Competition · AMC preparation · step 4 of 4

AMC 12 2021A: 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 2021A #1 grade 6+ arithmetic
AMC 12 2021A #2 grade 9+ algebra
Under what conditions is √(a²+b²)=a+b true, where a and b are real numbers?
AMC 12 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 12 2021A #4 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 12 2021A #5 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 12 2021A #6 grade 7+ probability
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is 1/3. When 4 b…
AMC 12 2021A #7 grade 8+ algebra
What is the least possible value of (xy-1)²+(x+y)² for real numbers x and y?
AMC 12 2021A #8 grade 4+ pattern
A sequence of numbers is defined by D₀=0,D₁=0,D₂=1 and D_n=D_(n-1)+D_(n-3) for n≥ 3. What are the parities (evenness or…
AMC 12 2021A #9 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 12 2021A #10 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 12 2021A #11 grade 8+ geometry-2d
A laser is placed at the point (3,5). The laser beam travels in a straight line. Larry wants the beam to hit and bounce…
AMC 12 2021A #12 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 12 2021A #13 grade 12+ algebra
Of the following complex numbers z, which one has the property that z⁵ has the greatest real part?
AMC 12 2021A #14 grade 11+ algebra
What is the value of (sum_(k=1)²⁰ log_(5^k) 3^k²)·(sum_(k=1)¹⁰⁰ log_(9^k) 25^k)?
AMC 12 2021A #15 grade 11+ counting
A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the…
AMC 12 2021A #16 grade 8+ arithmetic
In the following list of numbers, the integer n appears n times in the list for 1 ≤ n ≤ 200.1, 2, 2, 3, 3, 3, 4, 4, 4, 4…
AMC 12 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 12 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 12 2021A #19 grade 11+ algebra
How many solutions does the equation sin ( π/2 cos x)=cos ( π/2 sin x) have in the closed interval [0,π]?
AMC 12 2021A #20 grade 10+ geometry-2d
Suppose that on a parabola with vertex V and a focus F there exists a point A such that AF=20 and AV=21. What is the sum…
AMC 12 2021A #21 grade 11+ geometry-2d
The five solutions to the equation(z-1)(z²+2z+4)(z²+4z+6)=0 may be written in the form x_k+y_ki for 1≤ k≤ 5, where x_k a…
AMC 12 2021A #22 grade 11+ algebra
Suppose that the roots of the polynomial P(x)=x³+ax²+bx+c are cos 2π/7,cos 4π/7, and cos 6π/7, where angles are in radia…
AMC 12 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 12 2021A #24 grade 10+ geometry-2d
Semicircle Gamma has diameter AB of length 14. Circle Ω lies tangent to AB at a point P and intersects Gamma at points Q…
AMC 12 2021A #25 grade 11+ number-theory
Let d(n) denote the number of positive integers that divide n, including 1 and n. For example, d(1)=1,d(2)=2, and d(12)=…

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