence of the sequence

(Xn)n�l

defined by

Xl

E

(0,2)

and

Xn+l = 1 + - x�

for

n � 1.

5.

Consider the sequence of real numbers

(Xn)n�l

such that Prove that

Is the converse true?

2

2 2

lim

Xl

+

X2 + .. . + xn

₌

O.

n�oo

n

lim

Xl + X2 + ...

+

Xn

₌

O.

n�oo

n

6.

Let

(an)n�l

and

(bn)n�l

be sequences of positive numbers such that

an

>

nbn

for all

n

>

1.

Prove that if

(an)n�l

is increasing and

(bn)n�l

is unbounded, then the sequence

(Cn)n�l'

given by

Cn

=

an+l - an,

is also unbounded.

7.

Let

0

<

a

<

a

be real numbers and let

(Xn)n�l

be defined by

Xl = a

and

(a

+

l)Xn-l + 0'2

Xn

=

Xn-l + _(a + 1) , n � 2.

18

2

and

5. MATHEMATICAL ANALYSIS

Prove that the sequence is convergent and find its limit.

8.

Find a sequence

(an)n2:l

of positive real numbers such that lim

(an+l - an) = 00

n-+oo

lim -

�) = O.

n-+oo

9.

Let

(Xn)n2:l

be an increasing sequence of positive real numbers such that

1. _1m

xn _{2 = 0.}

n-+oo

n

Prove that there is a sequence

(

n

k

h

2:l

of positive integers such that lim

Xnk+l - Xnk

= o.

k-+oo nk

10.

Let a,

(3

be real numbers and let

(Xn)n2:l, (Yn)n2:l, (Zn)n2:l

be real sequences

such that

max

{x�

+

O'Yn, y�

+

(3xn} :::; Zn

for all

n ;::: l.

1

a) Prove that

Zn ;::: -8(0'2

+

(32)

for all n

;::: 1.

b) If

n

l

�� Zn =

_

�(O'2

+

(32),

prove that the sequences

(Xn)n2:b (Yn)n2:l

are

convergent and find their limits.

11.

The sequences

(xn)n2:l

and

(Yn)n>l

are defined by

Xl 2, Yl

1 and

Xn+l = x�

+

1, Yn+l

=

XnYn

for all n

;::: 1.

a) Prove that

Xn / Yn

< V7 for all n

;::: 1.

b) Prove that the sequence

(Zn)n2:l, Zn = xn/Yn,

is convergent and lim

_n-+oo Zn

<

V7.

12.

Let a

;::: 0

and

a f= 0

be real numbers and let

(Xn)n2:l

be an increasing

sequence of real numbers such that

lim

nC¥(xn+l - xn) = a.

n-+oo

P�ove that the sequence is bounded if and only if a

> 1.

13.

Evaluate

14.

Evaluate lim k(n - k)! + (k +

1)

n-+oo

(k + l)!(n - k)!

n (

k

�+l

lim -

n

-+

ooL _k=l

n2

)

5 . 1 . PROBLEMS 183

15.

Evaluate (i)

�

q > 1;

(ii)

�

(4n +

q > 1.

16.

Let

(Xn)n2:l

be an increasing sequence of positive integers such that

Xn+2

+

Xn > 2Xn+l

for all

n ;::: 1.

Prove that the number is irrational.

17.

Prove that is irrational for all

n ;::: 1.

00

1

-

n=l 10Xn

00

1

An = _L

_(k!)n

k=l

18.

Let k, s be positive integers and let

aI, a2,· .. , ak, bl, b2, .. . , bs

be positive real

numbers such that

y!al

+

y'a2

+ . . . +

ifiik = yb;

+

yb;

+ . . . +

\I'bs

for infinitely many integers n

;::: 2.

Prove that

1)

k = S;

2) ala2 ... ak = bl

b2 . . . bs·

19.

Let

(xn)n2:l

be a sequence with

Xl

=

1

and let

X

be a real number such that

Prove that

II

00

( 1

n ) = e-x•

n=l Xn+l

20.

Let

A f= ±1

be a real number. Find all functions f : R � R and 9

: (0,00)

� R such that

f(ln x + A ln y)

=

9

(y'X)

+ 9

(..fij)

for all

x, Y

E

(0,00).

21.

Let f be a continuous real-valued function on the interval

[a, b]

and let

ml, m2

be real numbers such that

ml m2 > O.

Prove that the equation

f(x) = � _{a - x b - x}

+

m2

184 5. MATHEMATICAL ANALYSIS

22.

Let

a

and

b

be real numbers in the interval

(0,1/2),

and let 9 be a continuous real-valued function such that

g(g(x))

=

ag(x)+bx

for all real

x.

Prove that

g(x)

= ex

for some constant e.

23.

Find all continuous functions

f

:

�

�

[0,00)

such that

f2(X + y) - f2(X - y)

=

4f(x)f(y)

for all real numbers

x, y.

24.

(i) Prove that if the continuous functions

f : �

�

(-00,0]

and

g : �

�

[0,

00) have a fixed point, then

f +

9 has a fixed point.

(ii) Prove that if the continuous functions

rp

:

�

�

[0,1]

and 'ljJ :

�

�

[1,00)

have a fixed point, then

rp'ljJ

has a fixed point.

25.

Let

rp

:

�

�

�

be a differentiable function at the origin and satisfying

rp(O)

=

0.

Evaluate

�� � [_{rp(x) + rp} (�) _{+ .. . + rp} (�)] ,

where

n

is a positive integer.

26.

Let

a

be a positive real number. Prove that there is a unique positive real number

f-£

such that

f-£X

>

aIL-x

for all

x

>

0.

XIL -

27.

Let

f : [a, b)

�

�

be a twice differentiable function on

[a, b)

such that

f(a)

=

f(b)

and

f'(a)

=

j'(b).

Prove that for any real number

,\

the equation

f"(x) - ,\(f'(X))2

= °

has at least a solution in the interval

(a, b).

28.

Find all functions

f

:

[0, 2]

�

(0,1]

that are differentiable at the origin and satisfies

f(2x)

=

2f2(x) - 1, x

E

[0, 1]

29.

Let ,\ be a positive integer. Prove that there is a unique positive real number

o such that

for all real number

x

>

0.

30.

Let

f

:

�

�

�

be a function continuous at the origin and let

'\, f-£

be two distinct positive real numbers.

Prove that the limit

5.1. PROBLEMS

1. _1m

- f(f-£X)

x�o x

exists and is finite if and only if

f

is differentiable at the origin.

31.

The sequence

(Xn)n�l

is defined by

Xl

<

0, Xn+l

₌

eXn - 1, n � 1.

Prove that lim

n�oo nXn

=

-2.

185

32.

Let

Xo

E

(0,1]

and

Xn+l

=

Xn -

arcsin(sin3

xn),

n

� 0.

Evaluate

n

l

�� Vnxn.

33.

Let

f

:

�

�

�

be a twice differentiable function with the second derivative nonnegati ve.

Prove that

f(x + f'(x)) � f(x), x

E

�.

34.

Let

a

<

b

be positive real numbers. Prove that the equation

( a b r+Y

=

a" bY

has at least a solution in the interval

(a, b).

35.

Find with proof if there are differentiable functions

rp

:

�

�

�

such that

rp( x )

and

rp'(x)

are integers only if

x

is integer.

36.

Let

f

:

[a, b)

�

�

be a differentiable function.

Prove that for any positive integer

n

there are numbers

01

<

O2

< . . . <

On

in the interval

(a, b)

such that

f(b) - f(a)

_

f'(OI) + f'(02) + .. . + f'(On)

b - a

n

37.

Let

f,

9 :

�

�

�

be differentiable functions with continuous derivatives such that

f (x)

+ 9

(x)

=

f' (x) - g' (x)

for all

x

E

�.

Prove that if

Xl, X2

are two consecutive real solutions of the equation

f(x) -g(x)

=

0,

then the equation

f(x) + g(x)

= ° has at least a solution in the interval

(Xl, X2).

38.

Let

f : [-�, �]

�

(-1,1)

be a differentiable function whose derivative

f'

is continuous and nonnegative. Prove that there exists

Xo

in

[-�, �]

such that

186

_5.

MATHEMATICAL ANALYSIS

39.

Prove that there are no positive real numbers

x

and

y

such that

x2Y + y2-X

=

x + y.

( ) n(n+1)

40.

a) Prove that if

x

�

y

�

₁

for some integer n � 2, then yX

+ n+�

� \IY

+ n+V'x.

b) Prove that

n

+ _{n+l n + 1 -}

>

2n + 1, n

_> 3.

41.

Let X l , X2 , . . . , Xn be positive real numbers such that Xl

+

X2

+ .. . +

Xn = 1.

Prove that

42.

Let f : lR � lR be a function with a noninjective antiderivative. Prove that

f

(

c

)

= ° for some

c

E lR.

43.

Let h ,

h, .. . , f n

: lR � lR be continuous functions. Prove that max(h

(x), h(x), .. . , fn(x))dx

is a deri vati ve and evaluate

44.

Evaluate

45.

Let p be a polynomial of odd degree such that p' has no multiple zero and

let f : lR � lR be a function such that f o p is a derivative.

Prove that f is a derivative.

46.

Let 1 =

(0,00)

and let

f

: I � I be a function with an antiderivative

F

that

satisfies the condition

F(x)f G)

=

x,

for all

x

in I. Prove that

g :

1 -+ 1R,

g(x)

=

F(x)F (�)

is a constant function and then find

f.

5.1.

PROBLEMS

187

47.

Let

n

> 1 be an integer and let

f : [0, 1]

� lR be a continuous function such that

11 _f(x)dx

1

1

=

1 + - + . . . + - .

a

2

n

Prove that there is a real number X

o

E

(0, 1),

such that

f(xo)

=

1 -_{1 - Xo} xg

48.

Consider the continuous functions

f,

9

: [

a,

b]

� _R

Prove that the equation

f(x) /." g(t)dt

=

g(x) l f(

t

)dt

has at least a solution in the interval (a,

b).

49.

Let f :

[a, b]

� lR be a continuous function such that

[ f(x)dx

oj O.

Prove that there are numbers a < a <

f3

<

b

such that

t f(x)dx

=

(b -a.)f({3)·

50.

Let a, c be nonnegative real numbers and let f

: [

a,

b]

�

[c, dJ

be a bijective

increasing function.

Prove that there is a unique real number f-£ E (a,

b)

such that

[

f(t)dt = (IJ

- a)c + (b - 1J)d.

51.

Let r.p : lR � lR be a continuous function such that

for all

x, y

E lR.

r.p (t)dt = r.p(t)dt,

Prove that r.p is a constant function.

52.

Let f : lR � lR be a differentiable function such that

for all real number

x

<

y.

!:.±lL

y

1.

2

f(t)dt

5

!!:.±lL

f (t)dt

2

Prove that

f

is a nondecreasing function.

188

5, MATHEMATICAL ANALYSIS

Prove that the function

F : (0,

00) � lR,

l1a:

F(x)

= -

x a f(t)dt

is monotone,

54.

Prove that �

1

lim

n2 xa:+1dx

= - ,

n-too

₂

55.

Prove that there are no Riemann integrable functions f : lR � lR \ {o} such

that

f(t)dt

=

f(x)

f(y) ,

for all real numbers

x

=J

y.

56.

Let

f : [0, 1]

� lR be a differentiable function with continuous derivative such that

[[/'(X}]2dX =

1. Prove that

If(l) - f(O)1

<

1.

57.

Find all continuous functions

f : [0, 1]

� lR such that

11 f(x)(x - f(x))dx =

- .

1

a

12

58.

Let

fa : [0, 1]

� lR be a continuous function and let the sequence

(f n)n> l

be

defined by -

fn(x) = { fn

-

I (t}dt, x E [0, 1].

Prove that if there i s an integer m 2:: ° such that

[ fm(t}dt

=

1}1'

then the function

fa

has a fixed point.

5,9.

Let

f : [-1, 1]

� lR be a differentiable function with nondecreasing derivative. Prove that

1 f(x)dx � f(-I) + f'(I).

60.

Let

f, g : [a,

b] � lR be continuous functions. Prove that there is a real number

In document 360 Problems for Mathematical Contests (Page 92-97)