Competition · AMC preparation · step 4 of 4

AMC 12 2006B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2006B #1 grade 7+ arithmetic
What is (-1)¹ + (-1)² + ... + (-1)²⁰⁰⁶ ?
AMC 12 2006B #2 grade 7+ algebra
For real numbers x and y, define x ♠ y = (x+y)(x-y). What is 3 ♠ (4 ♠ 5)?
AMC 12 2006B #3 grade 7+ algebra
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, a…
AMC 12 2006B #4 grade 6+ arithmetic
Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0…
AMC 12 2006B #5 grade 6+ rate-ratio
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour.…
AMC 12 2006B #6 grade 6+ rate-ratio
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calor…
AMC 12 2006B #7 grade 7+ counting
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other tw…
AMC 12 2006B #8 grade 8+ algebra
The lines x=1/4y+a and y=1/4x+b intersect at the point (1,2). What is a+b?
AMC 12 2006B #9 grade 7+ counting
How many even three-digit integers have the property that their digits, all read from left to right, are in strictly inc…
AMC 12 2006B #10 grade 7+ geometry-2d
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third s…
AMC 12 2006B #11 grade 6+ rate-ratio
Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounce…
AMC 12 2006B #12 grade 9+ algebra
The parabola y=ax²+bx+c has vertex (p,p) and y-intercept (0,-p), where p≠ 0. What is b?
AMC 12 2006B #13 grade 8+ geometry-2d
Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24 and ∠ BAD = 60°. What is the area of rhombus BFD…
AMC 12 2006B #14 grade 6+ number-theory
Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4 cents per glob and J b…
AMC 12 2006B #15 grade 8+ geometry-2d
Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B are on the cir…
AMC 12 2006B #16 grade 8+ geometry-2d
Regular hexagon ABCDEF has vertices A and C at (0,0) and (7,1), respectively. What is its area?
AMC 12 2006B #17 grade 7+ probability
For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6, on each die are in the ratio…
AMC 12 2006B #18 grade 7+ counting, geometry-2d
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right,…
AMC 12 2006B #19 grade 6+ number-theory
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with…
AMC 12 2006B #20 grade 11+ probability
Let x be chosen at random from the interval (0,1). What is the probability that ⌊log₁₀4x⌋ - ⌊log₁₀x⌋ = 0? Here ⌊ x⌋ deno…
AMC 12 2006B #21 grade 11+ geometry-2d
Rectangle ABCD has area 2006. An ellipse with area 2006π passes through A and C and has foci at B and D. What is the per…
AMC 12 2006B #22 grade 8+ number-theory
Suppose a, b and c are positive integers with a+b+c=2006, and a!b!c!=m· 10ⁿ, where m and n are integers and m is not div…
AMC 12 2006B #23 grade 9+ geometry-2d
Isosceles △ ABC has a right angle at C. Point P is inside △ ABC, such that PA=11, PB=7, and PC=6. Legs AC and BC have le…
AMC 12 2006B #24 grade 11+ algebra, geometry-2d
Let S be the set of all point (x,y) in the coordinate plane such that 0 ≤ x ≤ π/2 and 0 ≤ y ≤ π/2. What is the area of t…
AMC 12 2006B #25 grade 8+ number-theory, pattern
A sequence a₁,a₂,… of non-negative integers is defined by the rule a_(n+2)=|a_(n+1)-a_n| for n≥ 1. If a₁=999, a₂<999 and…

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