MathDB

2012 LMT

Part of LMT

Subcontests

(3)
Individual
1

2012 LMT Individual Round - Lexington Mathematical Tournament

p1. Evaluate 1!+2!+3!+4!+5!1! + 2! + 3! + 4! + 5! (where n!n! is the product of all integers from 11 to nn, inclusive).
p2. Harold opens a pack of Bertie Bott's Every Flavor Beans that contains 1010 blueberry, 1010 watermelon, 33 spinach and 22 earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored?
p3. Find the sum of the positive factors of 3232 (including 3232 itself).
p4. Carol stands at a flag pole that is 2121 feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked 1010 feet, her shadow passes the flag's shadow. Given that Carol is exactly 55 feet tall, how long in feet is her shadow?
p5. A solid metal sphere of radius 77 cm is melted and reshaped into four solid metal spheres with radii 11, 55, 66, and xx cm. What is the value of xx?
p6. Let A=(2,−2)A = (2,-2) and B=(−3,3)B = (-3, 3). If (a,0)(a,0) and (0,b)(0, b) are both equidistant from AA and BB, then what is the value of a+ba + b?
p7. For every flip, there is an x2x^2 percent chance of flipping heads, where xx is the number of flips that have already been made. What is the probability that my first three flips will all come up tails?
p8. Consider the sequence of letters Z  W  Y  X  VZ\,\,W\,\,Y\,\,X\,\,V. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order?
p9. A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is 169169, then find the side length of the square.
p10. If A=503A = 50\sqrt3, B=602B = 60\sqrt2, and C=85C = 85, then order AA, BB, and CC from least to greatest.
p11. How many ways are there to arrange the letters of the word RACECARRACECAR? (Identical letters are assumed to be indistinguishable.)
p12. A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let rr be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find r2r^2.
p13. Given that x2+x+1x+1x2=10x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10, find all possible values of x+1xx +\frac{1}{x} .
p14. Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions 3×2×23\times 2\times 2. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible.
p15. Triangle ABCABC has AB=4AB = 4, BC=3BC = 3, and AC=5AC = 5. Point BB is reflected across AC‾\overline{AC} to point B′B'. The lines that contain AB′AB' and BCBC are then drawn to intersect at point DD. Find ADAD.
p16. Consider a rectangle ABCDABCD with side lengths 55 and 1212. If a circle tangent to all sides of △ABD\vartriangle ABD and a circle tangent to all sides of △BCD\vartriangle BCD are drawn, then how far apart are the centers of the circles?
p17. An increasing geometric sequence a0,a1,a2,...a_0, a_1, a_2,... has a positive common ratio. Also, the value of a3+a2−a1−a0a_3 + a_2 - a_1 - a_0 is equal to half the value of a4−a0a_4 - a_0. What is the value of the common ratio?
p18. In triangle ABCABC, AB=9AB = 9, BC=11BC = 11, and AC=16AC = 16. Points EE and FF are on AB‾\overline{AB} and BC‾\overline{BC}, respectively, such that BE=BF=4BE = BF = 4. What is the area of triangle CEFCEF?
p19. Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After 88 minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in 1212 minutes. After how many seconds since the three began running did Zach first pass Yuna?
p20. How many unit fractions are there such that their decimal equivalent has a cycle of 66 repeating integers? Exclude fractions that repeat in cycles of 11, 22, or 33.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
6
Team Round
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2012 LMT Team Round - Potpourri - Lexington Math Tournament

p1. What is 7%7\% of one half of 11%11\% of 2000020000 ?
p2. Three circles centered at A,BA, B, and CC are tangent to each other. Given that AB=8AB = 8, AC=10AC = 10, and BC=12BC = 12, find the radius of circle A A.
p3. How many positive integer values of xx less than 20122012 are there such that there exists an integer yy for which 1x+22y+1=1y\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y} ?
p4. The positive difference between 8 8 and twice xx is equal to 1111 more than xx. What are all possible values of xx?
p5. A region in the coordinate plane is bounded by the equations x=0x = 0, x=6x = 6, y=0y = 0, and y=8y = 8. A line through (3,4)(3, 4) with slope 44 cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line?
p6. A polygon is composed of only angles of degrees 138138 and 150150, with at least one angle of each degree. How many sides does the polygon have?
p7. M,A,T,HM, A, T, H, and LL are all not necessarily distinct digits, with M≠0M \ne 0 and L≠0L \ne 0. Given that the sum MATH+LMTMATH +LMT, where each letter represents a digit, equals 20122012, what is the average of all possible values of the three-digit integer LMTLMT?
p8. A square with side length 10\sqrt{10} and two squares with side length 7\sqrt{7} share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square?
p9. Consider the sequence 2012,12012,20120,20121,...2012, 12012, 20120, 20121, .... This sequence is the increasing sequence of all integers that contain “20122012”. What is the 3030th term in this sequence?
p10. What is the coefficient of the x5x^5 term in the simplified expansion of (x+x+x3)10(x +\sqrt{x} +\sqrt[3]{x})^{10} ?
PS. You had better use hide for answers.