Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 2005B #2 grade 8+ arithmetic
A positive number x has the property that x% of x is 4. What is x?
AMC 10 2005B #3 grade 5+ rate-ratio
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third…
AMC 10 2005B #4 grade 8+ arithmetic
For real numbers a and b, define a diamond b = √(a² + b²). What is the value of (5 diamond 12) diamond ((-12) diamond (-…
AMC 10 2005B #5 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 10 2005B #6 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 10 2005B #7 grade 8+ geometry-2d
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this…
AMC 10 2005B #8 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 10 2005B #9 grade 7+ probability
One fair die has faces 1, 1, 2, 2, 3, 3 and another has faces 4, 4, 5, 5, 6, 6. The dice are rolled and the numbers on t…
AMC 10 2005B #10 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 10 2005B #11 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 10 2005B #12 grade 7+ probability
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
AMC 10 2005B #13 grade 6+ arithmetic
How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?
AMC 10 2005B #14 grade 8+ geometry-2d
Equilateral △ ABC has side length 2, M is the midpoint of AC, and C is the midpoint of BD. What is the area of △ CDM?
AMC 10 2005B #15 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 10 2005B #16 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 10 2005B #17 grade 8+ arithmetic
Suppose that 4^a = 5, 5^b = 6, 6^c = 7, and 7^d = 8. What is a · b· c · d?
AMC 10 2005B #18 grade 7+ arithmetic
All of David's telephone numbers have the form 555-abc-defg, where a, b, c, d, e, f, and g are distinct digits and in in…
AMC 10 2005B #19 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 10 2005B #20 grade 6+ arithmetic
What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exact…
AMC 10 2005B #21 grade 7+ probability
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on f…
AMC 10 2005B #22 grade 7+ number-theory
For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + … + n?
AMC 10 2005B #23 grade 8+ geometry-2d
In trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC, and F as the midpoint of DA. The area of ABEF is t…
AMC 10 2005B #24 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 10 2005B #25 grade 5+ arithmetic
A subset B of the set of integers from 1 to 100, inclusive, has the property that no two elements of B sum to 125. What…

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