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individual

Part of 2017 LMT

Problems(1)

2017 LMT Individual Round - Lexington Mathematical Tournament

Source:

9/16/2023
p1. Find the number of zeroes at the end of 201720^{17}.
p2. Express 120+17\frac{1}{\sqrt{20} +\sqrt{17}} in simplest radical form.
p3. John draws a square ABCDABCD. On side ABAB he draws point PP so that BPPA=120\frac{BP}{PA}=\frac{1}{20} and on side BCBC he draws point QQ such that BQQC=117\frac{BQ}{QC}=\frac{1}{17} . What is the ratio of the area of △PBQ\vartriangle PBQ to the area of ABCDABCD?
p4. Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints?
p5. Alex is playing a game with an unfair coin which has a 15\frac15 chance of flipping heads and a 45\frac45 chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins?
p6. Positive two-digit number ab‾\overline{ab} has 88 divisors. Find the number of divisors of the four-digit number abab‾\overline{abab}.
p7. Call a positive integer nn diagonal if the number of diagonals of a convex nn-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to 20172017.
p8. There are 44 houses on a street, with 22 on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors.
p9. Compute ∣2017−∣2016∣−∣2015−∣...∣3−∣2−1∣∣...∣∣∣∣.|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.
p10. Given points A,BA,B in the coordinate plane, let A⊕BA \oplus B be the unique point CC such that AC‾\overline{AC} is parallel to the xx-axis and BC‾\overline{BC} is parallel to the yy-axis. Find the point (x,y)(x, y) such that ((x,y)⊕(0,1))⊕(1,0)=(2016,2017)⊕(x,y)((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y).
p11. In the following subtraction problem, different letters represent different nonzero digits. \begin{tabular}{ccccc} & M & A & T & H \\ - & & H & A & M \\ \hline & & L & M & T \\ \end{tabular} How many ways can the letters be assigned values to satisfy the subtraction problem?
p12. If mm and nn are integers such that 17n+20m=201717n +20m = 2017, then what is the minimum possible value of ∣m−n∣|m-n|?
p13. Let f(x)=x4−3x3+2x2+7x−9f(x)=x^4-3x^3+2x^2+7x-9. For some complex numbers a,b,c,da,b,c,d, it is true that f(x)=(x2+ax+b)(x2+cx+d)f (x) = (x^2+ax+b)(x^2+cx +d) for all complex numbers xx. Find ab+cd\frac{a}{b}+ \frac{c}{d}.

p14. A positive integer is called an imposter if it can be expressed in the form 2a+2b2^a +2^b where a,ba,b are non-negative integers and a≠ba \ne b. How many almost positive integers less than 20172017 are imposters?
p15. Evaluate the infinite sum ∑n=1∞n(n+1)2n+1=12+34+68+1016+1532+...\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...
p16. Each face of a regular tetrahedron is colored either red, green, or blue, each with probability 13\frac13 . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors?
p17. Let (k,k)(k,\sqrt{k}) be the point on the graph of y=xy=\sqrt{x} that is closest to the point (2017,0)(2017,0). Find kk.
p18. Alice is going to place 20162016 rooks on a 2016×20162016 \times 2016 chessboard where both the rows and columns are labelled 11 to 20162016; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations.
p19. Let f(n)f (n) be a function defined recursively across the natural numbers such that f(1)=1f (1) = 1 and f(n)=nf(n−1)f (n) = n^{f (n-1)}. Find the sum of all positive divisors less than or equal to 1515 of the number f(7)−1f (7)-1.
p20. Find the number of ordered pairs of positive integers (m,n)(m,n) that satisfy gcd (m,n)+lcm (m,n)=2017.gcd \,(m,n)+ lcm \,(m,n) = 2017.
p21. Let △ABC\vartriangle ABC be a triangle. Let MM be the midpoint of ABAB and let PP be the projection of AA onto BCBC. If AB=20AB = 20, and BC=MC=17BC = MC = 17, compute BPBP.
p22. For positive integers nn, define the odd parent function, denoted op(n)op(n), to be the greatest positive odd divisor of nn. For example, op(4)=1op(4) = 1, op(5)=5op(5) = 5, and op(6)=3op(6) =3. Find ∑i=1256op(i).\sum^{256}_{i=1}op(i).
p23. Suppose △ABC\vartriangle ABC has sidelengths AB=20AB = 20 and AC=17AC = 17. Let XX be a point inside △ABC\vartriangle ABC such that BX⊥CXBX \perp CX and AX⊥BCAX \perp BC. If ∣BX4−CX4∣=2017|BX^4 -CX^4|= 2017, the compute the length of side BCBC.
p24. How many ways can some squares be colored black in a 6×66 \times 6 grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct.
p25. Let ABCDABCD be a convex quadrilateral with AB=BC=2AB = BC = 2, AD=4AD = 4, and ∠ABC=120o\angle ABC = 120^o. Let MM be the midpoint of BDBD. If ∠AMC=90o\angle AMC = 90^o, find the length of segment CDCD.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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