Competition · AMC preparation · step 4 of 4

AMC 12 2007B: 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 2007B #1 grade 4+ geometry-2d
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the wa…
AMC 12 2007B #2 grade 6+ rate-ratio
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return t…
AMC 12 2007B #3 grade 8+ geometry-2d
The point O is the center of the circle circumscribed about triangle ABC, with ∠ BOC = 120° and ∠ AOB = 140°, as shown.…
AMC 12 2007B #4 grade 6+ rate-ratio
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges co…
AMC 12 2007B #5 grade 6+ arithmetic
The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect resp…
AMC 12 2007B #6 grade 4+ geometry-2d
Triangle ABC has side lengths AB = 5, BC = 6, and AC = 7. Two bugs start simultaneously from A and crawl along the sides…
AMC 12 2007B #7 grade 8+ geometry-2d
All sides of the convex pentagon ABCDE are of equal length, and ∠ A= ∠ B = 90°. What is the degree measure of ∠ E?
AMC 12 2007B #8 grade 8+ algebra
Tom's age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of…
AMC 12 2007B #9 grade 9+ algebra
A function f has the property that f(3x-1)=x²+x+1 for all real numbers x. What is f(5)?
AMC 12 2007B #10 grade 6+ algebra
Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls…
AMC 12 2007B #11 grade 8+ geometry-2d
The angles of quadrilateral ABCD satisfy ∠ A=2 ∠ B=3 ∠ C=4 ∠ D. What is the degree measure of ∠ A, rounded to the neares…
AMC 12 2007B #12 grade 6+ arithmetic
A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the…
AMC 12 2007B #13 grade 7+ probability
A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then r…
AMC 12 2007B #14 grade 8+ geometry-2d
Point P is inside equilateral △ ABC. Points Q, R, and S are the feet of the perpendiculars from P to AB, BC, and CA, res…
AMC 12 2007B #15 grade 11+ algebra
The geometric series a+ar+ar²… has a sum of 7, and the terms involving odd powers of r have a sum of 3. What is a+r?
AMC 12 2007B #16 grade 8+ counting, geometry-3d
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable…
AMC 12 2007B #17 grade 11+ algebra
If a is a nonzero integer and b is a positive number such that ab²=log₁₀b, what is the median of the set {0,1,a,b,1/b}?
AMC 12 2007B #18 grade 9+ number-theory, algebra
Let a, b, and c be digits with a≠ 0. The three-digit integer abc lies one third of the way from the square of a positive…
AMC 12 2007B #19 grade 11+ geometry-3d
Rhombus ABCD, with side length 6, is rolled to form a cylinder of volume 6 by taping AB to DC. What is sin(∠ ABC)?
AMC 12 2007B #20 grade 9+ geometry-2d, number-theory
The parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18. The parallelogram bounded by the…
AMC 12 2007B #21 grade 7+ counting, number-theory
The first 2007 positive integers are each written in base 3. How many of these base-3 representations are palindromes? (…
AMC 12 2007B #22 grade 10+ geometry-2d
Two particles move along the edges of equilateral △ ABC in the direction A⇒ B⇒ C⇒ A, starting simultaneously and moving…
AMC 12 2007B #23 grade 9+ number-theory, geometry-2d
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 time…
AMC 12 2007B #24 grade 6+ number-theory
How many pairs of positive integers (a,b) are there such that gcd(a,b)=1 and a/b + 14b/9a is an integer?
AMC 12 2007B #25 grade 10+ geometry-3d
Points A,B,C,D and E are located in 3-dimensional space with AB=BC=CD=DE=EA=2 and ∠ ABC=∠ CDE=∠ DEA=90°. The plane of △…

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