MathDB

2017 LMT

Part of LMT

Subcontests

(5)
individual
1

2017 LMT Individual Round - Lexington Mathematical Tournament

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.
6
Radical Cent
1

2017 LMT Team Round - Radical center lies on OI - Lexington Math Tournament

Let PP be a point and ω\omega be a circle with center OO and radius rr . We define the power of the point PP with respect to the circle ω\omega to be OP2−r2OP^2 - r^2 , and we denote this by pow (P,ω)(P, \omega). We define the radical axis of two circles ω1\omega_1 and ω2\omega_2 to be the locus of all points P such that pow (P,ω1)=(P,\omega_1) = pow (P,ω2)(P,\omega_2). It turns out that the pairwise radical axes of three circles are either concurrent or pairwise parallel. The concurrence point is referred to as the radical center of the three circles.
In △ABC\vartriangle ABC, let II be the incenter, Γ\Gamma be the circumcircle, and OO be the circumcenter. Let A1,B1,C1A_1,B_1,C_1 be the point of tangency of the incircle of △ABC\vartriangle ABC with side BC,CA,ABBC,CA, AB, respectively. Let X1,X2∈ΓX_1,X_2 \in \Gamma such that X1,B1,C1,X2X_1,B_1,C_1,X_2 are collinear in this order. Let MAM_A be the midpoint of BCBC, and define ωA\omega_A as the circumcircle of △X1X2MA\vartriangle X_1X_2M_A. Define ωB\omega_B ,ωC\omega_C analogously. The goal of this problem is to show that the radical center of ωA\omega_A, ωB\omega_B, ωC\omega_C lies on line OIOI.
(a) LetA1′ A'_1 denote the intersection of B1C1B_1C_1 and BCBC. Show that A1BA1C=A1′BA1′C\frac{A_1B}{A_1C}=\frac{A'_1B}{A'_1C}. (b) Prove that A1A_1 lies on ωA\omega_A. (c) Prove that A1A_1 lies on the radical axis of ωB\omega_B and ωC\omega_C . (d) Prove that the radical axis of ωB\omega_B and ωC\omega_C is perpendicular to B1C1B_1C_1. (e) Prove that the radical center of ωA\omega_A, ωB\omega_B, ωC\omega_C is the orthocenter of △A1B1C1\vartriangle A_1B_1C_1. (f ) Conclude that the radical center of ωA\omega_A, ωB\omega_B, ωC\omega_C , OO, and II are collinear.
PS. You had better use hide for answers.
Max Area
1

2017 LMT Team Round - Max Area of Polygon - Lexington Math Tournament

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
Team Round
1

2017 LMT Team Round - Potpourri - Lexington Math Tournament

p1. Suppose that 20%20\% of a number is 1717. Find 20%20\% of 17%17\% of the number.
p2. Let A,B,C,DA, B, C, D represent the numbers 11 through 44 in some order, with A≠1A \ne 1. Find the maximum possible value of log⁡ABC+D\frac{\log_A B}{C +D}.
Here, log⁡AB\log_A B is the unique real number XX such that AX=BA^X = B.
p3. There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed.
p4. Let a,b,ca,b,c be real numbers which satisfy 2017a=a(b+c),2017b=b(a+c),2017c=c(a+b).\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b). Find the sum of all possible values of abcabc.
p5. Let aa and bb be complex numbers such that ab+a+b=(a+b+1)(a+b+3)ab + a +b = (a +b +1)(a +b +3). Find all possible values of a+1b+1\frac{a+1}{b+1}.
p6. Let △ABC\vartriangle ABC be a triangle. Let X,Y,ZX,Y,Z be points on lines BCBC, CACA, and ABAB, respectively, such that XX lies on segment BCBC, BB lies on segment AYAY , and CC lies on segment AZAZ. Suppose that the circumcircle of △XYZ\vartriangle XYZ is tangent to lines ABAB, BCBC, and CACA with center IAI_A. If AB=20AB = 20 and IAC=AC=17I_AC = AC = 17 then compute the length of segment BCBC.
p7. An ant makes 40344034 moves on a coordinate plane, beginning at the point (0,0)(0, 0) and ending at (2017,2017)(2017, 2017). Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region ∣x−y∣≤2|x - y| \le 2. Let N be the number of paths the ant can take. Find the remainder when NN is divided by 10001000.
p8. A 1010 digit positive integer a9a8a7...a1a0‾\overline{a_9a_8a_7...a_1a_0} with a9a_9 nonzero is called deceptive if there exist distinct indices i>ji > j such that aiaj‾=37\overline{a_i a_j} = 37. Find the number of deceptive positive integers.
p9. A circle passing through the points (2,0)(2, 0) and (1,7)(1, 7) is tangent to the yy-axis at (0,r)(0, r ). Find all possible values of r r.
p10. An ellipse with major and minor axes 2020 and 1717, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square.
PS. You had better use hide for answers.