MathDB

p1. David runs at

3

times the speed of Alice. If Alice runs

2

miles in

30

minutes, determine how many minutes it takes for David to run a mile.
p2. Al has

2019

red jelly beans. Bob has

2018

green jelly beans. Carl has

x

blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee

2

jelly beans of each color is

4041

. Compute

x

.
p3. Find the

7

-digit palindrome which is divisible by

7

and whose first three digits are all

2

.
p4. Determine the number of ways to put

5

indistinguishable balls in

6

distinguishable boxes.
p5. A certain reduced fraction

\frac{a}{b}

(with

a,b > 1

) has the property that when

2

is subtracted from the numerator and added to the denominator, the resulting fraction has

\frac16

of its original value. Find this fraction.
p6. Find the smallest positive integer

n

such that

|\tau(n +1)-\tau(n)| = 7

. Here,

\tau(n)

denotes the number of divisors of

n

.
p7. Let

\vartriangle ABC

be the triangle such that

AB = 3

AC = 6

and

\angle BAC = 120^o

. Let

D

be the point on

BC

such that

AD

bisect

\angle BAC

. Compute the length of

AD

.
p8.

26

points are evenly spaced around a circle and are labeled

A

through

Z

in alphabetical order. Triangle

\vartriangle LMT

is drawn. Three more points, each distinct from

L, M

, and

T

, are chosen to form a second triangle. Compute the probability that the two triangles do not overlap.
p9. Given the three equations

a +b +c = 0

a^2 +b^2 +c^2 = 2

a^3 +b^3 +c^3 = 19

find

abc

.
p10. Circle

\omega

is inscribed in convex quadrilateral

ABCD

and tangent to

AB

and

CD

P

and

Q

, respectively. Given that

AP = 175

BP = 147

CQ = 75

, and

AB \parallel CD

, find the length of

DQ

.
p11. Let

p

be a prime and m be a positive integer such that

157p = m^4 +2m^3 +m^2 +3

. Find the ordered pair

(p,m)

.
p12. Find the number of possible functions

f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}

that satisfy the following conditions. (1)

f (x) \ne f (y)

when

x \ne y

(2) There exists some

x

such that

f (x)^2 = x^2

p13. Let

p

be a prime number such that there exists positive integer

n

such that

41pn -42p^2 = n^3

. Find the sum of all possible values of

p

.
p14. An equilateral triangle with side length

1

is rotated

60

degrees around its center. Compute the area of the region swept out by the interior of the triangle.
p15. Let

\sigma (n)

denote the number of positive integer divisors of

n

. Find the sum of all

n

that satisfy the equation

\sigma (n) =\frac{n}{3}

.
p16. Let

C

be the set of points

\{a,b,c\} \in Z

for

0 \le a,b,c \le 10

. Alice starts at

(0,0,0)

. Every second she randomly moves to one of the other points in

C

that is on one of the lines parallel to the

x, y

, and

z

axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach

(10,10,10)

.
p17. Find the maximum possible value of

abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3

where

a,b,c

are real such that

a +b +c = 0

.
p18. Circle

\omega

with radius

6

is inscribed within quadrilateral

ABCD

\omega

is tangent to

AB

BC

CD

, and

DA

E, F, G

, and

H

respectively. If

AE = 3

BF = 4

and

CG = 5

, find the length of

DH

.
p19. Find the maximum integer

p

less than

1000

for which there exists a positive integer

q

such that the cubic equation

x^3 - px^2 + q x -(p^2 -4q +4) = 0

has three roots which are all positive integers.
p20. Let

\vartriangle ABC

be the triangle such that

\angle ABC = 60^o

\angle ACB = 20^o

. Let

P

be the point such that

CP

bisects

\angle ACB

and

\angle PAC = 30^o

. Find

\angle PBC

.
PS. You had better use hide for answers.

Problem Statement