MathDB

1

Prove this colouring sequential problem

d(n)

be the number of divisors of

n,

where

n

is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence

\left\{a_{1}, a_{2}, \cdots\right\}

if

\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}

is an nonconstant geometric series then

\left\{a_{1}, a_{2}, \cdots\right\}

does not bear same colour.

SAQ: P 5

1

Prove these two problems on Fibonacci sequence.

[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by

n

is periodic, where

n

is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number

n,

P(n)

is a prime.

SAQ: P 4

1

Prove that, there exists a x in S such that f(x)=x

Suppose

S

is a finite subset of

\mathbb{R}

. If

f: S \rightarrow S

is a function such that,

\left|f\left(s_{1}\right)-f\left(s_{2}\right)\right| \leq \frac{1}{2}\left|s_{1}-s_{2}\right|, \forall s_{1}, s_{2} \in S

Prove that, there exists a

x \in S

such that

f(x)=x

SAQ: P 3

1

Does there exist any continuous function f such that f(f(x))=-x^{2019}

Does there exist any continuous function

f

such that

f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}

SAQ: P 2

1

Prove that every coefficients of P(x) is rational numbers.

P(x)

is polynomial with real coefficients such that

\forall n \in \mathbb{Z}, P(n) \in \mathbb{Z}

. Prove that every coefficients of

P(x)

is rational numbers.

SAQ: P 1

1

Prove (x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0

Let

x, y, z

are real numbers such that

x<y<z .

Prove that

(x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0

MCQ: P 7

1

How many functions f: A-> A are there such that f(1)<f(2)<f(3)

A=\{1,2,3,4,5,6,7,8\} .

How many functions

f: A \rightarrow A

are there such that

f(1)<f(2)<f(3)

[*]

{{8}\choose{3}}

[*]

{{8}\choose{3}}5^{8}

[*]

{{8}\choose{3}} 8^{5}

[*]

\frac{8 !}{3 !}

MCQ: P6

1

In a class among 80 students number of boys is 40 and number of girls is 40

In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct?
[*] Only 10 boys use spectacles [*] Only 20 girls use spectacles [*] At most 25 boys do not use spectacles [*] At most 30 girls do not use spectacles

MCQ: P 5

1

f(x)=cos(x^2018)sin x If M and m are max and min find M+m

Let the maximum and minimum value of

f(x)=\cos \left(x^{2018}\right) \sin x

are

M

and

m

respectively where

x \in[-2 \pi, 2 \pi] .

Then

M+m=

[*]

\frac{1}{2}

[*]

-\frac{1}{\sqrt{2}}

[*]

\frac{1}{2018}

[*] Does not exists

MCQ: P4

1

Find the value of AC/OC

Take a point

O

inside

\Delta A B C

such that

\angle B O C=90^{\circ}

,

\angle C A O=\angle A B O

,

\angle B A O=\angle B C O .

Find the value of

\frac{A C}{O C}

[*]

\sqrt{2}

[*]

\sqrt{\frac{3}{2}}

[*] 2 [*] None of these

MCQ: P 3

1

Find sum_{n=3}^{\infty}(18+999F_n/(F_{n-1}X F_{n+1})

F_{n}

denotes the Fibonacci Sequence where

F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3

Find

\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}

[*] 2016 [*] 2017 [*] 2018 [*] None of these

MCQ: P 2

1

Find lim x->infinity [x]/tan x

\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}

where

[x]

is the greatest integer function
[*] -1 [*] 0 [*] 1 [*] Does not exists

MCQ: P 1

1

In how many cases 2 consecutive heads does not appear?

A coin is tossed 9 times. Hence

2^{9}

different outcomes are possible. In how many cases 2 consecutive heads does not appear?
[*] 34 [*] 55 [*] 89 [*] None of these

2018 Mathematical Talent Reward Programme

Subcontests

Prove this colouring sequential problem

Prove these two problems on Fibonacci sequence.

Prove that, there exists a x in S such that f(x)=x

Does there exist any continuous function f such that f(f(x))=-x^{2019}

Prove that every coefficients of P(x) is rational numbers.

Prove (x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0

How many functions f: A-> A are there such that f(1)<f(2)<f(3)

In a class among 80 students number of boys is 40 and number of girls is 40

f(x)=cos(x^2018)sin x If M and m are max and min find M+m

Find the value of AC/OC

Find sum_{n=3}^{\infty}(18+999F_n/(F_{n-1}X F_{n+1})

Find lim x->infinity [x]/tan x

In how many cases 2 consecutive heads does not appear?

2018 Mathematical Talent Reward Programme

Subcontests

Prove this colouring sequential problem

Prove these two problems on Fibonacci sequence.

Prove that, there exists a x in S such that f(x)=x

Does there exist any continuous function f such that f(f(x))=-x^{2019}

Prove that every coefficients of P(x) is rational numbers.

Prove (x-y)^{3}+(y-z)^{3}+(z-x)^{3}&gt;0

How many functions f: A-&gt; A are there such that f(1)&lt;f(2)&lt;f(3)

In a class among 80 students number of boys is 40 and number of girls is 40

f(x)=cos(x^2018)sin x If M and m are max and min find M+m

Find the value of AC/OC

Find sum_{n=3}^{\infty}(18+999F_n/(F_{n-1}X F_{n+1})

Find lim x-&gt;infinity [x]/tan x

In how many cases 2 consecutive heads does not appear?

Prove (x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0

How many functions f: A-> A are there such that f(1)<f(2)<f(3)

Find lim x->infinity [x]/tan x