Competition · AMC preparation · step 4 of 4

AMC 12 2003A: 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 2003A #1 grade 3+ counting
What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd countin…
AMC 12 2003A #2 grade 5+ arithmetic
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than…
AMC 12 2003A #3 grade 6+ geometry-3d
A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this…
AMC 12 2003A #4 grade 6+ rate-ratio
It takes Anna 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from scho…
AMC 12 2003A #5 grade 6+ arithmetic
The sum of the two 5-digit numbers AMC10 and AMC12 is 123422. What is A+M+C?
AMC 12 2003A #6 grade 6+ arithmetic
Define x ♡ y to be |x-y| for all real numbers x and y. Which of the following statements is not true? (A) x ♡ y = y ♡ x…
AMC 12 2003A #7 grade 7+ geometry-2d
How many non-congruent triangles with perimeter 7 have integer side lengths?
AMC 12 2003A #8 grade 7+ probability
What is the probability that a randomly drawn positive factor of 60 is less than 7?
AMC 12 2003A #9 grade 8+ geometry-2d
A set S of points in the xy-plane is symmetric about the origin, both coordinate axes, and the line y=x. If (2,3) is in…
AMC 12 2003A #10 grade 6+ rate-ratio
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a rati…
AMC 12 2003A #11 grade 8+ geometry-2d
A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the sq…
AMC 12 2003A #12 grade 4+ arithmetic
Sally has five red cards numbered 1 through 5 and four blue cards numbered 3 through 6. She stacks the cards so that the…
AMC 12 2003A #13 grade 6+ geometry-3d
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more cong…
AMC 12 2003A #14 grade 8+ geometry-2d
Points K, L, M, and N lie in the plane of the square ABCD such that AKB, BLC, CMD, and DNA are equilateral triangles. If…
AMC 12 2003A #15 grade 8+ geometry-2d
A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller s…
AMC 12 2003A #16 grade 8+ geometry-2d
A point P is chosen at random in the interior of equilateral triangle ABC. What is the probability that △ ABP has a grea…
AMC 12 2003A #17 grade 8+ geometry-2d
Square ABCD has sides of length 4, and M is the midpoint of CD. A circle with radius 2 and center M intersects a circle…
AMC 12 2003A #18 grade 6+ number-theory
Let n be a 5-digit number, and let q and r be the quotient and the remainder, respectively, when n is divided by 100. Fo…
AMC 12 2003A #19 grade 9+ algebra
A parabola with equation y=ax²+bx+c is reflected about the x-axis. The parabola and its reflection are translated horizo…
AMC 12 2003A #20 grade 11+ counting
How many 15-letter arrangements of 5 A's, 5 B's, and 5 C's have no A's in the first 5 letters, no B's in the next 5 lett…
AMC 12 2003A #21 grade 11+ algebra
The graph of the polynomial P(x) = x⁵ + ax⁴ + bx³ + cx² + dx + e has five distinct x-intercepts, one of which is at (0,0…
AMC 12 2003A #22 grade 7+ probability
Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts…
AMC 12 2003A #23 grade 8+ number-theory
How many perfect squares are divisors of the product 1! · 2! · 3! · hdots · 9!?
AMC 12 2003A #24 grade 11+ algebra
If a≥ b > 1, what is the largest possible value of log_a(a/b) + log_b(b/a)?
AMC 12 2003A #25 grade 9+ algebra
Let f(x)= √(ax²+bx). For how many real values of a is there at least one positive value of b for which the domain of f a…

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