CNCM Online Round 3

Part of CNCM Online

Subcontests

(7)

1

CNCM Online R3 P7

A subset of the positive integers

S

is said to be a \emph{configuration} if 200

\notin S

and for all nonnegative integers

x

x \in S

if and only if both 2

x\in S

\left \lfloor{\frac{x}{2}}\right \rfloor\in S

. Let the number of subsets of

\{1, 2, 3, \dots, 130\}

that are equal to the intersection of

\{1, 2, 3, \dots, 130\}

with some configuration

S

k

. Compute the remainder when

k

is divided by 1810.
Proposed Hari Desikan (HariDesikan)

1

CNCM Online R3 P6

ABC

has side lengths

AB=13, BC=14,

CA=15

\Gamma

denote the circumcircle of

\triangle ABC

H

be the orthocenter of

\triangle ABC

AH

\Gamma

D

A

BH

AC

F

\Gamma

G

B

DG

AC

X

. Compute the greatest integer less than or equal to the area of quadrilateral

HDXF

.
Proposed by Kenan Hasanaliyev (claserken)

1

CNCM Online R3 P5

How many positive integers

N

10^{1000}

N

x

digits when written in base ten and

\frac{1}{N}

x

digits after the decimal point when written in base ten? For example, 20 has two digits and

\frac{1}{20}

= 0.05 has two digits after the decimal point, so

20

is a valid N.
Proposed by Hari Desikan (HariDesikan)

1

CNCM Online R3 P4

Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots

0 < a < b < c < 1

. He knows the following three facts:

P(0) = -\frac{1}{8}

, the roots form a geometric progression in the order

a,b,c

\sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2}

a + b + c

can be expressed as

\frac{m}{n}

m,n

are relatively prime positive integers. Find

m + n

.
Proposed by Akshar Yeccherla (TopNotchMath)

1

CNCM Online R3 P3

a_1 = 1

a_{n+1} = a_n \cdot p_n

n \geq 1

p_n

n

th prime number, starting with

p_1 = 2

\tau(x)

be equal to the number of divisors of

x

. Find the remainder when

\sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d)

is divided by 91 for positive integers

d

d|a_n

d

a_n

.
Proposed by Minseok Eli Park (wolfpack)

1

CNCM Online R3 P2

Consider rectangle

ABCD

AB = 6

BC = 8

E, F, G, H

such that the angles

\angle AEB, \angle BFC, \angle CGD, \angle AHD

are all right. What is the largest possible area of quadrilateral

EFGH

?
Proposed by Akshar Yeccherla (TopNotchMath)

1

CNCM Online R3 P1

Harry, who is incredibly intellectual, needs to eat carrots

C_1, C_2, C_3

and solve Daily Challenge problems

D_1, D_2, D_3

. However, he insists that carrot

C_i

must be eaten only after solving Daily Challenge problem

D_i

. In how many satisfactory orders can he complete all six actions?
Proposed by Albert Wang (awang2004)