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AMC8 2027 - Self-practice Set 9

Question 1

In the illustrated grid of 3 columns and 4 rows, how many placements of the given L tetromino are possible using rotations but no reflections?

Question illustration
  1. 15
  2. 14
  3. 21
  4. 25
  5. 26

Question 2

Three bags contain 26,28,30 beans and 50%,25%,20% of each bag are yellow. All beans are mixed. Approximately what percentage are yellow?
  1. 31%
  2. 32%
  3. 33%
  4. 35%
  5. 95%

Question 3

How many integers from 1000 through 2000 are divisible by 15,20,and 25?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 4

One hundred students attended a summer basketball camp: $52$ boys and $48$ girls. Forty students came from Jones Middle School and sixty from Clay Middle School. Twenty girls came from Jones. How many boys came from Clay?
  1. 20
  2. 32
  3. 40
  4. 48
  5. 52

Question 5

How many integers from 99 through 999 contain exactly one digit 0?
  1. 72
  2. 90
  3. 144
  4. 162
  5. 180

Question 6

Pour out half the water, then one third of the remainder, then one quarter, then one fifth, and so on. After how many pours does one tenth of the original water remain?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10

Question 7

Findp/(3 q+1), wherep andq are primes and 3 p+5 q=31.
  1. $\frac 18$ and 1
  2. $\frac 17$ and 2
  3. $\frac 15$ and 11
  4. $\frac 1{13}$ and 17
  5. 5

Question 8

The six children listed below belong to two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Every pair of siblings shares at least one characteristic. Who are Jim's siblings?

| Name | Eye color | Hair color |
|---|---|---|
| Benjamin | Blue | Black |
| Jim | Brown | Blond |
| Nadeen | Brown | Black |
| Austin | Blue | Blond |
| Tevyn | Blue | Black |
| Sue | Blue | Blond |
  1. Nadeen, Austin
  2. Benjamin, Sue
  3. Benjamin, Austin
  4. Nadeen, Tevyn
  5. Austin, Sue

Question 9

Place at most one X in each cell of a $3\times3$ grid, with no complete row, column, or main diagonal of Xs. What is the maximum number of Xs?

Question illustration
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 10

In fall 1996,800 students join the school cleanup. Participation is expected to increase 50% in each of 1997,1998,1999. How many participants are expected in 1999?
  1. 1200
  2. 1500
  3. 2000
  4. 2400
  5. 2700

Question 11

Alice can pack all her soda cans into packs of 6,8,or 14 without leftovers. What is the least possible positive number of cans?

Question illustration
  1. 28
  2. 168
  3. 98
  4. 48
  5. 336

Question 12

A bag contains $1995$ black balls and $2000$ white balls. I remove two balls. If they have the same color, I put one white ball into the bag; if their colors differ, I put one black ball into the bag. I repeat until one ball remains. What color is it?
  1. The last two balls are black
  2. White
  3. Black
  4. One ball cannot remain
  5. Not enough information

Question 13

Georgie leaves a haunted house through one of 6 windows and returns through a different window. How many ordered choices are possible?
  1. 15
  2. 20
  3. 30
  4. 35
  5. 36

Question 14

Jenny has an important exam on December 31 this year. Today is August 5. What is the greatest number of days she has left to study?
  1. 365
  2. 200
  3. 150
  4. 148
  5. 120

Question 15

Gears A,B,C have 15,18,24 teeth respectively. How many complete turns must C make before all gears return to their shown orientations?

Question illustration
  1. 360
  2. 40
  3. 15
  4. 24
  5. 20

Question 16

A drawer contains red, white, and blue socks, with at least five of each color. A person takes some socks without looking. What is the minimum number that guarantees five socks of one color?
  1. 6
  2. 9
  3. 12
  4. 13
  5. 15

Question 17

Spin both fair spinners shown. What is the probability that their sum is even?

Question illustration
  1. $\frac{1}{6}$
  2. $\frac{3}{7}$
  3. $\frac{1}{2}$
  4. $\frac{2}{3}$
  5. $\frac{5}{7}$

Question 18

What is the remainder of 1999^2000 modulo 5?
  1. 4
  2. 3
  3. 2
  4. 1
  5. 0

Question 19

Several students compete in three races. In each race, first place earns $5$ points, second earns $3$, and third earns $1$. There are no ties. What is the smallest score after three races that guarantees no other student has a higher score?
  1. 9
  2. 10
  3. 11
  4. 13
  5. 15

Question 20

Five married couples attend a party. Each husband shakes hands with the wives of the other four husbands; no other handshakes occur. How many handshakes are there?
  1. 18
  2. 20
  3. 21
  4. 24
  5. 25