The AMC-8 is the most competitive math contest for middle school students. It tests students’ critical thinking, mathematical imagination, and problem-solving abilities with challenging problems. Here in this blog, we take a look at the 20 toughest problems in AMC-8 history, dissecting them with solutions, strategies, and tips to master them.
Top 20 Hardest AMC-8 Math Problems
Now, let’s dive into some of the toughest AMC-8 Math questions.
1. Sophia explores all possible ways to traverse a 6 × 6 hexagonal grid, moving from the bottom row to the top row. Each move must be either one unit northeast or one unit northwest, ensuring she never moves downward. The total number of such distinct paths is P.
For each path, she calculates the total enclosed area between the path and the rightmost boundary of the grid. Let S be the sum of these areas over all paths.
What is the value of S in terms of P?
A) \frac{\Large 5}{\Large 12}P
B) \frac{\Large 1}{\Large 3}P
C) \frac{\Large 7}{\Large 18}P
D) \frac{\Large 2}{\Large 5}P.
E) \frac{\Large 2}{\Large 5}P.
B) \frac{\Large 1}{\Large 3}P
C) \frac{\Large 7}{\Large 18}P
D) \frac{\Large 2}{\Large 5}P.
E) \frac{\Large 2}{\Large 5}P.
2. Jean has designed another piece of stained glass art with a mountain theme, as shown in the figure below. The peaks of the two mountains reach heights of 8 feet and 12 feet. Each peak forms a 90^\circ angle, and the sloped sides make a 45^\circ angle with the ground. The total width of the artwork is 30 feet. The bases of the mountains are aligned along the same horizontal ground level.
A small triangular piece at the bottom, formed by the intersection of the two mountain slopes, has been cut out to create a window. The area of this cut-out triangle is 9 square feet. How far above the ground is the bottom vertex of this cut-out triangle?
A small triangular piece at the bottom, formed by the intersection of the two mountain slopes, has been cut out to create a window. The area of this cut-out triangle is 9 square feet. How far above the ground is the bottom vertex of this cut-out triangle?
A) 2
B) 3\sqrt{2}
C) 3
D) 4
E) 4\sqrt{2}
B) 3\sqrt{2}
C) 3
D) 4
E) 4\sqrt{2}
3. A basketball league consists of two five-team divisions. Each team plays every other team in its division N times. Each team plays every team in the other division M times, where N>2M and M>4. Each team plays a 90-game schedule. How many games does a team play within its own division?
A) 48
B) 54
C) 60
D) 66
E) 72
B) 54
C) 60
D) 66
E) 72
4. A small theater has 5 rows of seats with 4 seats in each row. 10 audience members have already taken their seats, randomly distributed among all seats. A pair of friends arrives and wants to sit next to each other in the same row. What is the probability that there will be 2 adjacent empty seats in the same row for them to sit together?
A) \frac{\Large 5}{\Large 11}
B) \frac{\Large 9}{\Large 17}
C) \frac{\Large 12}{\Large 19}
D) \frac{\Large 14}{\Large 23}.
E) \frac{\Large 15}{\Large 26}.
B) \frac{\Large 9}{\Large 17}
C) \frac{\Large 12}{\Large 19}
D) \frac{\Large 14}{\Large 23}.
E) \frac{\Large 15}{\Large 26}.
5. Isosceles \triangle ABC has equal side lengths AB and BC. In the figure below, segments are drawn parallel to \overline{AC} so that the shaded portions of \triangle ABC have the same area. The heights of the two unshaded portions are 9 and 4 units, respectively. What is the height of h of \triangle ABC? (Diagram not drawn to scale.)
A) 12.6
B) 12.8
C) 13
D) 13.2
E) 13.4
B) 12.8
C) 13
D) 13.2
E) 13.4
6. Fourteen integers b1,b2,b3,…,b14 are arranged in order on a number line. The integers are equally spaced and have the property that 6≤b≤16, 22≤b≤32, 192≤b≤202.
What is the sum of the digits of b13?
A) 12
B) 13
C) 14
D) 15
E) 16
B) 13
C) 14
D) 15
E) 16
7. The figure below shows a polygon ABCDEFGH, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that AH=EF=8 and GH=14. What is the volume of the prism?
A) 112
B) 128
C) 192
D) 240
E) 288
B) 128
C) 192
D) 240
E) 288
8. Rectangles R1 and R2 and squares S1, S2 and S3 shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of in units?
A) 112
B) 128
C) 192
D) 240
E) 288
B) 128
C) 192
D) 240
E) 288
9. A large square region is paved with n^2 gray square tiles, each measuring s inches on a side. A border d inches wide surrounds each tile. The figure below shows the case for n=3 . When n=24, the 576 gray tiles cover 64\% of the area of the large square region. What is the ratio \frac{\Large d}{\Large s} for this larger value of n?
A) \frac{\Large 6}{\Large 25}
B) \frac{\Large 1}{\Large 4}
C) \frac{\Large 9}{\Large 25}
D) \frac{\Large 7}{\Large 16}.
E) \frac{\Large 9}{\Large 16}.
B) \frac{\Large 1}{\Large 4}
C) \frac{\Large 9}{\Large 25}
D) \frac{\Large 7}{\Large 16}.
E) \frac{\Large 9}{\Large 16}.
10. In trapezoid PQRS, angles Q and R measure 45^\circ, and PQ=SR. The side lengths are all positive integers, and the perimeter of PQRS is 24 units. How many non-congruent trapezoids satisfy all of these conditions?
A) 1
B) 2
C) 4
D) 5
E) 6
B) 2
C) 4
D) 5
E) 6
Want to know the Answers? Find it here
11. How many perfect fourth powers lie between 3^{6}+ 5 and 3^{14}+ 5, inclusive?
A) 6
B) 7
C) 8
D) 9
E) 11
B) 7
C) 8
D) 9
E) 11
12. Sophia has 40 apples. In how many ways can she share them with Becky, Chris, David, and Emma so that each of the five people gets at least three apples?
A) 19635
B) 21450
C) 22950
D) 24880
E) 26460
B) 21450
C) 22950
D) 24880
E) 26460
13. After Euclid High School's championship basketball game, it was determined that \frac{\large 2}{\large 9} of the team's total points were scored by Alexa, and \frac{\large 3}{\large 10} were scored by Brittany. Chelsea scored 18 points. None of the other 8 team members scored more than 3 points each. What was the total number of points scored by the other 8 team members?
A) 14
B) 16
C) 18
D) 24
E) 22
B) 16
C) 18
D) 24
E) 22
14. From a regular octagon, a quadrilateral is formed by connecting four randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the quadrilateral is also a side of the octagon?
A) \frac{\Large 1}{\Large 3}
B) \frac{\Large 5}{\Large 12}
C) \frac{\Large 2}{\Large 3}
D) \frac{\Large 7}{\Large 12}.
E) \frac{\Large 13}{\Large 14}.
B) \frac{\Large 5}{\Large 12}
C) \frac{\Large 2}{\Large 3}
D) \frac{\Large 7}{\Large 12}.
E) \frac{\Large 13}{\Large 14}.
15. In the figure shown, \overline{US} and \overline{UT} are line segments each of length 3, and \angle TUS = 45^\circ. Arcs \overset{\frown}{TR} and \overset{\frown}{SR} are each one-fourth of a circle with radius 3. What is the area of the region shown?
A) 6\sqrt{2} - \frac{\Large 3\pi}{\Large 2}
B) 9 - \frac{\Large 3\pi}{\Large 2}
C) 3\sqrt{2}
D) 6\sqrt{2} - \frac{\Large 3\pi}{\Large 4}.
E) 6 + \frac{\Large 3\pi}{\Large 2}.
B) 9 - \frac{\Large 3\pi}{\Large 2}
C) 3\sqrt{2}
D) 6\sqrt{2} - \frac{\Large 3\pi}{\Large 4}.
E) 6 + \frac{\Large 3\pi}{\Large 2}.
16. Mrs. Johnson has four grandchildren, who call her regularly. One calls her every 3 days, one calls her every 4 days, one calls her every 5 days, and one calls her every 6 days. All four called her on January 1, 2024. On how many days during the year 2024 did she not receive a phone call from any of her grandchildren?
A) 120
B) 127
C) 146
D) 152
E) 160
B) 127
C) 146
D) 152
E) 160
17. Each day for five days, Linda traveled for 90 minutes at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 4 minutes over the preceding day. Each of the five days, her distance traveled was also an integer number of miles. What was the total number of miles for the five trips?
A) 18
B) 22
C) 30
D) 45
E) 54
B) 22
C) 30
D) 45
E) 54
18. A semicircle is inscribed in an isosceles triangle with base 18 and height 20 so that the diameter of the semicircle is contained in the base of the triangle, as shown. What is the radius of the semicircle?
A) \frac{\Large 180}{\Large 29}
B) 5\sqrt{3}
C) \frac{\Large 19\sqrt{2}}{\Large 2}
D) 12.
E) \frac{\Large 19\sqrt{3}}{\Large 2}.
B) 5\sqrt{3}
C) \frac{\Large 19\sqrt{2}}{\Large 2}
D) 12.
E) \frac{\Large 19\sqrt{3}}{\Large 2}.
19. The digits 1,2,3,4,5,6 are each used once to write a six-digit number ABCDEF. The four-digit number ABCD is divisible by 4, the four-digit number BCDE is divisible by 5, and the four-digit number CDEF is divisible by 3. What is A?
A) 1
B) 2
C) 3
D) 4
E) 5
B) 2
C) 3
D) 4
E) 5
20. One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
A) 9
B) \frac{\Large 25}{\Large 2}
C) 15
D) \frac{\Large 31}{\Large 2}.
E) 17.
B) \frac{\Large 25}{\Large 2}
C) 15
D) \frac{\Large 31}{\Large 2}.
E) 17.
- B)
- B)
- C)
- C)
- C)
- E)
- C)
- (A
- (A
- (A
- (A
- (C
- (D
- (E
- (D
- (B
- (D
- (A
- (A
- (B
Strategies to Approach Challenging AMC-8 Problems
Tackling these tough problems requires more than just knowing formulas. Here are some tips to help you succeed:
- Understand the question thoroughly: Read carefully, and identify what is being asked before diving into the math.
- Eliminate obvious wrong answers: Use the process of elimination to make educated guesses, especially when time is limited.
- Practice, practice, practice: The more problems you solve, the more familiar you will become with the patterns in SAT Math questions.
How to Practice and Improve Your SAT Math Skills
Here are some important strategies to allow you to work through these challenging problems:
- Break Down the Problem
Identify what is asked and break it up into little chunks. Write down information you know and try to see patterns.
- Make Use of the Answer Choices
If possible, work backwards from the multiple-choice answers given. Plugging in values may be faster than working algebraically in some cases.
- Utilize Logical Reasoning
Certain problems can be solved through reasoning by elimination instead of higher math.
- Think Visually for Geometry Problems
With geometry, a diagram or using symmetry will make solutions more apparent.
- Practice with Timed Drills
Since AMC-8 is a timed test, practice under time constraints to build speed and accuracy.
Final Thoughts
To become proficient in the most challenging AMC-8 problems, you must spend time, strategic reasoning, and practice. By focusing on understanding concepts thoroughly and developing problem-solving skills, you can boost your confidence level and performance in the competition. Practice and stay curious!
Have a Question?
If you have any specific problems you’d like help with, comment below, and we’ll be happy to discuss solutions!
