Competition · AMC preparation · step 4 of 4

AMC 12 2005B: 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 2005B #1 grade 4+ arithmetic
A scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two fo…
AMC 12 2005B #2 grade 8+ arithmetic
A positive number x has the property that x% of x is 4. What is x?
AMC 12 2005B #3 grade 4+ rate-ratio
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth o…
AMC 12 2005B #4 grade 6+ arithmetic
At the beginning of the school year, Lisa's goal was to earn an A on at least 80% of her 50 quizzes for the year. She ea…
AMC 12 2005B #5 grade 7+ geometry-2d
An 8-foot by 10-foot bathroom floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consist…
AMC 12 2005B #6 grade 8+ geometry-2d
In △ ABC, we have AC=BC=7 and AB=2. Suppose that D is a point on line AB such that B lies between A and D and CD=8. What…
AMC 12 2005B #7 grade 8+ geometry-2d
What is the area enclosed by the graph of |3x|+|4y|=12?
AMC 12 2005B #8 grade 9+ algebra
For how many values of a is it true that the line y = x + a passes through the vertex of the parabola y = x² + a² ?
AMC 12 2005B #9 grade 6+ arithmetic
On a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and…
AMC 12 2005B #10 grade 6+ arithmetic
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. W…
AMC 12 2005B #11 grade 7+ probability
An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replace…
AMC 12 2005B #12 grade 8+ algebra
The quadratic equation x²+mx+n has roots twice those of x²+px+m, and none of m,n, and p is zero. What is the value of n/…
AMC 12 2005B #13 grade 11+ algebra
Suppose that 4^x₁=5, 5^x₂=6, 6^x₃=7, ... , 127^x₁₂₄=128. What is x₁x₂...x₁₂₄?
AMC 12 2005B #14 grade 10+ geometry-2d
A circle having center (0,k), with k>6, is tangent to the lines y=x, y=-x and y=6. What is the radius of this circle?
AMC 12 2005B #15 grade 8+ number-theory
The sum of four two-digit numbers is 221. None of the eight digits is 0 and no two of them are the same. Which of the fo…
AMC 12 2005B #16 grade 10+ geometry-3d
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest…
AMC 12 2005B #17 grade 11+ algebra, number-theory
How many distinct four-tuples (a,b,c,d) of rational numbers are there with a·log₁₀2+b·log₁₀3+c·log₁₀5+d·log₁₀7=2005?
AMC 12 2005B #18 grade 10+ geometry-2d
Let A(2,2) and B(7,7) be points in the plane. Define R as the region in the first quadrant consisting of those points C…
AMC 12 2005B #19 grade 8+ arithmetic
Let x and y be two-digit integers such that y is obtained by reversing the digits of x. The integers x and y satisfy x²…
AMC 12 2005B #20 grade 9+ algebra
Let a,b,c,d,e,f,g and h be distinct elements in the set {-7,-5,-3,-2,2,4,6,13}. What is the minimum possible value of (a…
AMC 12 2005B #21 grade 8+ number-theory
A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n?
AMC 12 2005B #22 grade 12+ algebra
A sequence of complex numbers z₀, z₁, z₂, ... is defined by the rule z_(n+1) = (iz_n)/(z_n), where z_n is the complex co…
AMC 12 2005B #23 grade 11+ algebra
Let S be the set of ordered triples (x,y,z) of real numbers for which log₁₀(x+y) = z and log₁₀(x²+y²) = z+1. There are r…
AMC 12 2005B #24 grade 11+ geometry-2d
All three vertices of an equilateral triangle are on the parabola y = x², and one of its sides has a slope of 2. The x-c…
AMC 12 2005B #25 grade 7+ probability
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultan…

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