2020 CMIMC Team

Part of 2020 CMIMC

Subcontests

(16)

1

2020 Team 11

Find the number of ordered triples of integers

(a,b,c)

1

64

a^2 + b^2 \equiv c^2\pmod{64}.

1

2020 Team 12

Determine the maximum possible value of

\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})

x\in [0,1]

1

2020 Team 10

ABC

be a triangle. The incircle

\omega

\triangle ABC

, which has radius

3

, is tangent to

\overline{BC}

D

. Suppose the length of the altitude from

A

\overline{BC}

15

BD^2 + CD^2 = 33

BC

1

2020 Team 13

10

points arranged in a equilateral triangular grid of side length

4

, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?

1

2020 Team 9

Over all natural numbers

n

with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of

s(n)/n

s(n)

denotes the sum of the divisors of

n

. What is the value of

d(d(n))

d(n)

is the the number of divisors of

n

1

2020 Team 14

a_0=1

n\ge 1

a_n

be the smaller root of the equation

4^{-n}x^2-x+a_{n-1} = 0.

a_n

approaches a value

L

n

goes to infinity, what is the value of

L

1

2020 Team 15

ABC

be an acute triangle with

AB = 3

AC = 4

M

is the midpoint of segment

\overline{BC}

N

is the midpoint of

\overline{AM}

E

F

are the feet of the altitudes of

M

\overline{AB}

\overline{AC}

, respectively. Further suppose

BC

NE

S

NF

T

X

Y

be the circumcenters of

\triangle MES

\triangle MFT

, respectively. If

XY

is tangent to the circumcircle of

\triangle ABC

, what is the area of

\triangle ABC

1

2020 Team 8

\dbinom{2020}{1010}\dbinom{1010}{1010}+\dbinom{2019}{1010}\dbinom{1011}{1010}+\cdots+\dbinom{1011}{1010}\dbinom{2019}{1010} + \dbinom{1010}{1010}\dbinom{2020}{1010}.

1

2020 Team: Estimation

(x,y)

in the square bounded by

(0,0), (0,1), (1,0)

(1,1)

. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form

(0.abcd, 0.efgh)

a, b, c, d, e, f, g, h

are decimal digits.

1

2020 Team 7

P

Q

lie on a circle

\omega

. The tangents to

\omega

P

Q

intersect at point

T

R

\omega

T

R

lie on opposite sides of

PQ

\angle PQR = \angle PTQ

RT

\omega

for the second time at point

S

PQ = 12

TR = 28

PS

1

2020 Team 6

Misha is currently taking a Complexity Theory exam, but he seems to have forgotten a lot of the material! In the question, he is asked to fill in the following boxes with

\subseteq

\subsetneq

to identify the relationship between different complexity classes:

\mathsf{NL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{NP}\ \fbox{\phantom{tt}}\ \mathsf{PH}\ \fbox{\phantom{tt}}\ \mathsf{PSPACE}\ \fbox{\phantom{tt}}\ \mathsf {EXP}

\mathsf{coNL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{coNP}\ \fbox{\phantom{tt}}\ \mathsf{PH}

Luckily, he remembers that

\mathsf{P} \neq \mathsf{EXP}

\mathsf{NL} \neq \mathsf{PSPACE}

\mathsf{coNL} \neq \mathsf{PSPACE}

\mathsf{NP} \neq \mathsf{coNP}\implies \mathsf{P}\neq \mathsf{NP} \land \mathsf{P}\neq \mathsf{coNP}

. How many ways are there for him to fill in the boxes so as not to contradict what he remembers?

1

2020 Team 5

We say that a binary string

s

contains another binary string

t

if there exist indices

i_1,i_2,\ldots,i_{|t|}

i_1 < i_2 < \ldots < i_{|t|}

s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.

(In other words,

t

is found as a not necessarily contiguous substring of

s

.) For example,

110010

111

. What is the length of the shortest string

s

which contains the binary representations of all the positive integers less than or equal to

2048

1

2020 Team 4

n=2020

6

n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}

from least to greatest. Give your answer as a 6 digit permutation of the string "123456", where the number

i

corresponds to the

i

-th expression in the list, from left to right.

1

2020 Team 3

ABC

be a triangle with centroid

G

BC = 3

ABC

GAB

, compute the area of

ABC

1

2020 Team 2

Find all sets of five positive integers whose mode, mean, median, and range are all equal to

5

1

2020 Team 1

In a game of ping-pong, the score is

4-10

. Six points later, the score is

10-10

. You remark that it was impressive that I won the previous

6

points in a row, but I remark back that you have won

n

points in a row. What the largest value of

n

such that this statement is true regardless of the order in which the points were distributed?