MathDB

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names

R1. What is

11^2 - 9^2

?
R2. Write

\frac{9}{15}

as a decimal.
R3. A

90^o

sector of a circle is shaded, as shown below. What percent of the circle is shaded?
R4. A fair coin is flipped twice. What is the probability that the results of the two flips are different?
R5. Wayne Dodson has

55

pounds of tungsten. If each ounce of tungsten is worth

75

cents, and there are

16

ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth?
R6. Tenley Towne has a collection of

28

sticks. With these

28

sticks he can build a tower that has

1

stick in the top row,

2

in the next row, and so on. Let

n

be the largest number of rows that Tenley Towne’s tower can have. What is n?
R7. What is the sum of the four smallest primes?
R8 / P1. Let

ABC

be an isosceles triangle such that

\angle B = 42^o

. What is the sum of all possible degree measures of angle

A

?
R9. Consider a line passing through

(0, 0)

and

(4, 8)

. This line passes through the point

(2, a)

. What is the value of

a

?
R10 / P2. Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled?
R11. Guang chooses

4

distinct integers between

0

and

9

, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number?
R12 / P4. David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex

n

-gon, and it so happens that every angle he assigned is less than

144

degrees. He tells Pratik the value of

n

and the degree measures in the

n

-gon, and to David’s dismay, Pratik claims that such an

n

-gon does not exist. What is the smallest value of

n \ge 3

such that Pratik’s claim is necessarily true?
R13 / P3. Consider a triangle

ABC

with side lengths of

5

5

, and

2\sqrt5

. There exists a triangle with side lengths of

5, 5

, and

x

(

x \ne 2\sqrt5

) which has the same area as

ABC

. What is the value of

x

?
R14 / P5. A mother has

11

identical apples and

9

identical bananas to distribute among her

3

kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana?
R15 / P7. Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes.
P6. Srinivasa Ramanujan has the polynomial

P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12

. His friend Hardy tells him that

3

is one of the roots of

P(x)

. What is the sum of the other roots of

P(x)

?
P8.

ABC

is an equilateral triangle with side length

10

. Let

P

be a point which lies on ray

\overrightarrow{BC}

such that

PB = 20

. Compute the ratio

\frac{PA}{PC}

.
P9. Let

ABC

be a triangle such that

AB = 10

BC = 14

, and

AC = 6

. The median

CD

and angle bisector

CE

are both drawn to side

AB

. What is the ratio of the area of triangle

CDE

to the area of triangle

ABC

?
P10. Find all integer values of

x

between

0

and

2017

inclusive, which satisfy

2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).

P11. Let

x^2 + ax + b

be a quadratic polynomial with positive integer roots such that

a^2 - 2b = 97

. Compute

a + b

.
P12. Let

S

be the set

\{2, 3, ... , 14\}

. We assign a distinct number from

S

to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct)
P13. In triangle

ABC

AB = 10

BC = 21

, and

AC = 17

D

is the foot of the altitude from

A

BC

E

is the foot of the altitude from

D

AB

, and

F

is the foot of the altitude from

D

AC

. Find the area of the smallest circle that contains the quadrilateral

AEDF

.
P14. What is the greatest distance between any two points on the graph of

3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11

?
P15. For a positive integer

n

\tau (n)

is defined to be the number of positive divisors of

n

. Given this information, find the largest positive integer

n

less than

1000

such that

\sum_{d|n} \tau (d) = 108.

In other words, we take the sum of

\tau (d)

for every positive divisor

d

n

, which has to be

108

.

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement