Competition · AMC preparation · step 4 of 4

AMC 10 2007A: 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 2007A #1 grade 6+ arithmetic
One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buy…
AMC 10 2007A #2 grade 7+ arithmetic
Define a@b = ab - b² and a#b = a + b - ab². What is (6@2)/(6#2)?
AMC 10 2007A #3 grade 5+ geometry-3d
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a…
AMC 10 2007A #4 grade 6+ algebra
The larger of two consecutive odd integers is three times the smaller. What is their sum?
AMC 10 2007A #5 grade 8+ arithmetic
The school store sells 7 pencils and 8 notebooks for mathdollar 4.15. It also sells 5 pencils and 3 notebooks for mathdo…
AMC 10 2007A #6 grade 7+ arithmetic
At Euclid High School, the number of students taking the AMC 10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 a…
AMC 10 2007A #7 grade 7+ algebra
Last year Mr. Jon Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of wh…
AMC 10 2007A #8 grade 8+ geometry-2d
Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside triangle ABC, angle ABC measures 40 degrees,…
AMC 10 2007A #9 grade 8+ arithmetic
Real numbers a and b satisfy the equations 3^a = 81^(b + 2) and 125^b = 5^(a - 3). What is ab?
AMC 10 2007A #10 grade 7+ arithmetic
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20,…
AMC 10 2007A #11 grade 3+ geometry-3d
The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each f…
AMC 10 2007A #12 grade 4+ arithmetic
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but…
AMC 10 2007A #13 grade 7+ rate-ratio
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he…
AMC 10 2007A #14 grade 8+ geometry-2d
A triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle with radius 3. What is the area of the tria…
AMC 10 2007A #15 grade 8+ geometry-2d
Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2, as sh…
AMC 10 2007A #16 grade 7+ probability
Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. Wha…
AMC 10 2007A #17 grade 8+ arithmetic
Suppose that m and n are positive integers such that 75m = n³. What is the minimum possible value of m + n?
AMC 10 2007A #18 grade 8+ geometry-2d
Consider the 12-sided polygon ABCDEFGHIJKL, as shown. Each of its sides has length 4, and each two consecutive sides for…
AMC 10 2007A #19 grade 8+ geometry-2d
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area o…
AMC 10 2007A #20 grade 8+ algebra
Suppose that the number a satisfies the equation 4 = a + a^(- 1). What is the value of a⁴ + a^(- 4)?
AMC 10 2007A #21 grade 8+ geometry-3d
A sphere is inscribed in a cube that has a surface area of 24 square meters. A second cube is then inscribed within the…
AMC 10 2007A #22 grade 6+ number-theory
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively…
AMC 10 2007A #23 grade 6+ arithmetic
How many ordered pairs (m,n) of positive integers, with m ≥ n, have the property that their squares differ by 96?
AMC 10 2007A #24 grade 8+ geometry-2d
Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB, and OA = 2√2. Segments OC and O…
AMC 10 2007A #25 grade 4+ number-theory
For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n))…

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