The quadratic polynomial P(x) has the following properties: P(x) \(\ge\) 0 for all real numbers x, P(1) = 0, and P(2) = 2. What is the value of P(0) + P(4)?

 An equilateral triangle and a regular hexagon have equal perimeters. What is the ratio of the area of the triangle to the area of the hexagon? 

Suppose f(x) is a polynomial with integer coeffcients for which 3 and 13 are both roots. Which of the following could possibly be the value of f(10)?
(a) 3      (b) 10      (c) 14      (d) 39       (e) 42

Tim's car gets 3 more miles per gallon during highway driving than it does during city driving. On a recent trip, Tim drove 136 miles on the highway and 155 miles in the city, using a total of 9 gallons of gasoline. How many miles per gallon does Tim's car get during city driving?

A bowl contains 100 pieces of colored candy: 28 green, 20 red, 12 yellow, 10 blue, 20 brown, and 10 orange. If you are blindfolded as you pick and eat candy from this bowl, then how many pieces must you eat in order to guarantee that you have eaten at least 15 of the same color?

One solution of x³ + 5x² - 2x - 4 = 0 is x= 1. Find other solutions/

 Let x₁, x₂, x₃, be the numbers which can be written as a sum of one or more different powers of 3 with x₁ < x₂ < x₃< ...  

For example, x₁ = 3⁰ = 1, x₂ = 3¹ = 3, and x₃ = 3⁰ + 3¹ = 4.

What is the value of x₁₀₀?

Suppose that AB = AC = 4, and \(\angle\)CAB is a right angle.  P is the midpoint of AB. M is in BC, N is in AC. What is the smallest possible value of the perimeter of \(\Delta PMN\) ?

 Let two 8x12 rectangles share a common corner and overlap. The distance from the bottom right corner of one rectangle to the intersection point along the right edge of that rectangle is 7. What is the area of the shaded region?

Suppose that the numbers 1, 2, 4, 8, 16, 32, 64, 128, 256 are placed into the nine squares in such away that the product of the numbers appearing in any row, column or diagonal is the same. What is the value of this common product?
(a) 512    (b) 4096    (c) 8192     (d) 16384      (e) 32768

Suppose that the angle between the minute hand and hour hand of a clock is 60^o. If the minute hand is 16 inches long and the hour hand is 10 inches long, then what is the distance between the tip ends of the hands in inches?

 Suppose that f(x)=x⁵+ax⁴ + bx³ + cx² + dx + e and that f(1) = f(2) = f(3) = f(4) = f(5).

Then a =
(a) 8    (b) 10    (c)15    (d) 22    (e) 35

 Dave can answer each problem on a certain test in 6 minutes. Michael can answer each problem in 1 minute. Suppose Michael rests for two hours in the middle of answering the problems but Daveworks straight through the test without stopping. Suppose further that they nish the test at the same time. How long did it take Dave to answer all the problems?

Given that f(x) = (x⁵ - 1)(x³ + 1),  g(x) = (x² - 1)(x² - x + 1), and h(x) is a polynomial such that f(x)=g(x)h(x), what is the value of h(1)?

Let s(n) denote the sum of the digits of n. For example, s(197) = 1+9+7=17.

Let s²(n) = s(s(n))  ,     s³(n)=s(s(s(n))), and so on.

What is the value of s¹⁹⁹⁶(1996)?

(a) 18      (b) 4     (c) 12      (d) 7      (e) 0

At a party, every two people shook hands once. How many people attended the party if there were 45 handshakes?

Which of the following numbers is the largest?
(a) 10000¹⁰⁰     (b) 2¹⁰⁰⁰⁰         (c) 1000¹⁰⁰⁰    (d) 5⁴⁰⁰⁰     (e) 3²⁰⁰⁰

In the figure below, suppose that AB = AC = CD and AD = BD. What is the measure of \(\angle\)ADC in degrees?

 In an isosceles triangle, the inscribed circle has radius 2. Another circle of radius 1 is tangent to the inscribed circle and the two equal sides. What is the area of the triangle?

Beginning at 5:00 P.M., how many hours must elapse before the hourhand and minute-hand of a clock are perpendicular to each other?

(a) 1/5    (b) 2/11    (c) 5/22    (d) 4/23         (e) 7/30