MathDB

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names
Set 3
P3.11 Find all possible values of

c

in the following system of equations:

a^2 + ab + c^2 = 31

b^2 + ab - c^2 = 18

a^2 - b^2 = 7

P3.12 / R5.25 In square

ABCD

with side length

13

, point

E

lies on segment

CD

. Segment

AE

divides

ABCD

into triangle

ADE

and quadrilateral

ABCE

. If the ratio of the area of

ADE

to the area of

ABCE

4 : 11

, what is the ratio of the perimeter of

ADE

to the perimeter of

ABCE

?
P3.13 Thomas has two distinct chocolate bars. One of them is

1

5

and the other one is

1

3

. If he can only eat a single

1

1

piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars?
P3.14 In triangle

ABC

AB = 13

BC = 14

, and

CA = 15

. The entire triangle is revolved about side

BC

. What is the volume of the swept out region?
P3.15 Find the number of ordered pairs of positive integers

(a, b)

that satisfy the equation

a(a -1) + 2ab + b(b - 1) = 600

.
Set 4
P4.16 Compute the sum of the digits of

(10^{2017} - 1)^2

.
P4.17 A right triangle with area

210

is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle.

2

semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the

2

semicircles but outside the first semicircle?
P4.18 Find the smallest positive integer

n

such that exactly

\frac{1}{10}

of its positive divisors are perfect squares.
P4.19 One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius

1

inch. Each player begins with

14

oranges. Jamie builds his tower by making a

3

3

base, placing a

2

2

square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats

4

of his oranges. With his remaining

10

oranges, he builds a similar tower, forming an equilateral triangle with

3

oranges on each side, placing another equilateral triangle with

2

oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers?
P4.20 Let

r, s

, and

t

be the roots of the polynomial

x^3 - 9x + 42

. Compute the value of

(rs)^3 + (st)^3 + (tr)^3

.
Set 5
P5.21 For all integers

k > 1

\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}

. There exists a sequence of integers

j_0, j_1, ...

such that

\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3

for all integers

k > 1

. Find

j_{10}

.
P5.22 Nimi is a triangle with vertices located at

(-1, 6)

(6, 3)

, and

(7, 9)

. His center of mass is tied to his owner, who is asleep at

(0, 0)

, using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through?
P5.23 The polynomial

x^{19} - x - 2

has

19

distinct roots. Let these roots be

a_1, a_2, ..., a_{19}

. Find

a^{37}_1 + a^{37}_2+...+a^{37}_{19}

.
P5.24 I start with a positive integer

n

. Every turn, if

n

is even, I replace

n

with

\frac{n}{2}

, otherwise I replace

n

with

n-1

. Let

k

be the most turns required for a number

n < 500

to be reduced to

1

. How many values of

n < 500

require k turns to be reduced to

1

?
P5.25 In triangle

ABC

AB = 13

BC = 14

, and

AC = 15

. Let

I

and

O

be the incircle and circumcircle of

ABC

, respectively. The altitude from

A

intersects

I

at points

P

and

Q

, and

O

at point

R

, such that

Q

lies between

P

and

R

. Find

PR

.
PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here, and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement