Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2006B #1 grade 7+ arithmetic
What is (-1)¹ + (-1)² + ... + (-1)²⁰⁰⁶ ?
AMC 10 2006B #2 grade 7+ algebra
For real numbers x and y, define x ♠ y = (x+y)(x-y). What is 3 ♠ (4 ♠ 5)?
AMC 10 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 10 2006B #4 grade 7+ geometry-2d
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside…
AMC 10 2006B #5 grade 4+ geometry-2d
A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any point, and the sides of…
AMC 10 2006B #6 grade 7+ geometry-2d
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure 2/π, as shown. What is…
AMC 10 2006B #7 grade 8+ arithmetic
Which of the following is equivalent to √(x/(1-(x-1)/x)) when x < 0?
AMC 10 2006B #8 grade 8+ geometry-2d
A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?
AMC 10 2006B #9 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 10 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 10 2006B #11 grade 5+ arithmetic
What is the tens digit in the sum 7!+8!+9!+...+2006!
AMC 10 2006B #12 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 10 2006B #13 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 10 2006B #14 grade 7+ algebra
Let a and b be the roots of the equation x²-mx+2=0. Suppose that a+1/b and b+1/a are the roots of the equation x²-px+q=0…
AMC 10 2006B #15 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 10 2006B #16 grade 4+ arithmetic
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
AMC 10 2006B #17 grade 7+ probability
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice r…
AMC 10 2006B #18 grade 6+ arithmetic
Let a₁ , a₂ , ... be a sequence for which a₁=2 , a₂=3, and a_n=(a_(n-1))/(a_(n-2)) for each positive integer n ≥ 3. What…
AMC 10 2006B #19 grade 8+ geometry-2d
A circle of radius 2 is centered at O. Square OABC has side length 1. Sides AB and CB are extended past B to meet the ci…
AMC 10 2006B #20 grade 8+ geometry-2d
In rectangle ABCD, we have A=(6,-22), B=(2006,178), D=(8,y), for some integer y. What is the area of rectangle ABCD?
AMC 10 2006B #21 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 10 2006B #22 grade 6+ number-theory
Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4cent per glob and J blo…
AMC 10 2006B #23 grade 7+ geometry-2d
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite…
AMC 10 2006B #24 grade 8+ geometry-2d
Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B on the circle…
AMC 10 2006B #25 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…

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