MathDB

Team Round

Part of 2019 LMT Spring

Problems(1)

2019 LMT Spring Team Round - Lexington Math Tournament

Source:

1/13/2022
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.
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