MathDB

2014 LMT

Part of LMT

Subcontests

(3)
Individual
1

2014 LMT Individual Round - Lexington Mathematical Tournament

p1. What is 6×7+4×7+6×3+4×36\times 7 + 4 \times 7 + 6\times 3 + 4\times 3?
p2. How many integers nn have exactly n\sqrt{n} factors?
p3. A triangle has distinct angles 3x+103x+10, 2x+202x+20, and x+30x+30. What is the value of xx?
p4. If 44 people of the Math Club are randomly chosen to be captains, and Henry is one of the 3030 people eligible to be chosen, what is the probability that he is not chosen to be captain?
p5. a,b,c,da, b, c, d is an arithmetic sequence with difference xx such that a,c,da, c, d is a geometric sequence. If bb is 1212, what is xx? (Note: the difference of an aritmetic sequence can be positive or negative, but not 00)
p6. What is the smallest positive integer that contains only 00s and 55s that is a multiple of 2424.
p7. If ABCABC is a triangle with side lengths 1313, 1414, and 1515, what is the area of the triangle made by connecting the points at the midpoints of its sides?
p8. How many ways are there to order the numbers 1,2,3,4,5,6,7,81,2,3,4,5,6,7,8 such that 11 and 88 are not adjacent?
p9. Find all ordered triples of nonnegative integers (x,y,z)(x, y, z) such that x+y+z=xyzx + y + z = xyz.
p10. Noah inscribes equilateral triangle ABCABC with area 3\sqrt3 in a cricle. If BRBR is a diameter of the circle, then what is the arc length of Noah's ARCARC?
p11. Today, 4/12/144/12/14, is a palindromic date, because the number without slashes 4121441214 is a palindrome. What is the last palindromic date before the year 30003000?
p12. Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?
p13. How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?
p14. Find all functions ff from R→RR \to R such that f(x+y)+f(x−y)=x2+y2f(x + y) + f(x - y) = x^2 + y^2.
p15. What are the last four digits of 1(1!)+2(2!)+3(3!)+...+2013(2013!)1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)/
p16. In how many distinct ways can a regular octagon be divided up into 66 non-overlapping triangles?
p17. Find the sum of the solutions to the equation 1x−3+1x−5+1x−7+1x−9=2014\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014 .
p18. How many integers nn have the property that (n+1)(n+2)(n+3)(n+4)(n+1)(n+2)(n+3)(n+4) is a perfect square of an integer?
p19. A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?
p20. In blindfolded solitary tic-tac-toe, a player starts with a blank 33-by-33 tic-tac-toe board. On each turn, he randomly places an "XX" in one of the open spaces on the board. The game ends when the player gets 33 XXs in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only XXs are used, not OOs). What fraction of games will run the maximum 77 amount of moves?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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Team Round
1

2014 LMT Team Round - Potpourri - Lexington Math Tournament

p1. Let A%B=BA−B−A+1A\% B = BA - B - A + 1. How many digits are in the number 1%(3%(3%7))1\%(3\%(3\%7)) ?
p2. Three circles, of radii 1,21, 2, and 33 are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle?
p3. Express 13\frac13 in base 22 as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.)
p4. Isosceles trapezoid ABCDABCD with ABAB parallel to CDCD is constructed such that DB=DCDB = DC. If AD=20AD = 20, AB=14AB = 14, and PP is the point on ADAD such that BP+CPBP + CP is minimized, what is AP/DPAP/DP?
p5. Let f(x)=5x−6x−2f(x) = \frac{5x-6}{x-2} . Define an infinite sequence of numbers a0,a1,a2,....a_0, a_1, a_2,.... such that ai+1=f(ai)a_{i+1} = f(a_i) and aia_i is always an integer. What are all the possible values for a2014a_{2014} ?
p6. MATHMATH and TEAMTEAM are two parallelograms. If the lengths of MHMH and AEAE are 1313 and 1515, and distance from AMAM to TT is 1212, find the perimeter of AMHEAMHE.
p7. How many integers less than 10001000 are there such that nn+nn^n + n is divisible by 55 ?
p8. 1010 coins with probabilities of 1,1/2,1/3,...,1/101, 1/2, 1/3 ,..., 1/10 of coming up heads are flipped. What is the probability that an odd number of them come up heads?
p9. An infinite number of coins with probabilities of 1/4,1/9,1/16,...1/4, 1/9, 1/16, ... of coming up heads are all flipped. What is the probability that exactly 1 1 of them comes up heads?
p10. Quadrilateral ABCDABCD has side lengths AB=10AB = 10, BC=11BC = 11, and CD=13CD = 13. Circles O1O_1 and O2O_2 are inscribed in triangles ABDABD and BDCBDC. If they are both tangent to BDBD at the same point EE, what is the length of DADA ?
PS. You had better use hide for answers.