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2009年ARML Team题


Alternate Solution: Recall that tan(a + b) = tan(3a) = tan(2a + a) =
2 tan a 1?tan2 a + 2 tan a ? 1? tan2 a

tan a+tan b 1?tan a tan b ,

thus tan(2a) =

2

tan a 1?tan2 a

and

tan a · tan a

1

=

3 tan a ? tan3 a 2 tan a + tan a ? tan3 a = . 2 2 1 ? tan a ? 2 tan a 1 ? 3 tan2 a
π 2.

Back to the problem at hand, divide both sides by 2 to obtain 3 tan?1 x + 2 tan?1 (3x) = tangent of the left side yields tan
π 2

Taking the

is unde?ned, thus 1 = tan(3 tan

tan(3 tan x)+tan(2 tan (3x)) 1?tan(3 tan?1 x) tan(2 tan?1 (3x)) . ?1 ?1

?1

?1

We know that the denominator must be 0 since
3x?x3 1?3x2

x) tan(2 tan

(3x)) =

·

2·3x 1?(3x)2

and hence (1 ? 3x2 )(1 ? 9x2 ) =


8 3 (3x ? x3 )(6x). Simplifying yields 33x4 ? 30x2 + 1 = 0, and applying the quadratic formula gives x2 = 15± . 33 √ 15+8 3 The “+” solution is extraneous: as noted in the previous solution, x = 33 yields a value of 3π for the left √ 8 3 side of the equation), so we are left with x2 = 15? . 33

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Team Problems

Problem 1. Let N be a six-digit number formed by an arrangement of the digits 1, 2, 3, 3, 4, 5. Compute the smallest value of N that is divisible by 264. Problem 2. In triangle ABC , AB = 4, BC = 6, and AC = 8. Squares ABQR and BCST are drawn external to and lie in the same plane as ABC . Compute QT . Problem 3. The numbers 1, 2, . . . , 8 are placed in the 3 × 3 grid below, leaving exactly one blank square. Such a placement is called okay if in every pair of adjacent squares, either one square is blank or the di?erence between the two numbers is at most 2 (two squares are considered adjacent if they share a common side). If re?ections, rotations, etc. of placements are considered distinct, compute the number of distinct okay placements.

Problem 4. An ellipse in the ?rst quadrant is tangent to both the x-axis and y -axis. One focus is at (3, 7), and the other focus is at (d, 7). Compute d. Problem 5. Let A1 A2 A3 A4 A5 A6 A7 A8 be a regular octagon. Let u be the vector from A1 to A2 and let v be the vector from A1 to A8 . The vector from A1 to A4 can be written as au + bv for a unique ordered pair of real numbers (a, b). Compute (a, b). Problem 6. 1024. Compute the integer n such that 2009 < n < 3009 and the sum of the odd positive divisors of n is

Problem 7. Points A, R, M , and L are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point A are (11, 5). If points R, M , and L are all lattice points, and R is in Quadrant I, compute the number of possible ordered pairs (x, y ) of coordinates for point R.

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Problem 8.

The taxicab distance between points (x1 , y1 , z1 ) and (x2 , y2 , z2 ) is given by d((x1 , y1 , z1 ), (x2 , y2 , z2 )) = |x1 ? x2 | + |y1 ? y2 | + |z1 ? z2 |.

The region R is obtained by taking the cube {(x, y, z ) : 0 ≤ x, y, z ≤ 1} and removing every point whose 3 taxicab distance to any vertex of the cube is less than 5 . Compute the volume of R. Problem 9. Let a and b be real numbers such that a3 ? 15a2 + 20a ? 50 = 0 Compute a + b. Problem 10. For a positive integer n, de?ne s(n) to be the sum of n and its digits. For example, s(2009) = 2009 + 2 + 0 + 0 + 9 = 2020. Compute the number of elements in the set {s(0), s(1), s(2), . . . , s(9999)}. and 8b3 ? 60b2 ? 290b + 2575 = 0.

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