Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Calculus Subject Test a point is chosen randomly with uniform distribution in the interior of a circle of radius 1. What is its expected distance from the center of the circle? 2. a particle moves along the -axis in such a way that its velocity at position is given by the formula v(ar)=2+ sin r. What is its acceleration at a 3. What is the area of the region bounded by the curves y= x2003 and y=x/2003 and ing above the x-axis 4. The sequence of real numbers 21, 2, T3,. satisfies limn-oo(a2n+ 2n+1)= 315 and limn-oo(2n +I2n-1=2003. Evaluate limn-oo(a2n/ 2n+1) 5. Find the minimum distance from the point (0, 5/2)to the graph of y=x4/8 6. For n an integer, evaluate 3(√n2-02+√m2 7. For what value of a> 1 is 1 minimum? 8. A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone.(For example, when the height of the water in the cone is 4 inches, water Hows out at a rate of 4 cubic inches per second. Determine how many seconds it will take for all of the water to How out of the cone 9. Two differentiable real functions f(ar) and g(ar) satisfy f(x)-9(x) for all r, and f(0)=g(2003)= 1. Find the largest constant c such that f(2003)>c for all such functions f, g valuate d 1
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Calculus Subject Test 1. A point is chosen randomly with uniform distribution in the interior of a circle of radius 1. What is its expected distance from the center of the circle? 2. A particle moves along the x-axis in such a way that its velocity at position x is given by the formula v(x) = 2 + sin x. What is its acceleration at x = π 6 ? 3. What is the area of the region bounded by the curves y = x 2003 and y = x 1/2003 and lying above the x-axis? 4. The sequence of real numbers x1, x2, x3, . . . satisfies limn→∞(x2n + x2n+1) = 315 and limn→∞(x2n + x2n−1) = 2003. Evaluate limn→∞(x2n/x2n+1). 5. Find the minimum distance from the point (0, 5/2) to the graph of y = x 4/8. 6. For n an integer, evaluate limn→∞ µ 1 √ n2 − 0 2 + 1 √ n2 − 1 2 + · · · + 1 q n2 − (n − 1)2 ¶ . 7. For what value of a > 1 is Z a 2 a 1 x log x − 1 32 dx minimum? 8. A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of the water in the cone is 4 inches, water flows out at a rate of 4 cubic inches per second.) Determine how many seconds it will take for all of the water to flow out of the cone. 9. Two differentiable real functions f(x) and g(x) satisfy f 0 (x) g 0 (x) = e f(x)−g(x) for all x, and f(0) = g(2003) = 1. Find the largest constant c such that f(2003) > c for all such functions f, g. 10. Evaluate Z ∞ −∞ 1 − x 2 1 + x 4 dx. 1
Harvard-MIT Mathematics Tournament March 15. 2003 Individual round: General Test Part 1 1. 10 people are playing musical chairs with n chairs in a circle. They can be seated in 7! ways(assuming only one person fits on each chair, of course), where different arrangements of the same people on chairs, even rotations, are considered different Find n and triangle TEN has area 10. What is the length of a side of the square, area 62 2. OPEN is a square, and T is a point on side NO, such that triangle TOP has 3. There are 16 members on the Height-Measurement Matching Team. Each member was asked, How many other people on the team not counting yourself- are exactly the same height as you? The answers included six 1's, six 2s, and three 3s. What was the sixteenth answer?(Assume that everyone answered truthfully. 4. How many 2-digit positive integers have an even number of positive divisors? 5. A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12m. What is the area (in m) of the room? 6. In how many ways can 3 bottles of ketchup and 7 bottles of mustard be arranged in a row so that no bottle of ketchup is immediately between two bottles of mustard?(The bottles of ketchup are mutually indistinguishable, as are the bottles of mustard. 7. Find the real value of a such that x3+3.x2+3x+7=0 8. A broken calculator has the and x keys switched. For how many ordered pairs(a, b) of integers will it correctly calculate a b using the labelled key? 9. Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonals intersecting only at vertices. What is the smallest possible number of obtuse triangles in the triangulation? 10. Bessie the cow is trying to navigate her way through a field. She can travel only from lattice point to adjacent lattice point, can turn only at lattice points, and can travel only to the east or north. (A lattice point is a point whose coordinates are both egers ) (0, 0)is the southwest corner of the field. (5, 5) is the northeast corner of the field. Due to large rocks, Bessie is unable to walk on the points(1, 1),(2, 3),or (3, 2). How many ways are there for Bessie to travel from(0, 0)to(5, 5) under these constraints?
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: General Test, Part 1 1. 10 people are playing musical chairs with n chairs in a circle. They can be seated in 7! ways (assuming only one person fits on each chair, of course), where different arrangements of the same people on chairs, even rotations, are considered different. Find n. 2. OP EN is a square, and T is a point on side NO, such that triangle T OP has area 62 and triangle T EN has area 10. What is the length of a side of the square? 3. There are 16 members on the Height-Measurement Matching Team. Each member was asked, “How many other people on the team — not counting yourself — are exactly the same height as you?” The answers included six 1’s, six 2’s, and three 3’s. What was the sixteenth answer? (Assume that everyone answered truthfully.) 4. How many 2-digit positive integers have an even number of positive divisors? 5. A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12m. What is the area (in m2 ) of the room? 6. In how many ways can 3 bottles of ketchup and 7 bottles of mustard be arranged in a row so that no bottle of ketchup is immediately between two bottles of mustard? (The bottles of ketchup are mutually indistinguishable, as are the bottles of mustard.) 7. Find the real value of x such that x 3 + 3x 2 + 3x + 7 = 0. 8. A broken calculator has the + and × keys switched. For how many ordered pairs (a, b) of integers will it correctly calculate a + b using the labelled + key? 9. Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonals intersecting only at vertices. What is the smallest possible number of obtuse triangles in the triangulation? 10. Bessie the cow is trying to navigate her way through a field. She can travel only from lattice point to adjacent lattice point, can turn only at lattice points, and can travel only to the east or north. (A lattice point is a point whose coordinates are both integers.) (0, 0) is the southwest corner of the field. (5, 5) is the northeast corner of the field. Due to large rocks, Bessie is unable to walk on the points (1, 1), (2, 3), or (3, 2). How many ways are there for Bessie to travel from (0, 0) to (5, 5) under these constraints? 1
Harvard-MIT Mathematics Tournament March 15. 2003 Individual round: General Test Part 2 A compact disc has the shape of a circle of diameter 5 inches with a l-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the 2. You have a list of real numbers, whose sum is 40. If you replace every number a on the list by 1- the sum of the new numbers will be 20. If instead you had replaced every number r by 1+T, what would the sum then be? 3. How many positive rational numbers less than T have denominator at most 7 when written in lowest terms?(Integers have denominator 1.) 4. In triangle ABC with area 51, points D and E trisect AB and points F and G trisect BC. Find the largest possible area of quadrilateral DEFG 5. You are given a 10 x 2 grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent? 6. The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, and diagonals of a 3 x 3 square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list? 7. Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that he will win the game? 8. If c>0,y>0 are integers, randomly chosen with the constraint a +y< 10, what is the probability that t +y is even? 9. In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one(i.e. move one desk forward, back left or right ). In how many ways can this reassignment be made? 10. Several positive integers are given, not necessarily all different. Their sum is 2003 Suppose that ni of the given numbers are equal to 1, n2 of them are equal to 2, 72003 of them are equal to 2003. Find the largest possible value of +2n3+3m4+…+20027
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: General Test, Part 2 1. A compact disc has the shape of a circle of diameter 5 inches with a 1-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity? 2. You have a list of real numbers, whose sum is 40. If you replace every number x on the list by 1 − x, the sum of the new numbers will be 20. If instead you had replaced every number x by 1 + x, what would the sum then be? 3. How many positive rational numbers less than π have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) 4. In triangle ABC with area 51, points D and E trisect AB and points F and G trisect BC. Find the largest possible area of quadrilateral DEF G. 5. You are given a 10 × 2 grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent? 6. The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, and diagonals of a 3 × 3 square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list? 7. Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that he will win the game? 8. If x ≥ 0, y ≥ 0 are integers, randomly chosen with the constraint x + y ≤ 10, what is the probability that x + y is even? 9. In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways can this reassignment be made? 10. Several positive integers are given, not necessarily all different. Their sum is 2003. Suppose that n1 of the given numbers are equal to 1, n2 of them are equal to 2, . . ., n2003 of them are equal to 2003. Find the largest possible value of n2 + 2n3 + 3n4 + · · · + 2002n2003. 1
Harvard-MIT Mathematics Tournament March 15. 2003 HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 1.5] Simplify 2y2/1-35.y89+15 2. [5] The graph of r=12y? is a union of n different lines. What is the value of n? 3. 5 If a and b are positive integers that can each be written as a sum of two squares then ab is also a sum of two squares. Find the smallest positive integer c such that c=ab, where a=x3+y and b=23+y each have solutions in integers(, y),but c=r+y does not HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 4.[6]Letz=1-2. Find+是+是+… 5. [6 Compute the surface area of a cube inscribed in a sphere of surface area T 6. [6 Define the Fibonacci numbers by Fo=0, Fi=l, Fn=Fn-1+ Fn-2 for n 22. For how many n,0≤n≤100, is Fn a multiple of13? HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 7. 6 a and b are integers such that a+ vb=v15+v216. Compute a/b 8. [6 How many solutions in nonnegative integers(a, b, c) are there to the equation 9. [6 For z a real number, let f(r)=0 if x I and f(a)=2. c-2 if x >1. How many solutions are there to the equation f(f((f(a)=c
Harvard-MIT Mathematics Tournament March 15, 2003 Guts Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 1. [5] Simplify 2003q 2 √ 11 − 3 √ 5 · 4006q 89 + 12√ 55. 2. [5] The graph of x 4 = x 2 y 2 is a union of n different lines. What is the value of n? 3. [5] If a and b are positive integers that can each be written as a sum of two squares, then ab is also a sum of two squares. Find the smallest positive integer c such that c = ab, where a = x 3 + y 3 and b = x 3 + y 3 each have solutions in integers (x, y), but c = x 3 + y 3 does not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 4. [6] Let z = 1 − 2i. Find 1 z + 2 z 2 + 3 z 3 + · · ·. 5. [6] Compute the surface area of a cube inscribed in a sphere of surface area π. 6. [6] Define the Fibonacci numbers by F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2. For how many n, 0 ≤ n ≤ 100, is Fn a multiple of 13? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 7. [6] a and b are integers such that a + √ b = q 15 + √ 216. Compute a/b. 8. [6] How many solutions in nonnegative integers (a, b, c) are there to the equation 2 a + 2b = c! ? 9. [6] For x a real number, let f(x) = 0 if x < 1 and f(x) = 2x − 2 if x ≥ 1. How many solutions are there to the equation f(f(f(f(x)))) = x? 1
HARVARD-MIT MATHEMATICS TOURNAMENT. MARCH 15. 2003- GUTS ROUND 10. 7 Suppose that A, B, C, D are four points in the plane, and let Q, R, S, T, U, V be the respective midpoints of AB, AC, AD, BC, BD, CD. If QR=2001, SU= 2002, TV 2003, find the distance between the midpoints of QU and RV. 11. [7] Find the smallest positive integer n such that 12+22+32+42+.+n2is divisible by100 12.[7 As shown in the figure, a circle of radius 1 has two equal circles whose diameters cover a chosen diameter of the larger circle. In each of these smaller circles we similarly draw three equal circles, then four in each of those, and so on. Compute the area of the region enclosed by a positive even number of circles HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003- GUTS ROUND 13.[7] If ry=5 and 2+y2=21, compute z 4+y 14. 7 A positive integer will be called"sparkly"if its smallest(positive) divisor, other than 1, equals the total number of divisors(including 1). How of the numbers 2,3,….,2003are 15. 7 The product of the digits of a 5-digit number is 180. How many such numbers exist?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 10. [7] Suppose that A, B, C, D are four points in the plane, and let Q, R, S, T, U, V be the respective midpoints of AB, AC, AD, BC, BD, CD. If QR = 2001, SU = 2002, T V = 2003, find the distance between the midpoints of QU and RV . 11. [7] Find the smallest positive integer n such that 12 + 22 + 32 + 42 +· · ·+n 2 is divisible by 100. 12. [7] As shown in the figure, a circle of radius 1 has two equal circles whose diameters cover a chosen diameter of the larger circle. In each of these smaller circles we similarly draw three equal circles, then four in each of those, and so on. Compute the area of the region enclosed by a positive even number of circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND 13. [7] If xy = 5 and x 2 + y 2 = 21, compute x 4 + y 4 . 14. [7] A positive integer will be called “sparkly” if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers 2, 3, . . . , 2003 are sparkly? 15. [7] The product of the digits of a 5-digit number is 180. How many such numbers exist? 2