Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Algebra Subject Test Find the smallest value of r such that a>14 a for all nonnegative a 2. Compute an 2000-5in-0203) 3. Find the smallest n such that n! ends in 290 zeroes 4. Simplify:2V15+V2-(1.5+ 5. 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 n2003 of them are equal to 2003. Find the largest possible value of +2n3+3 6. Let a1=1, and let an=Ln /an-1l 7. Let a, b, c be the three roots of p(r)=x+12-333 -1001. Find a+b+c Find the value of 3+1+4+2 9. For how many integers n, for 1<ns1000, is the number 2(an)even? 10.S P()is a poly such that P(1) P(2x) P(x+1) x+7 for all real a for which both sides are defined. Find P(1)
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Algebra Subject Test 1. Find the smallest value of x such that a ≥ 14√ a − x for all nonnegative a. 2. Compute tan2 (20◦)−sin2 (20◦) tan2(20◦) sin2 (20◦) . 3. Find the smallest n such that n! ends in 290 zeroes. 4. Simplify: 2q 1.5 + √ 2 − (1.5 + √ 2). 5. 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. 6. Let a1 = 1, and let an = bn 3/an−1c for n > 1. Determine the value of a999. 7. Let a, b, c be the three roots of p(x) = x 3 + x 2 − 333x − 1001. Find a 3 + b 3 + c 3 . 8. Find the value of 1 3 2+1 + 1 4 2+2 + 1 5 2+3 + · · ·. 9. For how many integers n, for 1 ≤ n ≤ 1000, is the number 1 2 ³ 2n n ´ even? 10. Suppose P(x) is a polynomial such that P(1) = 1 and P(2x) P(x + 1) = 8 − 56 x + 7 for all real x for which both sides are defined. Find P(−1). 1
Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Geometry Subject Test 1. AD and bC are both perpendicular to AB, and CD is perpendicular to AC. If AB=4 and BC=3. find CD B 2. As shown, U and C are points on the sides of triangle MNH such that MU =s UN=6, NC=20, CH=S, HM=25. If triangle UNC and quadrilateral MUCH have equal areas, what is s 3. 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? 4. Farmer John is inside of an ellipse with reflective sides, given by the equation x2/a2+ y / b2=l, with a>b>0. He is standing at the point (3, 0), and he shines a laser pointer in the y-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair(a, b)?
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Geometry Subject Test 1. AD and BC are both perpendicular to AB, and CD is perpendicular to AC. If AB = 4 and BC = 3, find CD. C A D B 2. As shown, U and C are points on the sides of triangle MNH such that MU = s, UN = 6, NC = 20, CH = s, HM = 25. If triangle UNC and quadrilateral MUCH have equal areas, what is s? M N C H U 6 s 25 20 s 3. 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? 4. Farmer John is inside of an ellipse with reflective sides, given by the equation x 2/a2 + y 2/b2 = 1, with a > b > 0. He is standing at the point (3, 0), and he shines a laser pointer in the y-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair (a, b)? 1
5. 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? 6. Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center Take the remaining bead"and mold it into a new sphere. What is this sphere's radius? 7. Let RSTUV be a regular pentagon. Construct an equilateral triangle PRS with point P inside the pentagon. Find the measure(in degrees) of angle PTV 8. Let ABC be an equilateral triangle of side length 2. Let w be its circumcircle, and let A, WB, wc be circles congruent to w centered at each of its vertices. Let R be the set of all points in the plane contained in exactly two of these four circles. what is the area of R? 9. In triangle ABC,∠ABC=50°and∠ACB=709. Let d be the midpoint of side BC. A circle is tangent to BC at b and is also tangent to segment AD nstersects AB again at P. Another circle is tangent to bc at C and is also tangent to segment AD; this circle intersects AC again at Q. Find LAPQ (in degrees 10. Convex quadrilateral MATH is given with HM/MT=3/4, and LATM=LMAT= LAHM =60. N is the midpoint of MA, and O is a point on TH such that lines MT AH NO are concurrent. Find the ratio HO/OT
5. 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? 6. Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining “bead” and mold it into a new sphere. What is this sphere’s radius? 7. Let RST UV be a regular pentagon. Construct an equilateral triangle P RS with point P inside the pentagon. Find the measure (in degrees) of angle P T V . 8. Let ABC be an equilateral triangle of side length 2. Let ω be its circumcircle, and let ωA, ωB, ωC be circles congruent to ω centered at each of its vertices. Let R be the set of all points in the plane contained in exactly two of these four circles. What is the area of R? 9. In triangle ABC, 6 ABC = 50◦ and 6 ACB = 70◦ . Let D be the midpoint of side BC. A circle is tangent to BC at B and is also tangent to segment AD; this circle instersects AB again at P. Another circle is tangent to BC at C and is also tangent to segment AD; this circle intersects AC again at Q. Find 6 AP Q (in degrees). 10. Convex quadrilateral MAT H is given with HM/MT = 3/4, and 6 ATM = 6 MAT = 6 AHM = 60◦ . N is the midpoint of MA, and O is a point on T H such that lines MT, AH, NO are concurrent. Find the ratio HO/OT. 2
Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Combinatorics Subject Test You have 2003 switches, numbered from 1 to 2003, arranged in a circle. Initially, each switch is either on or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch S, if the two switches next to S were tially in the same position, then you set s to oN; otherwise, you set s to OFF What is the probability that all switches will now be ON? 2. 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? 3. 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? 4. In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every aide serves on 3 committees. How many committees are there altogether? 5. We wish to color the integers 1, 2, 3, .., 10 in red, green, and blue, so that no two numbers a and b, with a-b odd, have the same color. We do not require that all three colors be used. )In how many ways can this be done? 6. 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? 7. You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled 1, 2, ... Each ball has probability 1/2n of being put into box n. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls? 8. For any subset SC1, 2,., 151, a number n is called an "anchor"for S if n and n+S are both members of S, where S denotes the number of members of S. Find the average number of anchors over all possible subsets SC(1, 2,., 15) At a certain college, there are 10 clubs and some number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three different students, there is some club such that either exactly one or all three belong to that club. What is the largest possible number of students?
Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Combinatorics Subject Test 1. You have 2003 switches, numbered from 1 to 2003, arranged in a circle. Initially, each switch is either ON or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch S, if the two switches next to S were initially in the same position, then you set S to ON; otherwise, you set S to OFF. What is the probability that all switches will now be ON? 2. 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? 3. 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? 4. In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators, of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every aide serves on 3 committees. How many committees are there altogether? 5. We wish to color the integers 1, 2, 3, . . . , 10 in red, green, and blue, so that no two numbers a and b, with a − b odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done? 6. 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? 7. You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled 1, 2, . . .. Each ball has probability 1/2 n of being put into box n. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls? 8. For any subset S ⊆ {1, 2, . . . , 15}, a number n is called an “anchor” for S if n and n + |S| are both members of S, where |S| denotes the number of members of S. Find the average number of anchors over all possible subsets S ⊆ {1, 2, . . . , 15}. 9. At a certain college, there are 10 clubs and some number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three different students, there is some club such that either exactly one or all three belong to that club. What is the largest possible number of students? 1
10. A calculator has a display, which shows a nonnegative integer N, and a button, which replaces N by a random integer chosen uniformly from the set 10, 1,..., N-1,pro- vided that N>0. Initially, the display holds the number N= 2003. If the button is pressed repeatedly until N=0, what is the probability that the numbers 1, 10, 100 and 1000 will each show up on the display at some point?
10. A calculator has a display, which shows a nonnegative integer N, and a button, which replaces N by a random integer chosen uniformly from the set {0, 1, . . . , N − 1}, provided that N > 0. Initially, the display holds the number N = 2003. If the button is pressed repeatedly until N = 0, what is the probability that the numbers 1, 10, 100, and 1000 will each show up on the display at some point? 2
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