MathDB

1

limit of probability , first n tosses of unbiased coin

Consider an unbiased coin which is tossed infinitely many times. Let

A_n

be the event that no two successive heads occur in the first

n

tosses of this experiment. Then which of the following is incorrect : (A)

\lim_{n \to \infty} P(A_n)=0

(B)

\lim_{n \to \infty}3^n P(A_n)=0

(C)

2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}

(D)

\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}

is lesser than

1.2

16

1

lim frac{f^{(2n)}1}{(2n)!}, f(x)=\frac{e^x}{x}

The

n^{th}

derivative of a function

f(x)

(if it exists) is denoted by

f^{(n)}(x)

. Let

f(x)=\frac{e^x}{x}

. Suppose

f

is differentiable infinitely many times in

(0,\infty)

. Then find

\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}

18

1

n \choose 1 sin (a) + n \choose 2 sin (2a) +...+ n \choose n sin (na)

Evaluate the following sum:

n \choose 1

\sin (a) +

n \choose 2

\sin (2a) +...+

n \choose n

\sin (na)

(A)

2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)

(B)

2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)

(C)

2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)

(D)

2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)

17

1

triangle with sidelengths 4n^2, 4n^4+1 , 4n^4-1 , angle limit wanted

Let

a_n

denote the angle opposite to the side of length

4n^2

units in an integer right angled triangle with lengths of sides of the triangle being

4n^2, 4n^4+1

and

4n^4-1

where

n \in N

. Then find the value of

\lim_{p \to \infty} \sum_{n=1}^p a_n

(A)

\pi/2

(B)

\pi/4

(C)

\pi

(D)

\pi/3

20

1

sequence with countably infinite distinct subsequences

Let

\{a_n \}_n

be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A)

\{a_n \}_n

is bounded (B)

\{a_n \}_n

is unbounded (C) The set of convergent subsequence

\{a_n \}_n

is countable (D) None of these

13

1

d(n), d(d(n)), d(d(d(n))), ... , sum of digits , limit

For every

n \in N

, let

d(n)

denote the sum of digits of

n

. It is easy to see that the sequence

d(n), d(d(n))

,

d(d(d(n))), ...

will eventually become a constant integer between

1

and

9

(both inclusive). This number is called the digital root of

n

. Denote it by

b(n)

. Then for how many natural numbers

k<1000 , \lim_{n \to \infty} b(k^n)

exists.

14

1

n=\sum_{d|n} f(d), f(100)

Let

f: N \to N

satisfy

n=\sum_{d|n} f(d), \forall n \in N

. Then sum of all possible values of

f(100)

is?

15

1

x^2+y^2=9999(x-y) diophantine

How many integer pairs

(x,y)

satisfies

x^2+y^2=9999(x-y)

?

12

1

GCD of all elements of {k^{19}-k: 1<k<20, k\in N}

Let

A

be the set

\{k^{19}-k: 1<k<20, k\in N\}

. Let

G

be the GCD of all elements of

A

. Then the value of

G

is?

11

1

Point inside Isosceles right triangle

\triangle PQR

is isosceles and right angled at

R

. Point

A

is inside

\triangle PQR

, such that

PA=11, QA=7

, and

RA=6

. Legs

\overline{PR}

and

\overline{QR}

have length

s=\sqrt{a+b\sqrt{2}}

, where

a

and

b

are positive integers. What is

a+b

?

10

1

Trigonometry in a triangle

In a triangle

\triangle XYZ

,

\tan(x)\tan(z)=2

,

\tan(y)\tan(z)=18

. Then what is

\tan^2(z)

?

9

1

Pairwise distance less than radius

Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is

\frac{1}{K}

,

K\in\mathbb{N}

. Find

K

.

8

1

Independent sets and probability

Let

S

be a finite set of size

s\geq 1

defined with a uniform probability

\mathbb{P}

( i.e. for any subset

X\subset S

of size

x

,

\mathbb{P}(x)=\frac{x}{s}

). Suppose

A

and

B

are subsets of

S

. They are said to be independent iff

\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)

. Which if these is sufficient for independence?
(A)

|A\cup B|=|A|+|B|

(B)

|A\cap B|=|A|+|B|

(C)

|A\cup B|=|A|\cdot |B|

(D)

|A\cap B|=|A|\cdot |B|

7

1

Length-Bash geometry

A circle

\mathfrak{D}

is drawn through the vertices

A

and

B

of

\triangle ABC

. If

\mathfrak{D}

intersects

AC

at a point

M

and

BC

at

P

and

MP

contains the incenter of

\triangle ABC

, then the length

MP

is (in standard notation, where

t=\frac{1}{a+b+c}

):
(A)

at(b+c)

(B)

ct(b+a)

(C)

bct

(D)

abt

6

1

Roots of Real valued function

Let

f(x)

be a real-valued function satisfying

af(x)+bf(-x)=px^2+qx+r

.

a

and

b

are distinct real numbers and

p,q,r

are non-zero real numbers. Then

f(x)=0

will have real solutions when
(A)

\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}

(B)

\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}

(C)

\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}

(D)

\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}

5

1

Area of hexagon

Regular hexagon

ABCDEF

has vertices

A

and

C

at

(0,0)

and

(7,1)

respectively. What is its area?
(A)

20\sqrt{3}

(B)

20\sqrt{2}

(C)

25\sqrt{3}

(D)None of these

4

1

Fibonacci mod 3

Define the sequence

\{a_n\}_{n\geq 1}

as

a_n=n-1

,

n\leq 2

and

a_n=

remainder left by

a_{n-1}+a_{n-2}

when divided by

3

\forall n\geq 2

. Then

\sum_{i=2018}^{2025}a_i=

?
(A)

6

(B)

7

(C)

8

(D)

9

3

1

No real roots, quadratic in disguise

Given that the equation

(m^2-12)x^4-8x^2-4=0

has no real roots, then the largest value of

m

is

p\sqrt{q}

, where

p

and

q

are natural numbers,

q

is square-free. Determine

p+q

.
(A)

4

(B)

5

(C)

3

(D)

6

2

1

g:R^4-->R F.E. in 6 var

The number of functions

g:\mathbb{R}^4\to\mathbb{R}

such that,

\forall a,b,c,d,e,f\in\mathbb{R}

:
(i)

g(1,0,0,1)=1

(ii)

g(ea,b,ec,d)=eg(a,b,c,d)

(iii)

g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)

(iv)

g(a,b,c,d)+g(b,a,d,c)=0

is :
(A)

1

(B)

0

(C)

\text{infinitely many}

(D)

\text{None of these}

[Hide=Hint(given in question)] Think of matrices

1

Number of involutions

Find the number of

f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}

such that

f(f(x))=x

(A)

26

(B)

41

(C)

120

(D)

60

2020 LIMIT Category 2

Subcontests

limit of probability , first n tosses of unbiased coin

lim frac{f^{(2n)}1}{(2n)!}, f(x)=\frac{e^x}{x}

n \choose 1 sin (a) + n \choose 2 sin (2a) +...+ n \choose n sin (na)

triangle with sidelengths 4n^2, 4n^4+1 , 4n^4-1 , angle limit wanted

sequence with countably infinite distinct subsequences

d(n), d(d(n)), d(d(d(n))), ... , sum of digits , limit

n=\sum_{d|n} f(d), f(100)

x^2+y^2=9999(x-y) diophantine

GCD of all elements of {k^{19}-k: 1<k<20, k\in N}

Point inside Isosceles right triangle

Trigonometry in a triangle

Pairwise distance less than radius

Independent sets and probability

Length-Bash geometry

Roots of Real valued function

Area of hexagon

Fibonacci mod 3

No real roots, quadratic in disguise

g:R^4-->R F.E. in 6 var

Number of involutions

2020 LIMIT Category 2

Subcontests

limit of probability , first n tosses of unbiased coin

lim frac{f^{(2n)}1}{(2n)!}, f(x)=\frac{e^x}{x}

n \choose 1 sin (a) + n \choose 2 sin (2a) +...+ n \choose n sin (na)

triangle with sidelengths 4n^2, 4n^4+1 , 4n^4-1 , angle limit wanted

sequence with countably infinite distinct subsequences

d(n), d(d(n)), d(d(d(n))), ... , sum of digits , limit

n=\sum_{d|n} f(d), f(100)

x^2+y^2=9999(x-y) diophantine

GCD of all elements of {k^{19}-k: 1&lt;k&lt;20, k\in N}

Point inside Isosceles right triangle

Trigonometry in a triangle

Pairwise distance less than radius

Independent sets and probability

Length-Bash geometry

Roots of Real valued function

Area of hexagon

Fibonacci mod 3

No real roots, quadratic in disguise

g:R^4--&gt;R F.E. in 6 var

Number of involutions

GCD of all elements of {k^{19}-k: 1<k<20, k\in N}

g:R^4-->R F.E. in 6 var