MathDB

2019 LMT Spring

Part of LMT

Subcontests

(3)
Individual
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2019 Spring LMT Individual Round - Lexington Mathematical Tournament

p1. Compute 2020⋅(2(0⋅1)+9−(201)8)2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right).
p2. Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make?
p3. Let ABCDABCD be a rhombus such that △ABD\vartriangle ABD and △BCD\vartriangle BCD are equilateral triangles. Find the angle measure of ∠ACD\angle ACD in degrees.
p4. Find the units digit of 201920192019^{2019}.
p5. Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same.
p6. Kathy rolls two fair dice numbered from 11 to 66. At least one of them comes up as a 44 or 55. Compute the probability that the sumof the numbers of the two dice is at least 1010.
p7. Find the number of ordered pairs of positive integers (x,y)(x, y) such that 20x+19y=201920x +19y = 2019.
p8. Let pp be a prime number such that both 2p−12p -1 and 10p−110p -1 are prime numbers. Find the sum of all possible values of pp.
p9. In a square ABCDABCD with side length 1010, let EE be the intersection of ACAC and BDBD. There is a circle inscribed in triangle ABEABE with radius rr and a circle circumscribed around triangle ABEABE with radius RR. Compute R−rR -r .
p10. The fraction 1337⋅77\frac{13}{37 \cdot 77} can be written as a repeating decimal 0.a1a2...an−1an0.a_1a_2...a_{n-1}a_n with nn digits in its shortest repeating decimal representation. Find a1+a2+...+an−1+ana_1 +a_2 +...+a_{n-1}+a_n.
p11. Let point EE be the midpoint of segment ABAB of length 1212. Linda the ant is sitting at AA. If there is a circle OO of radius 33 centered at EE, compute the length of the shortest path Linda can take from AA to BB if she can’t cross the circumference of OO.
p12. Euhan and Minjune are playing tennis. The first one to reach 2525 points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a 34\frac34 chance of making the right call when the ball is in, meaning that he has a 14\frac14 chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan.
p13. Find the number of subsets of {1,2,3,4,5,6,7}\{1, 2, 3, 4, 5, 6,7\} which contain four consecutive numbers.
p14. Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes.
p15. There are 1010 distinct subway lines in Boston, each of which consists of a path of stations. Using any 99 lines, any pair of stations are connected. However, among any 88 lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through?
p16. There exist positive integers kk and 3∤m3\nmid m for which 1−12+13−14+...+153−154+155=3k×m28×29×...×54×55.1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}. Find the value kk.
p17. Geronimo the giraffe is removing pellets from a box without replacement. There are 55 red pellets, 1010 blue pellets, and 1515 white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed.
p18. Find the remainder when 70!(14×67+15×66+...+166×5+167×4)70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right) is divided by 7171.
p19. Let A1A2...A12A_1A_2...A_{12} be the regular dodecagon. Let XX be the intersection of A1A2A_1A_2 and A5A11A_5A_{11}. Given that XA2⋅A1A2=10X A_2 \cdot A_1A_2 = 10, find the area of dodecagon.
p20. Evaluate the following infinite series: ∑n=1∞∑m=1∞nsec⁡2m−mtan⁡2n3m+n(m+n)\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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Team Round
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2019 LMT Spring Team Round - Lexington Math Tournament

p1. David runs at 33 times the speed of Alice. If Alice runs 22 miles in 3030 minutes, determine how many minutes it takes for David to run a mile.
p2. Al has 20192019 red jelly beans. Bob has 20182018 green jelly beans. Carl has xx blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee 22 jelly beans of each color is 40414041. Compute xx.
p3. Find the 77-digit palindrome which is divisible by 77 and whose first three digits are all 22.
p4. Determine the number of ways to put 55 indistinguishable balls in 66 distinguishable boxes.
p5. A certain reduced fraction ab\frac{a}{b} (with a,b>1a,b > 1) has the property that when 22 is subtracted from the numerator and added to the denominator, the resulting fraction has 16\frac16 of its original value. Find this fraction.
p6. Find the smallest positive integer nn such that ∣τ(n+1)−τ(n)∣=7|\tau(n +1)-\tau(n)| = 7. Here, τ(n)\tau(n) denotes the number of divisors of nn.
p7. Let △ABC\vartriangle ABC be the triangle such that AB=3AB = 3, AC=6AC = 6 and ∠BAC=120o\angle BAC = 120^o. Let DD be the point on BCBC such that ADAD bisect ∠BAC\angle BAC. Compute the length of ADAD.
p8. 2626 points are evenly spaced around a circle and are labeled AA through ZZ in alphabetical order. Triangle △LMT\vartriangle LMT is drawn. Three more points, each distinct from L,ML, M, and TT , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap.
p9. Given the three equations a+b+c=0a +b +c = 0 a2+b2+c2=2a^2 +b^2 +c^2 = 2 a3+b3+c3=19a^3 +b^3 +c^3 = 19 find abcabc.
p10. Circle ω\omega is inscribed in convex quadrilateral ABCDABCD and tangent to ABAB and CDCD at PP and QQ, respectively. Given that AP=175AP = 175, BP=147BP = 147,CQ=75CQ = 75, and AB∥CDAB \parallel CD, find the length of DQDQ.
p11. Let pp be a prime and m be a positive integer such that 157p=m4+2m3+m2+3157p = m^4 +2m^3 +m^2 +3. Find the ordered pair (p,m)(p,m).
p12. Find the number of possible functions f:{−2,−1,0,1,2}→{−2,−1,0,1,2}f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\} that satisfy the following conditions. (1) f(x)≠f(y)f (x) \ne f (y) when x≠yx \ne y (2) There exists some xx such that f(x)2=x2f (x)^2 = x^2
p13. Let pp be a prime number such that there exists positive integer nn such that 41pn−42p2=n341pn -42p^2 = n^3. Find the sum of all possible values of pp.
p14. An equilateral triangle with side length 1 1 is rotated 6060 degrees around its center. Compute the area of the region swept out by the interior of the triangle.
p15. Let σ(n)\sigma (n) denote the number of positive integer divisors of nn. Find the sum of all nn that satisfy the equation σ(n)=n3\sigma (n) =\frac{n}{3}.
p16. Let CC be the set of points {a,b,c}∈Z\{a,b,c\} \in Z for 0≤a,b,c≤100 \le a,b,c \le 10. Alice starts at (0,0,0)(0,0,0). Every second she randomly moves to one of the other points in CC that is on one of the lines parallel to the x,yx, y, and zz axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach (10,10,10)(10,10,10).
p17. Find the maximum possible value of abc(1a+1b+1c)3abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3 where a,b,ca,b,c are real such that a+b+c=0a +b +c = 0.
p18. Circle ω\omega with radius 66 is inscribed within quadrilateral ABCDABCD. ω\omega is tangent to ABAB, BCBC, CDCD, and DADA at E,F,GE, F, G, and HH respectively. If AE=3AE = 3, BF=4BF = 4 and CG=5CG = 5, find the length of DHDH.
p19. Find the maximum integer pp less than 10001000 for which there exists a positive integer qq such that the cubic equation x3−px2+qx−(p2−4q+4)=0x^3 - px^2 + q x -(p^2 -4q +4) = 0 has three roots which are all positive integers.
p20. Let △ABC\vartriangle ABC be the triangle such that ∠ABC=60o\angle ABC = 60^o,∠ACB=20o\angle ACB = 20^o. Let PP be the point such that CPCP bisects ∠ACB\angle ACB and ∠PAC=30o\angle PAC = 30^o. Find ∠PBC\angle PBC.
PS. You had better use hide for answers.