MathDB

p1. The English alphabet consists of

21

consonants and

5

vowels. (We count

y

as a consonant.) (a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be

4

consecutive consonants. (b) Give a list to show that there need not be

5

consecutive consonants. (c) Suppose that all the letters are arranged in a circle. Prove that there must be

5

consecutive consonants.
p2. From the set

\{1,2,3,... , n\}

k

distinct integers are selected at random and arranged in numerical order (lowest to highest). Let

P(i, r, k, n)

denote the probability that integer

i

is in position

r

. For example, observe that

P(1, 2, k, n) = 0

. (a) Compute

P(2, 1,6,10)

. (b) Find a general formula for

P(i, r, k, n)

.
p3. (a) Write down a fourth degree polynomial

P(x)

such that

P(1) = P(-1)

but

P(2) \ne P(-2)

(b) Write down a fifth degree polynomial

Q(x)

such that

Q(1) = Q(-1)

and

Q(2) = Q(-2)

but

Q(3) \ne Q(-3)

. (c) Prove that, if a sixth degree polynomial

R(x)

satisfies

R(1) = R(-1)

R(2) = R(-2)

, and

R(3) = R(-3)

, then

R(x) = R(-x)

for all

x

.
p4. Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number

N

of distinct values among these sums. (a) Give an example of five numbers yielding the smallest possible value of

N

. What is this value? (b) Give an example of five numbers yielding the largest possible value of

N

. What is this value? (c) Prove that the values of

N

you obtained in (a) and (b) are the smallest and largest possible ones.
p5. Let

A_1A_2A_3

be a triangle which is not a right triangle. Prove that there exist circles

C_1

C_2

, and

C_3

such that

C_2

is tangent to

C_3

A_1

C_3

is tangent to

C_1

A_2

, and

C_1

is tangent to

C_2

A_3

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

Problem Statement