1975 words - 8 pages

2018 Mock AMC 10

mathchampion1, kcbhatraju, kootrapali, chocolatelover111

November 2018

Special thanks to scrabbler94 for proofreading, ktong for testsolving, kcbha-

traju for compiling Asymptote, and mathchampion1 for compiling LATEX for

the Overleaf document.

1 Introduction

The same rules as any standard AMC 10 apply. As usual, there are 25 problems

with 75 total minutes to solve them, questions approximately ordered in terms

of difficulty.

Important. Problems in this test may contain the use of “power towers”, i.e.

exponents stacked on top of one another, the lowest of which is stacked on the

base, to form a large tower, such as ab

c

. By exponential convention, power

towers are “collapsed” from top to bottom, and in our example, ab

c

= a(b

c).

We greatly enjoyed developing this test, so hopefully you enjoy solving the

problems as well. The test begins on the next page. Good luck.

1

2 Problems

1. What is the value of (1! · 2! · 3! · 4!)

(

1!

2! · 3! +

2!

3! · 4!

)

?

(A) 4 (B) 8 (C) 14 (D) 28 (E) 56

2. Provided that it is positive, what is the difference between the square and

the cube of the answer to this problem?

(A)

√

5

2

(B)

1 +

√

5

2

(C) 2 (D)

2 +

√

5

2

(E)

3 +

√

5

2

3. A cyclist bikes 10 miles at a speed of 5 mph, starting from his home. He

then stops for 1 hour to take rest, resuming at twice the speed - a staggering

10 mph. He eventually starts his journey back home, impressively maintaining

this exact speed of 10 mph (nonstop) the entire time until he gets home. If the

entire trip took him 10 hours, then how many miles did the cyclist travel in total?

(A) 65 (B) 70 (C) 75 (D) 80 (E) 85

4. Silvia has 16 plots, arranged in a 4 × 4 square. She wishes to plant ei-

ther a poppy or a toyon in each plot, such that every row and column contains

exactly three poppies. In how many ways can this be done? Two ways that

differ only by rotation or reflection are considered distinct.

(A) 3 (B) 6 (C) 12 (D) 24 (E) 48

5. Six regular hexagons that each share exactly one distinct side of a seventh,

central regular hexagon are placed to form an octadecagon (18 sides). The adja-

cent centers of the surrounding six hexagons are then connected to form another

hexagon. What is the ratio of the area of this new hexagon to the area of the

octadecagon?

(A)

1

7

(B)

2

7

(C)

3

7

(D)

4

7

(E)

5

7

2

6. In a classroom of n students, the teacher forms a committee of two chairmen

at random, and then chooses 1 out of 7 possible elected class presidents. In

another classroom, this time with n − 3 students, the teacher needs to choose

three chairmen, and then choose 1 out of 9 possible class presidents. If the total

number of ways to choose the chairmen and class president for both classes is

the same, then what is the total number of ways for either class? Assume that

chairmen can also be presidents.

(A) 35 (B) 72 (C) 121 (D) 315 (E) 420

7. When the sum

(1 · 2 · 3) + (2 · 3 · 4) + (3 · 4 · 5) + · · ·+ (2018 · 2019 · 2020)

is evaluated, what is the units digit of the...

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