MathDB

p1. What is the maximum number of

T

-shaped polyominos (shown below) that we can put into a

6 \times 6

grid without any overlaps. The blocks can be rotated. https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png
p2. In triangle

\vartriangle ABC

\angle A = 30^o

D

is a point on

AB

such that

CD \perp AB

E

is a point on

AC

such that

BE \perp AC

. What is the value of

\frac{DE}{BC}

?
p3. Given that f(x) is a polynomial such that

2f(x) + f(1 - x) = x^2

. Find the sum of squares of the coefficients of

f(x)

.
p4. For each positive integer

n

, there exists a unique positive integer an such that

a^2_n \le n < (a_n + 1)^2

. Given that

n = 15m^2

, where

m

is a positive integer greater than

1

. Find the minimum possible value of

n - a^2_n

.
p5. What are the last two digits of

\lfloor (\sqrt5 + 2)^{2016}\rfloor

?
Note

\lfloor x \rfloor

is the largest integer less or equal to x.
p6. Let

f

be a function that satisfies

f(2^a3^b)) = 3a+ 5b

. What is the largest value of f over all numbers of the form

n = 2^a3^b

where

n \le 10000

and

a, b

are nonnegative integers.
p7. Find a multiple of

21

such that it has six more divisors of the form

4m + 1

than divisors of the form

4n + 3

where m, n are integers. You can keep the number in its prime factorization form.
p8. Find

\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor

where

\lfloor x \rfloor

is the largest integer less or equal to x.
p9. Let

A, B

be two randomly chosen subsets of

\{1, 2, . . . 10\}

. What is the probability that one of the two subsets contains the other?
p10. We want to pick

5

-person teams from a total of

m

people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint:

m

is determined by these conditions).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement