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