[Soln] 2013 AIME I Problem 12

Posted on 15 Jun 2016 by kjytay

2013 AIME I 12. Let \Delta PQR be a triangle with \angle P = 75^\circ and \angle Q = 60^\circ. A regular hexagon ABCDEF with side length 1 is drawn inside \Delta PQR so that side \overline{AB} lies on \overline{PQ}, side \overline{CD} lies on \overline{QR}, and one of the remaining vertices lies on \overline{RP}. There are positive integers a, b, c, and d such that the area of \Delta PQR can be expressed in the form \frac{a + b\sqrt{c}}{d}, where a and d are relatively prime and c is not divisible by the square of any prime. Find a + b + c + d.

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2013 AIME I Problem 12

Posted on 15 Jun 2016 by kjytay

2013 AIME I 12. Let \Delta PQR be a triangle with \angle P = 75^\circ and \angle Q = 60^\circ. A regular hexagon ABCDEF with side length 1 is drawn inside \Delta PQR so that side \overline{AB} lies on \overline{PQ}, side \overline{CD} lies on \overline{QR}, and one of the remaining vertices lies on \overline{RP}. There are positive integers a, b, c, and d such that the area of \Delta PQR can be expressed in the form \frac{a + b\sqrt{c}}{d}, where a and d are relatively prime and c is not divisible by the square of any prime. Find a + b + c + d.

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[Soln] 2013 AIME I Problem 11

Posted on 14 Jun 2016 by kjytay

2013 AIME I 11. Ms. Math’s kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions:

(a) If 16, 15, or 14 students are present, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers 0 < x < y < z < 14 such that when x, y or z students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.

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2013 AIME I Problem 11

Posted on 14 Jun 2016 by kjytay

2013 AIME I 11. Ms. Math’s kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions:

(a) If 16, 15, or 14 students are present, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers 0 < x < y < z < 14 such that when x, y or z students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.

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[Soln] 2013 AIME I Problem 10

Posted on 13 Jun 2016 by kjytay

2013 AIME I 10. There are nonzero integers a, b, r, and s such that the complex number r + si is a zero of the polynomial P(x) = x^3 - ax^2 + bx - 65. For each possible combination of a and b, let p_{a,b} be the sum of the zeroes of P(x).Find the sum of the p_{a,b}‘s for all possible combinations of a and b.

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2013 AIME I Problem 10

Posted on 13 Jun 2016 by kjytay

2013 AIME I 10. There are nonzero integers a, b, r, and s such that the complex number r + si is a zero of the polynomial P(x) = x^3 - ax^2 + bx - 65. For each possible combination of a and b, let p_{a,b} be the sum of the zeroes of P(x).Find the sum of the p_{a,b}‘s for all possible combinations of a and b.

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[Soln] 2013 AIME I Problem 9

Posted on 12 Jun 2016 by kjytay

2013 AIME I 9. A paper equilateral triangle ABC has side length 12. The paper triangle is folded so that vertex A touches a point on side \overline{BC} a distance 9 from point B. The length of the line segment along which the triangle is folded can be written as \displaystyle\frac{m\sqrt{p}}{n}, where m, n, and p are positive integers, $m$ and $n$ are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.

AIME_I_2016_9_01

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2013 AIME I Problem 9

Posted on 12 Jun 2016 by kjytay

2013 AIME I 9. A paper equilateral triangle ABC has side length 12. The paper triangle is folded so that vertex A touches a point on side \overline{BC} a distance 9 from point B. The length of the line segment along which the triangle is folded can be written as \displaystyle\frac{m\sqrt{p}}{n}, where m, n, and p are positive integers, $m$ and $n$ are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.

AIME_I_2016_9_01

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[Soln] 2013 AIME I Problem 8

Posted on 11 Jun 2016 by kjytay

2013 AIME I 8. The domain of the function f(x) = \arcsin \left( \log_m (nx) \right) is a closed interval of length \displaystyle\frac{1}{2013}, where m and n are positive integers and m > 1. Find the remainder when the smallest possible sum m + n is divided by 1000.

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2013 AIME I Problem 8

Posted on 11 Jun 2016 by kjytay

2013 AIME I 8. The domain of the function f(x) = \arcsin \left( \log_m (nx) \right) is a closed interval of length \displaystyle\frac{1}{2013}, where m and n are positive integers and m > 1. Find the remainder when the smallest possible sum m + n is divided by 1000.

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