Competition · AMC preparation · step 4 of 4

AMC 10 2014A: 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 2014A #1 grade 8+ arithmetic
What is 10·(1/2+1/5+1/10)⁻¹?
AMC 10 2014A #2 grade 6+ arithmetic
Roy's cat eats 1/3 of a can of cat food every morning and 1/4 of a can of cat food every evening. Before feeding his cat…
AMC 10 2014A #3 grade 5+ algebra
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for textdollar 2.50 each. In the…
AMC 10 2014A #4 grade 2+ logic, counting
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house…
AMC 10 2014A #5 grade 6+ arithmetic
On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest score…
AMC 10 2014A #6 grade 7+ rate-ratio
Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days?
AMC 10 2014A #7 grade 7+ arithmetic
Nonzero real numbers x, y, a, and b satisfy x < a and y < b. How many of the following inequalities must be true? (I) x+…
AMC 10 2014A #8 grade 8+ number-theory
Which of the following numbers is a perfect square?
AMC 10 2014A #9 grade 8+ geometry-2d
The two legs of a right triangle, which are altitudes, have lengths 2√3 and 6. How long is the third altitude of the tri…
AMC 10 2014A #10 grade 6+ algebra
Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that st…
AMC 10 2014A #11 grade 7+ algebra
A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10% off the…
AMC 10 2014A #12 grade 8+ geometry-2d
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, cre…
AMC 10 2014A #13 grade 8+ geometry-2d
Equilateral △ ABC has side length 1, and squares ABDE, BCHI, CAFG lie outside the triangle. What is the area of hexagon…
AMC 10 2014A #14 grade 8+ geometry-2d
The y-intercepts, P and Q, of two perpendicular lines intersecting at the point A(6,8) have a sum of zero. What is the a…
AMC 10 2014A #15 grade 7+ rate-ratio
David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he…
AMC 10 2014A #16 grade 8+ geometry-2d
In rectangle ABCD, AB=1, BC=2, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midp…
AMC 10 2014A #17 grade 7+ probability
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value…
AMC 10 2014A #18 grade 8+ geometry-2d
A square in the coordinate plane has vertices whose y-coordinates are 0, 1, 4, and 5. What is the area of the square?
AMC 10 2014A #19 grade 8+ geometry-3d
Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XY contained in t…
AMC 10 2014A #20 grade 6+ number-theory
The product (8)(888…8), where the second factor has k digits, is an integer whose digits have a sum of 1000. What is k?
AMC 10 2014A #21 grade 8+ arithmetic
Positive integers a and b are such that the graphs of y=ax+5 and y=3x+b intersect the x-axis at the same point. What is…
AMC 10 2014A #22 grade 8+ geometry-2d
In rectangle ABCD, AB=20 and BC=10. Let E be a point on CD such that ∠ CBE=15°. What is AE?
AMC 10 2014A #23 grade 8+ geometry-2d
A rectangular piece of paper whose length is √3 times the width has area A. The paper is divided into three equal sectio…
AMC 10 2014A #24 grade 6+ arithmetic
A sequence of natural numbers is constructed by listing the first 4, then skipping one, listing the next 5, skipping 2,…
AMC 10 2014A #25 grade 8+ number-theory, counting
The number 5⁸⁶⁷ is between 2²⁰¹³ and 2²⁰¹⁴. How many pairs of integers (m,n) are there such that 1≤ m≤ 2012 and 5ⁿ<2^m<2…

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