MathDB

p1. Find all real numbers that satisfy the equation

(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}

p2. Let

A

be an integer and

A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}

Find the last seven digits of

A

, in order from millions to units.
p3. In triangle

ABC

, points

P

and

Q

are on sides of

BC

so that the length of

BP

is equal to

CQ

\angle BAP = \angle CAQ

and

\angle APB

is acute. Is triangle

ABC

isosceles? Write down your reasons.
p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four

0

s (zero) and five

1

s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?
p5. Fulan keeps

100

turkeys with the weight of the

i

-th turkey, being

x_i

for

i\in\{1, 2, 3, ... , 100\}

. The weight of the

i

-th turkey in grams is assumed to follow the function

x_i(t) = S_it + 200 - i

where

t

represents the time in days and

S_i

is the

i

-th term of an arithmetic sequence where the first term is a positive number

a

with a difference of

b =\frac15

. It is known that the average data on the weight of the hundred turkeys at

t = a

150.5

grams. Calculate the median weight of the turkey at time

t = 20

days.

day 1

Problems(1)