Competition · AMC preparation · step 4 of 4

AMC 10 2020B: 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 2020B #1 grade 7+ arithmetic
What is the value of 1-(-2)-3-(-4)-5-(-6)?
AMC 10 2020B #2 grade 5+ geometry-3d
Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of…
AMC 10 2020B #3 grade 6+ rate-ratio
The ratio of w to x is 4:3, the ratio of y to z is 3:2, and the ratio of z to x is 1:6. What is the ratio of w to y?
AMC 10 2020B #4 grade 8+ geometry-2d
The acute angles of a right triangle are a° and b°, where a>b and both a and b are prime numbers. What is the least poss…
AMC 10 2020B #5 grade 7+ arithmetic
How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a r…
AMC 10 2020B #6 grade 4+ rate-ratio
Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the…
AMC 10 2020B #7 grade 5+ arithmetic
How many positive even multiples of 3 less than 2020 are perfect squares?
AMC 10 2020B #8 grade 8+ geometry-2d
Points P and Q lie in a plane with PQ=8. How many locations for point R in this plane are there such that the triangle w…
AMC 10 2020B #9 grade 6+ algebra
How many ordered pairs of integers (x, y) satisfy the equation x²⁰²⁰+y²=2y?
AMC 10 2020B #10 grade 8+ geometry-3d
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral su…
AMC 10 2020B #11 grade 7+ probability
Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this li…
AMC 10 2020B #12 grade 8+ arithmetic
The decimal representation of 1/20²⁰ consists of a string of zeros after the decimal point, followed by a 9 and then sev…
AMC 10 2020B #13 grade 6+ rate-ratio
Andy the Ant lives on a coordinate plane and is currently at (-20, 20) facing east (that is, in the positive x-direction…
AMC 10 2020B #14 grade 7+ geometry-2d
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the di…
AMC 10 2020B #15 grade 6+ arithmetic
Steve wrote the digits 1, 2, 3, 4, and 5 in order repeatedly from left to right, forming a list of 10,000 digits, beginn…
AMC 10 2020B #16 grade 5+ logic
Bela and Jenn play the following game on the closed interval [0, n] of the real number line, where n is a fixed integer…
AMC 10 2020B #17 grade 4+ counting
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 pe…
AMC 10 2020B #18 grade 7+ probability
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the follo…
AMC 10 2020B #19 grade 6+ arithmetic
In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (u…
AMC 10 2020B #20 grade 8+ geometry-3d
Let B be a right rectangular prism (box) with edges lengths 1, 3, and 4, together with its interior. For real r≥0, let S…
AMC 10 2020B #21 grade 8+ geometry-2d
In square ABCD, points E and H lie on AB and DA, respectively, so that AE=AH. Points F and G lie on BC and CD, respectiv…
AMC 10 2020B #22 grade 8+ number-theory
What is the remainder when 2²⁰² +202 is divided by 2¹⁰¹+2⁵¹+1?
AMC 10 2020B #23 grade 8+ geometry-2d
Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the foll…
AMC 10 2020B #24 grade 8+ number-theory
How many positive integers n satisfy (n+1000)/70 = ⌊ √n ⌋?(Recall that ⌊ x⌋ is the greatest integer not exceeding x.)
AMC 10 2020B #25 grade 7+ counting
Let D(n) denote the number of ways of writing the positive integer n as a productn = f₁· f₂… f_k,where k≥1, the f_i are…

A parent dashboard for the family lives at sensimlab.com.