Competition · AMC preparation · step 4 of 4

AMC 10 2021A: all 24 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2021A #1 grade 6+ arithmetic
What is the value of (2²-2)-(3²-3)+(4²-4)
AMC 10 2021A #2 grade 4+ arithmetic
Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 stude…
AMC 10 2021A #3 grade 6+ number-theory
The sum of two natural numbers is 17,402. One of the two numbers is divisible by 10. If the units digit of that number i…
AMC 10 2021A #4 grade 6+ rate-ratio
A cart rolls down a hill, travelling 5 inches the first second and accelerating so that during each successive 1-second…
AMC 10 2021A #5 grade 6+ arithmetic
The quiz scores of a class with k > 12 students have a mean of 8. The mean of a collection of 12 of these quiz scores is…
AMC 10 2021A #6 grade 6+ rate-ratio
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. C…
AMC 10 2021A #7 grade 5+ logic
Tom has a collection of 13 snakes, 4 of which are purple and 5 of which are happy. He observes that all of his happy sna…
AMC 10 2021A #8 grade 8+ arithmetic
When a student multiplied the number 66 by the repeating decimal, 1.a b a b…=1.a b, where a and b are digits, he did not…
AMC 10 2021A #9 grade 8+ algebra
What is the least possible value of (xy-1)²+(x+y)² for real numbers x and y?
AMC 10 2021A #10 grade 8+ algebra
Which of the following is equivalent to (2+3)(2²+3²)(2⁴+3⁴)(2⁸+3⁸)(2¹⁶+3¹⁶)(2³²+3³²)(2⁶⁴+3⁶⁴)?
AMC 10 2021A #11 grade 6+ number-theory
For which of the following integers b is the base-b number 2021_b - 221_b not divisible by 3?
AMC 10 2021A #12 grade 8+ geometry-3d
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The r…
AMC 10 2021A #13 grade 8+ geometry-3d
What is the volume of tetrahedron ABCD with edge lengths AB = 2, AC = 3, AD = 4, BC = √13, BD = 2√5, and CD = 5 ?
AMC 10 2021A #14 grade 8+ algebra
All the roots of the polynomial z⁶-10z⁵+Az⁴+Bz³+Cz²+Dz+16 are positive integers, possibly repeated. What is the value of…
AMC 10 2021A #15 grade 8+ counting
Values for A,B,C, and D are to be selected from {1, 2, 3, 4, 5, 6} without replacement (i.e. no two letters have the sam…
AMC 10 2021A #17 grade 8+ geometry-2d
Trapezoid ABCD has AB∥CD,BC=CD=43, and AD⊥BD. Let O be the intersection of the diagonals AC and BD, and let P be the mid…
AMC 10 2021A #18 grade 7+ number-theory
Let f be a function defined on the set of positive rational numbers with the property that f(a· b)=f(a)+f(b) for all pos…
AMC 10 2021A #19 grade 8+ geometry-2d
The area of the region bounded by the graph ofx²+y² = 3|x-y| + 3|x+y|is m+nπ, where m and n are integers. What is m + n?
AMC 10 2021A #20 grade 6+ counting
In how many ways can the sequence 1,2,3,4,5 be rearranged so that no three consecutive terms are increasing and no three…
AMC 10 2021A #21 grade 8+ geometry-2d
Let ABCDEF be an equiangular hexagon. The lines AB, CD, and EF determine a triangle with area 192√3, and the lines BC, D…
AMC 10 2021A #22 grade 6+ arithmetic
Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2, t…
AMC 10 2021A #23 grade 7+ probability
Frieda the frog begins a sequence of hops on a 3 × 3 grid of squares, moving one square on each hop and choosing at rand…
AMC 10 2021A #24 grade 8+ geometry-2d
The interior of a quadrilateral is bounded by the graphs of (x+ay)² = 4a² and (ax-y)² = a², where a is a positive real n…
AMC 10 2021A #25 grade 5+ counting
How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable…

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