MathDB

p1. Suppose that

20\%

of a number is

17

. Find

20\%

17\%

of the number.
p2. Let

A, B, C, D

represent the numbers

1

through

4

in some order, with

A \ne 1

. Find the maximum possible value of

\frac{\log_A B}{C +D}

.
Here,

\log_A B

is the unique real number

X

such that

A^X = B

.
p3. There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed.
p4. Let

a,b,c

be real numbers which satisfy

\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).

Find the sum of all possible values of

abc

.
p5. Let

a

and

b

be complex numbers such that

ab + a +b = (a +b +1)(a +b +3)

. Find all possible values of

\frac{a+1}{b+1}

.
p6. Let

\vartriangle ABC

be a triangle. Let

X,Y,Z

be points on lines

BC

CA

, and

AB

, respectively, such that

X

lies on segment

BC

B

lies on segment

AY

, and

C

lies on segment

AZ

. Suppose that the circumcircle of

\vartriangle XYZ

is tangent to lines

AB

BC

, and

CA

with center

I_A

. If

AB = 20

and

I_AC = AC = 17

then compute the length of segment

BC

.
p7. An ant makes

4034

moves on a coordinate plane, beginning at the point

(0, 0)

and ending at

(2017, 2017)

. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region

|x - y| \le 2

. Let N be the number of paths the ant can take. Find the remainder when

N

is divided by

1000

.
p8. A

10

digit positive integer

\overline{a_9a_8a_7...a_1a_0}

with

a_9

nonzero is called deceptive if there exist distinct indices

i > j

such that

\overline{a_i a_j} = 37

. Find the number of deceptive positive integers.
p9. A circle passing through the points

(2, 0)

and

(1, 7)

is tangent to the

y

-axis at

(0, r )

. Find all possible values of

r

.
p10. An ellipse with major and minor axes

20

and

17

, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square.
PS. You had better use hide for answers.

Problem Statement