[Soln] 2013 AIME I Problem 7

Posted on 10 Jun 2016 by kjytay

2013 AIME I 7. A rectangular box has width 12 inches, length 16 inches, and height \frac{m}{n} inches, where m and n are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of 30 square inches. Find m+n.

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2013 AIME I Problem 7

Posted on 10 Jun 2016 by kjytay

2013 AIME I 7. A rectangular box has width 12 inches, length 16 inches, and height \frac{m}{n} inches, where m and n are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of 30 square inches. Find m+n.

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[Soln] 2013 AIME I Problem 6

Posted on 9 Jun 2016 by kjytay

2013 AIME I 6. Melinda has three empty boxes and 12 textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as \displaystyle\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

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2013 AIME I Problem 6

Posted on 9 Jun 2016 by kjytay

2013 AIME I 6. Melinda has three empty boxes and 12 textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as \displaystyle\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

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[Soln] 2013 AIME I Problem 5

Posted on 8 Jun 2016 by kjytay

2013 AIME I 5. The real root of the equation 8x^3 - 3x^2 - 3x - 1 = 0 can be written in the form \displaystyle\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}, where a, b, and c are positive integers. Find a+b+c.

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2013 AIME I Problem 5

Posted on 8 Jun 2016 by kjytay

2013 AIME I 5. The real root of the equation 8x^3 - 3x^2 - 3x - 1 = 0 can be written in the form \displaystyle\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}, where a, b, and c are positive integers. Find a+b+c.

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[Soln] 2013 AIME I Problem 4

Posted on 7 Jun 2016 by kjytay

2013 AIME I 4. In the array of 13 squares shown below, 8 squares are coloured red, and the remaining 5 squares are coloured blue. If one of all possible such colourings is chosen at random, the probability that the chosen coloured array appears the same when rotated 90^\circ around the central square is \frac{1}{n}, where n is a positive integer. Find n.

AIME_I_2013_04_01

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2013 AIME I Problem 4

Posted on 7 Jun 2016 by kjytay

2013 AIME I 4. In the array of 13 squares shown below, 8 squares are coloured red, and the remaining 5 squares are coloured blue. If one of all possible such colourings is chosen at random, the probability that the chosen coloured array appears the same when rotated 90^\circ around the central square is \frac{1}{n}, where n is a positive integer. Find n.

AIME_I_2013_04_01

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[Soln] 2013 AIME I Problem 3

Posted on 6 Jun 2016 by kjytay

2013 AIME I 3. Let ABCD be a square, and let E and F be points on \overline{AB} and \overline{BC}, respectively. The line through E parallel to \overline{BC} and the line through F parallel to \overline{AB} divide ABCD into two squares and two non-square rectangles. The sum of the areas of the two squares is \frac{9}{10} of the area of square ABCD. Find \displaystyle\frac{AE}{EB} + \frac{EB}{AE}.

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2013 AIME I Problem 3

Posted on 6 Jun 2016 by kjytay

2013 AIME I 3. Let ABCD be a square, and let E and F be points on \overline{AB} and \overline{BC}, respectively. The line through E parallel to \overline{BC} and the line through F parallel to \overline{AB} divide ABCD into two squares and two non-square rectangles. The sum of the areas of the two squares is \frac{9}{10} of the area of square ABCD. Find \displaystyle\frac{AE}{EB} + \frac{EB}{AE}.

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