MathDB

p1. The total cost of

1

football,

3

tennis balls and

7

golf balls is

\$14

, while that of

1

football,

4

tennis balls and

10

golf balls is

\$17

.If one has

\$20

to spend, is this sufficient to buy a)

3

footballs and

2

tennis balls? b)

2

footballs and

3

tennis balls?
p2. Let

\overline{AB}

and

\overline{CD}

be two chords in a circle intersecting at a point

P

(inside the circle). a) Prove that

AP \cdot PB = CP\cdot PD

. b) If

\overline{AB}

is perpendicular to

\overline{CD}

and the length of

\overline{AP}

2

, the length of

\overline{PB}

6

, and the length of

\overline{PD}

3

, find the radius of the circle.
p3. A polynomial

P(x)

of degree greater than one has the remainder

2

when divided by

x-2

and the remainder

3

when divided by

x-3

. Find the remainder when

P(x)

is divided by

x^2-5x+6

.
p4. Let

x_1= 2

and

x_{n+1}=x_n+ (3n+2)

for all

n

greater than or equal to one. a) Find a formula expressing

x_n

as a function of

n

. b) Prove your result.
p5. The point

M

is the midpoint of side

\overline{BC}

of a triangle

ABC

. a) Prove that

AM \le \frac12 AB + \frac12 AC

.
b) A fly takes off from a certain point and flies a total distance of

4

meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement