{"id":35189,"date":"2025-03-13T15:27:18","date_gmt":"2025-03-13T15:27:18","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=35189"},"modified":"2025-03-15T16:18:43","modified_gmt":"2025-03-15T16:18:43","slug":"hardest-amc-problems","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/hardest-amc-problems\/","title":{"rendered":"Hardest Problems in AMC-8 Math Test"},"content":{"rendered":"\t\t
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\n\t\t\t\tThe 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.\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Top 20 Hardest AMC-8 Math Problems<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Now, let’s dive into some of the toughest<\/span> AMC-8 Math questions.\u00a0<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\n\t\t\t 1. Sophia explores all possible ways to traverse a \\(6 \u00d7 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\\).\r\nFor 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.\r\nWhat is the value of \\(S\\) in terms of \\(P\\)? <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\r\n A) \\(\\frac{\\Large 5}{\\Large 12}P\\)

\r\n B) \\(\\frac{\\Large 1}{\\Large 3}P\\)

\r\n C) \\(\\frac{\\Large 7}{\\Large 18}P\\)

\r\n D) \\(\\frac{\\Large 2}{\\Large 5}P\\).

\r\n E) \\(\\frac{\\Large 2}{\\Large 5}P\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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.
\r\nA 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? <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(2\\)<\/span>
\r\nB) \\(3\\sqrt{2}\\)<\/span>
\r\nC) \\(3\\)<\/span>
\r\nD) \\(4\\)<\/span>
\r\nE) \\(4\\sqrt{2}\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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? <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(48\\)<\/span>
\r\nB) \\(54\\)<\/span>
\r\nC) \\(60\\)<\/span>
\r\nD) \\(66\\)<\/span>
\r\nE) \\(72\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n<\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\r\n A) \\(\\frac{\\Large 5}{\\Large 11}\\)

\r\n B) \\(\\frac{\\Large 9}{\\Large 17}\\)

\r\n C) \\(\\frac{\\Large 12}{\\Large 19}\\)

\r\n D) \\(\\frac{\\Large 14}{\\Large 23}\\).

\r\n E) \\(\\frac{\\Large 15}{\\Large 26}\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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.)\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(12.6\r\n\\)<\/span>
\r\nB) \\(12.8\\)<\/span>
\r\nC) \\(13\\)<\/span>
\r\nD) \\(13.2\\)<\/span>
\r\nE) \\(13.4\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 6. Fourteen integers \\(b1,b2,b3,\u2026,b14\\) are arranged in order on a number line. The integers are equally spaced and have the property that \\(6\u2264b\u226416\\), \\(22\u2264b\u226432\\), \\(192\u2264b\u2264202\\).\r\nWhat is the sum of the digits of \\(b13\\)?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(12\\)<\/span>
\r\nB) \\(13\\)<\/span>
\r\nC) \\(14\\)<\/span>
\r\nD) \\(15\\)<\/span>
\r\nE) \\(16\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(112\r\n\\)<\/span>
\r\nB) \\(128\\)<\/span>
\r\nC) \\(192\\)<\/span>
\r\nD) \\(240\\)<\/span>
\r\nE) \\(288\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n\r\n\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(112\r\n\\)<\/span>
\r\nB) \\(128\\)<\/span>
\r\nC) \\(192\\)<\/span>
\r\nD) \\(240\\)<\/span>
\r\nE) \\(288\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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\\)?\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\r\n A) \\(\\frac{\\Large 6}{\\Large 25}\\)

\r\n B) \\(\\frac{\\Large 1}{\\Large 4}\\)

\r\n C) \\(\\frac{\\Large 9}{\\Large 25}\\)

\r\n D) \\(\\frac{\\Large 7}{\\Large 16}\\).

\r\n E) \\(\\frac{\\Large 9}{\\Large 16}\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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? <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(1\\)<\/span>
\r\nB) \\(2\\)<\/span>
\r\nC) \\(4\\)<\/span>
\r\nD) \\(5\\)<\/span>
\r\nE) \\(6\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Want to know the Answers? Find it here<\/a><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 11. How many perfect fourth powers lie between \\(3^{6}+ 5\\) and \\(3^{14}+ 5\\), inclusive?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(6\\)<\/span>
\r\nB) \\(7\\)<\/span>
\r\nC) \\(8\\)<\/span>
\r\nD) \\(9\\)<\/span>
\r\nE) \\(11\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(19635\\)<\/span>
\r\nB) \\(21450\\)<\/span>
\r\nC) \\(22950\\)<\/span>
\r\nD) \\(24880\\)<\/span>
\r\nE) \\(26460\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\tA) \\(14\\)<\/span>
\r\nB) \\(16\\)<\/span>
\r\nC) \\(18\\)<\/span>
\r\nD) \\(24\\)<\/span>
\r\nE) \\(22\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\r\n A) \\(\\frac{\\Large 1}{\\Large 3}\\)

\r\n B) \\(\\frac{\\Large 5}{\\Large 12}\\)

\r\n C) \\(\\frac{\\Large 2}{\\Large 3}\\)

\r\n D) \\(\\frac{\\Large 7}{\\Large 12}\\).

\r\n E) \\(\\frac{\\Large 13}{\\Large 14}\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\r\n A) \\(6\\sqrt{2} - \\frac{\\Large 3\\pi}{\\Large 2}\\)
\r\n B) \\(9 - \\frac{\\Large 3\\pi}{\\Large 2}\\)
\r\n C) \\(3\\sqrt{2}\\)
\r\n D) \\(6\\sqrt{2} - \\frac{\\Large 3\\pi}{\\Large 4}\\).
\r\n E) \\(6 + \\frac{\\Large 3\\pi}{\\Large 2}\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\tA) \\(120\\)<\/span>
\r\nB) \\(127\\)<\/span>
\r\nC) \\(146\\)<\/span>
\r\nD) \\(152\\)<\/span>
\r\nE) \\(160\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\tA) \\(18\\)<\/span>
\r\nB) \\(22\\)<\/span>
\r\nC) \\(30\\)<\/span>
\r\nD) \\(45\\)<\/span>
\r\nE) \\(54\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\r\n A) \\(\\frac{\\Large 180}{\\Large 29}\\)
\r\n B) \\(5\\sqrt{3}\\)
\r\n C) \\( \\frac{\\Large 19\\sqrt{2}}{\\Large 2}\\)
\r\n D) \\(12\\).
\r\n E) \\(\\frac{\\Large 19\\sqrt{3}}{\\Large 2}\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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\\)?\r\n <\/b> \r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\tA) \\(1\\)<\/span>
\r\nB) \\(2\\)<\/span>
\r\nC) \\(3\\)<\/span>
\r\nD) \\(4\\)<\/span>
\r\nE) \\(5\\)<\/span>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t 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?\r\n<\/b>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\t\t\t\t\t\t\"Math\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\r\n A) \\(9\\)
\r\n B) \\(\\frac{\\Large 25}{\\Large 2}\\)
\r\n C) \\( 15\\)
\r\n D) \\(\\frac{\\Large 31}{\\Large 2}\\).
\r\n E) \\(17\\).\r\n<\/span>\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t\n\n \n \n Answers<\/title>\n <style>\n .table-container {\n display: none;\n margin-top: 10px;\n border: 1px solid #ddd;\n padding: 10px;\n border-radius: 5px;\n background-color: #f9f9f9;\n }\n .toggle-button {\n cursor: pointer;\n background-color: #007BFF;\n color: white;\n font-size: 22px; \/* Increased font size *\/\n padding: 16px 20px; \/* Increased padding *\/\n border: none;\n border-radius: 5px;\n display: inline-block;\n margin-bottom: 10px;\n }\n .table-container li {\n font-size: 24px; \/* Increased answer text size *\/\n font-weight: bold; \/* Make it bold *\/\n margin-bottom: 5px;\n }\n <\/style>\n<\/head>\n<body>\n <button class=\"toggle-button\" onclick=\"toggleTable()\"><a id=\"1\">Show <b>ANSWERS OF 20 QUESTIONS GIVEN ABOVE<\/b><\/a><\/button>\n <div class=\"table-container\" id=\"mathTable\">\n <ol>\n <li>B)<\/li>\n <li>B)<\/li>\n <li>C)<\/li>\n <li>C)<\/li>\n <li>C)<\/li>\n <li>E)<\/li>\n <li>C)<\/li>\n <li>(A<\/li>\n <li>(A<\/li>\n <li>(A<\/li>\n <li>(A<\/li>\n <li>(C<\/li>\n <li>(D<\/li>\n <li>(E<\/li>\n <li>(D<\/li>\n <li>(B<\/li>\n <li>(D<\/li>\n <li>(A<\/li>\n <li>(A<\/li>\n <li>(B<\/li>\n \n <\/ol>\n <\/div>\n\n <script>\n function toggleTable() {\n var table = document.getElementById(\"mathTable\");\n if (table.style.display === \"none\" || table.style.display === \"\") {\n table.style.display = \"block\";\n } else {\n table.style.display = \"none\";\n }\n }\n <\/script>\n<\/body>\n<\/html>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-7a1db2c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7a1db2c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2b5055f\" data-id=\"2b5055f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-93233c7 elementor-widget elementor-widget-text-editor\" data-id=\"93233c7\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Strategies to Approach Challenging AMC-8 Problems<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c22c3f1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c22c3f1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2afcb3e\" data-id=\"2afcb3e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-98dd66c elementor-widget elementor-widget-text-editor\" data-id=\"98dd66c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Tackling these tough problems requires more than just knowing formulas. Here are some tips to help you succeed:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Understand the question thoroughly:<\/b><span style=\"font-weight: 400;\"> Read carefully, and identify what is being asked before diving into the math.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Eliminate obvious wrong answers:<\/b><span style=\"font-weight: 400;\"> Use the process of elimination to make educated guesses, especially when time is limited.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Practice, practice, practice:<\/b><span style=\"font-weight: 400;\"> The more problems you solve, the more familiar you will become with the patterns in SAT Math questions.<\/span><\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-0f76f4c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0f76f4c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-76c5fca\" data-id=\"76c5fca\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-df1a78b elementor-widget elementor-widget-text-editor\" data-id=\"df1a78b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>How to Practice and Improve Your SAT Math Skills<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-9bf5d4b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9bf5d4b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d720312\" data-id=\"d720312\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a678db2 elementor-widget elementor-widget-text-editor\" data-id=\"a678db2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Here<\/span><span style=\"font-weight: 400;\"> are some <\/span><span style=\"font-weight: 400;\">important<\/span><span style=\"font-weight: 400;\"> strategies to <\/span><span style=\"font-weight: 400;\">allow<\/span><span style=\"font-weight: 400;\"> you to <\/span><span style=\"font-weight: 400;\">work<\/span> <span style=\"font-weight: 400;\">through <\/span><span style=\"font-weight: 400;\">these <\/span><span style=\"font-weight: 400;\">challenging problems:<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-1e0dc1e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1e0dc1e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-69d58cc\" data-id=\"69d58cc\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4f64be5 elementor-widget elementor-widget-text-editor\" data-id=\"4f64be5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<ol><li><span style=\"color: #000080;\"><b> Break Down the Problem<\/b><\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">Identify<\/span><span style=\"font-weight: 400;\"> what is <\/span><span style=\"font-weight: 400;\">asked<\/span><span style=\"font-weight: 400;\"> and <\/span><span style=\"font-weight: 400;\">break<\/span><span style=\"font-weight: 400;\"> it <\/span><span style=\"font-weight: 400;\">up <\/span><span style=\"font-weight: 400;\">into <\/span><span style=\"font-weight: 400;\">little<\/span> <span style=\"font-weight: 400;\">chunks<\/span><span style=\"font-weight: 400;\">. <\/span><span style=\"font-weight: 400;\">Write<\/span><span style=\"font-weight: 400;\"> down information <\/span><span style=\"font-weight: 400;\">you<\/span> <span style=\"font-weight: 400;\">know <\/span><span style=\"font-weight: 400;\">and <\/span><span style=\"font-weight: 400;\">try<\/span><span style=\"font-weight: 400;\"> to <\/span><span style=\"font-weight: 400;\">see<\/span><span style=\"font-weight: 400;\"> patterns.<\/span><\/p><ol start=\"2\"><li><span style=\"color: #000080;\"><b> Make Use of the Answer Choices<\/b><\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">If <\/span><span style=\"font-weight: 400;\">possible<\/span><span style=\"font-weight: 400;\">, work backwards <\/span><span style=\"font-weight: 400;\">from<\/span><span style=\"font-weight: 400;\"> the multiple-choice answers<\/span><span style=\"font-weight: 400;\"> given<\/span><span style=\"font-weight: 400;\">. Plugging in values <\/span><span style=\"font-weight: 400;\">may<\/span><span style=\"font-weight: 400;\"> be <\/span><span style=\"font-weight: 400;\">faster<\/span><span style=\"font-weight: 400;\"> than <\/span><span style=\"font-weight: 400;\">working<\/span><span style=\"font-weight: 400;\"> algebraically <\/span><span style=\"font-weight: 400;\">in<\/span> <span style=\"font-weight: 400;\">some cases<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><ol start=\"3\"><li><span style=\"color: #000080;\"><b> Utilize Logical Reasoning<\/b><\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">Certain<\/span> <span style=\"font-weight: 400;\">problems<\/span> <span style=\"font-weight: 400;\">can<\/span><span style=\"font-weight: 400;\"> be <\/span><span style=\"font-weight: 400;\">solved through reasoning by elimination instead<\/span> <span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">higher math.<\/span><\/p><ol start=\"4\"><li><span style=\"color: #000080;\"><b> Think Visually for Geometry Problems<\/b><\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">With geometry, <\/span><span style=\"font-weight: 400;\">a diagram or <\/span><span style=\"font-weight: 400;\">using<\/span><span style=\"font-weight: 400;\"> symmetry will make <\/span><span style=\"font-weight: 400;\">solutions<\/span><span style=\"font-weight: 400;\"> more <\/span><span style=\"font-weight: 400;\">apparent.<\/span><\/p><ol start=\"5\"><li><span style=\"color: #000080;\"><b> Practice with Timed Drills<\/b><\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">Since AMC-8 is a timed test<\/span><span style=\"font-weight: 400;\">, practice under time <\/span><span style=\"font-weight: 400;\">constraints<\/span><span style=\"font-weight: 400;\"> to <\/span><span style=\"font-weight: 400;\">build speed and accuracy.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cee8133 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cee8133\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2ce60e6\" data-id=\"2ce60e6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bc21674 elementor-widget elementor-widget-text-editor\" data-id=\"bc21674\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>Final Thoughts<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f397a24 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f397a24\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-85d5798\" data-id=\"85d5798\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ac51e41 elementor-widget elementor-widget-text-editor\" data-id=\"ac51e41\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">To become<\/span> <span style=\"font-weight: 400;\">proficient in <\/span><span style=\"font-weight: 400;\">the <\/span><span style=\"font-weight: 400;\">most<\/span> <span style=\"font-weight: 400;\">challenging <\/span><span style=\"font-weight: 400;\">AMC-8 problems, you <\/span><span style=\"font-weight: 400;\">must<\/span> <span style=\"font-weight: 400;\">spend<\/span><span style=\"font-weight: 400;\"> time, strategic <\/span><span style=\"font-weight: 400;\">reasoning<\/span><span style=\"font-weight: 400;\">, and practice. By <\/span><span style=\"font-weight: 400;\">focusing<\/span><span style=\"font-weight: 400;\"> on <\/span><span style=\"font-weight: 400;\">understanding<\/span> <span style=\"font-weight: 400;\">concepts<\/span> <span style=\"font-weight: 400;\">thoroughly<\/span><span style=\"font-weight: 400;\"> and <\/span><span style=\"font-weight: 400;\">developing problem-solving skills, you can boost your confidence level and performance in the competition. Practice and stay curious!<\/span><\/p><p style=\"text-align: center;\"><em><span style=\"color: #993300;\"><b>Have a Question?<\/b><\/span><\/em><\/p><p style=\"text-align: center;\"><span style=\"font-weight: 400;\">If you have any specific<\/span><span style=\"font-weight: 400;\"> problems you’d like <\/span><span style=\"font-weight: 400;\">help with, comment below, and we’ll be happy<\/span><span style=\"font-weight: 400;\"> to <\/span><span style=\"font-weight: 400;\">discuss<\/span><span style=\"font-weight: 400;\"> solutions!<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>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 […]<\/p>\n","protected":false},"author":840,"featured_media":35191,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[1019,1018],"tags":[998,1022,1021,1023,1020,1024,995,990,993,989],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/35189"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/users\/840"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/comments?post=35189"}],"version-history":[{"count":139,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/35189\/revisions"}],"predecessor-version":[{"id":35342,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/35189\/revisions\/35342"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/35191"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=35189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=35189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=35189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}