Competition · AMC preparation · step 4 of 4

AMC 12 2004B: 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 2004B #1 grade 4+ arithmetic
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her…
AMC 12 2004B #2 grade 6+ algebra
In the expression c· a^b-d, the values of a, b, c, and d are 0, 1, 2, and 3, although not necessarily in that order. Wha…
AMC 12 2004B #3 grade 6+ number-theory
If x and y are positive integers for which 2^x3^y=1296, what is the value of x+y? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
AMC 12 2004B #4 grade 7+ probability
An integer x, with 10≤ x≤ 99, is to be chosen. If all choices are equally likely, what is the probability that at least…
AMC 12 2004B #5 grade 7+ arithmetic
On a trip from the United States to Canada, Isabella took d U.S. dollars. At the border she exchanged them all, receivin…
AMC 12 2004B #6 grade 8+ arithmetic
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown…
AMC 12 2004B #7 grade 7+ geometry-2d
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the uni…
AMC 12 2004B #8 grade 4+ arithmetic
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row abov…
AMC 12 2004B #9 grade 8+ geometry-2d
The point (-3,2) is rotated 90° clockwise around the origin to point B. Point B is then reflected over the line x=y to p…
AMC 12 2004B #10 grade 8+ geometry-2d
An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c, with b…
AMC 12 2004B #11 grade 7+ algebra
All the students in an algebra class took a 100-point test. Five students scored 100, each student scored at least 60, a…
AMC 12 2004B #12 grade 6+ arithmetic
In the sequence 2001, 2002, 2003, … , each term after the third is found by subtracting the previous term from the sum o…
AMC 12 2004B #13 grade 11+ algebra
If f(x) = ax+b and f⁻¹(x) = bx+a with a and b real, what is the value of a+b?
AMC 12 2004B #14 grade 10+ geometry-2d
In △ ABC, AB=13, AC=5, and BC=12. Points M and N lie on AC and BC, respectively, with CM=CN=4. Points J and K are on AB…
AMC 12 2004B #15 grade 6+ arithmetic
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be…
AMC 12 2004B #16 grade 11+ algebra
A function f is defined by f(z) = iz, where i=√(-1) and z is the complex conjugate of z. How many values of z satisfy bo…
AMC 12 2004B #17 grade 11+ algebra
For some real numbers a and b, the equation 8x³ + 4ax² + 2bx + a = 0 has three distinct positive roots. If the sum of th…
AMC 12 2004B #18 grade 8+ geometry-2d, algebra
Points A and B are on the parabola y=4x²+7x-1, and the origin is the midpoint of AB. What is the length of AB?
AMC 12 2004B #19 grade 10+ geometry-3d
A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface o…
AMC 12 2004B #20 grade 7+ geometry-3d
Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined indep…
AMC 12 2004B #21 grade 11+ algebra, geometry-2d
The graph of 2x² + xy + 3y² - 11x - 20y + 40 = 0 is an ellipse in the first quadrant of the xy-plane. Let a and b be the…
AMC 12 2004B #22 grade 8+ geometry-2d
The square 50 b c d e f g h 2 is a multiplicative magic square. That is, the product of the numbers in each row, column,…
AMC 12 2004B #23 grade 11+ algebra
The polynomial x³ - 2004 x² + mx + n has integer coefficients and three distinct positive zeros. Exactly one of these is…
AMC 12 2004B #24 grade 11+ geometry-2d
In △ ABC, AB = BC, and BD is an altitude. Point E is on the extension of AC such that BE = 10. The values of tan ∠ CBE,…
AMC 12 2004B #25 grade 9+ number-theory, counting
Given that 2²⁰⁰⁴ is a 604-digit number whose first digit is 1, how many elements of the set S = {2⁰,2¹,2²,… ,2²⁰⁰³} have…

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