MathDB

p1. Let

x_1 = 0

x_2 = 1/2

and for

n >2

, let

x_n

be the average of

x_{n-1}

and

x_{n-2}

. Find a formula for

a_n = x_{n+1} - x_{n}

n = 1, 2, 3, \dots

. Justify your answer.
p2. Given a triangle

ABC

. Let

h_a, h_b, h_c

be the altitudes to its sides

a, b, c,

respectively. Prove:

\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}

Is it possible to construct a triangle with altitudes

7

11

, and

20

? Justify your answer.
p3. Does there exist a polynomial

P(x)

with integer coefficients such that

P(0) = 1

P(2) = 3

and

P(4) = 9

? Justify your answer.
p4. Prove that if

\cos \theta

is rational and

n

is an integer, then

\cos n\theta

is rational. Let

\alpha=\frac{1}{2010}

. Is

\cos \alpha

rational ? Justify your answer.
p5. Let function

f(x)

be defined as

f(x) = x^2 + bx + c

, where

b, c

are real numbers. (A) Evaluate

f(1) -2f(5) + f(9)

. (B) Determine all pairs

(b, c)

such that

|f(x)| \le 8

for all

x

in the interval

[1, 9]

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

Problem Statement