2020 LIMIT Category 1

Part of 2020 LIMIT

Subcontests

(20)

1

Divisible by 11^3

How many integers

n

|n|<2020

and the equation

11^3|n^3+3n^2-107n+1

0

101

367

368

1

Comparing exponents and factorials

a=2019^{1009}, b=2019!

c=1010^{2019}

, then which of the following is true?
(A)

c<b<a

a<b<c

b<a<c

b<c<a

1

Squares inscribed in right triangle

\triangle ABC

be a right triangle with

\angle C=90^{\circ}

S_1

S_2

are inscribed in the triangle

ABC

S_1

ABC

share a common vertex

C

S_2

has one of its sides on

AB

\text{Area}(S_1)=1+\text{Area}(S_2)=441

, then calculate

AC+BC

400

420

441

462

1

Sum of k consecutive integers

k

consecutive integers is

90

. Then the sum of all possible values of

k

89

179

168

119

1

Picking balls

28

20

19

13

11

white balls and

9

black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that atleast

15

balls of a single colour will be drawn?

1

4*4 Chessboard non-attacking rook and bishop

4\times 4

chessboard, in how many ways can you place

3

rooks and one bishop such that none of these pieces threaten another piece?

1

Cubic diophantine

(m,n)

be the pairs of integers satisfying

2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437

. Find the sum of all possible values of

mn

1

Computational Geo

AC

of an acute triangle

\triangle ABC

D

is taken such that

2AD=CD=2, BD\perp AC

. A circle of radius

2

A

D

and is tangent to the circumcircle of

\triangle BDC

[\text{Area}(\triangle ABC)]

[.]

is the greatest integer function.

1

Least fraction closest to 31/17

q

is the smallest rational number having the following properties:
(i)

q>\frac{31}{17}

q

is written in its reduced form

\frac{a}{b}

b<17

As in part (ii) above, find

a+b

1

Circle tangent to side at vertex

\triangle ABC

\angle A=30^{\circ}

BC=13

2

\gamma_1, \gamma_2

r_1,r_2

A

BC

B

C

respectively. Find

r_1r_2

1

Base change of a rational

For natural number

t

, the repeating base-

t

representation of the (base-ten) rational number

\frac{7}{51}

0.\overline{23}_t=0.232323..._t

t

1

Sum of first n squares|(2n)!

What is the sum of all two-digit positive integer

n<50

for which the sum of the squares of first

n

positive integers is not a divisor of

(2n)!

2

LIMIT 2020 P8

Kunal and Arnab play a game as follows. Initially there are

2

piles of coins with

x

y

coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of

(x,y)

which ensure Kunal's victory against Arnab given both os them play optimally. \\ You are required to find an exhaustive set of solutions

Greatest integer not exceeding a fraction

Find the greatest integer which doesn't exceed

\frac{3^{100}+2^{100}}{3^{96}+2^{96}}

81

80

79

82

2

LIMIT 2020 P7

\triangle{ABC}

is equailateral.

E

is any point on

\overline{AC}

produced and the equilateral

\triangle{ECD}

M

N

are the midpoints of

\overline{AD}

\overline{EB}

respectively then show that

\triangle{CMN}

is equilateral.

Exploiting Vieta's relations

P(x)=x^6-x^5-x^3-x^2-x

a,b,c

d

be the roots of the equation

x^4-x^3-x^2-1=0

, then determine the value of

P(a)+P(b)+P(c)+P(d)

5

6

7

8

2

Sum of digits even

What is the number of

4

digit natural numbers such that the sum of digits is even?
(A)

4999

5000

5050

4500

LIMIT 2020 P6

m=2+2\sqrt{44n^2+1}

is an integer then show that it is also a perfect square. Here

n

is a natural number.

2

LIMIT 2020 P5

P(x),Q(x)

be monic polynomials with integer coeeficients. Let

a_n=n!+n

for all natural numbers

n

\frac{P(a_n)}{Q(a_n)}

is an integer for all positive integer

n

\frac{P(n)}{Q(n)}

is an integer for every integer

n\neq0

. \\ Hint (given in question): Try applying division algorithm for polynomials

Intersection of lines

Rohit is counting the minimum number of lines

m

, that can be drawn so that the number of distinct points of intersection exceeds

2020

m

63

64

65

66

2

LIMIT 2020 P4

a_1,a_2,...,a_n

be n real numbers. Show that there exists some real number

\alpha

such that

a_1+\alpha,a_2+\alpha,...,a_n+\alpha

are all irrational. \\ color=#000 Prove that such a satetement is not valid if all these are rquired to be rational. \\ Hint (given in question) : Use Pigeon Hole Principle

Same old product diophantine

The total number of solutions of

xyz=2520

2520

2160

540

(D)None of these

2

LIMIT 2020 P3

\overline{AC}

\overline{BD}

of a quaderilateral

ABCD

O

s_1

\triangle{AOB}

s_2

\triangle{OCD}

. Then show that

\sqrt{s_1}+\sqrt{s_2} \leq \sqrt{s}

Also find a geometrical condition for equality to hold (By geometrical condition we mean something like parallel lines, perpendicular lines,bisecting lines etc.)

Sum of Digit sum and digit product

2

n=ab

a

b

are digits) have the property that

n=a+b+a\times b

20

15

9

8

2

LIMIT 2020 P2

Prove that any integer has a multiple consisting of all ten digits

\{0,1,2,3,4,5,6,7,8,9\}

. \\ Note: Any digit can be repeated any number of times

Area of CDEF

ABCD

2

E

is the midpoint of

AD

F

BE

CF\perp BE

, then the quadrilateral

CDEF

has an area of
(A)

2

2.2

\sqrt{5}

(D)None of these

2

LIMIT 2020 P1

Find all polynomial

P(x)

\leq n

and non negative coefficients such that

P(x)P(\frac{1}{x})\leq P(1)^2

for all positive

x

n

is a natuaral number

Easy rationality

a

is a rational number and

b

is an irrational number such that

ab

is rational, then which of the following is false?
(A)

ab^2

is irrational (B)

a^2b

is rational (C)

\sqrt{ab}

is rational (D)

a+b