HMMT 二月 2000 · 冲刺赛 · 第 39 题

Guts HMMT 2000 School: Problem Gu1 [4] e The sum of 3 real numbers is known to be zero. If the sum of their cubes is π , what is their product equal to? Problem Gu2 [5] 2 3 2 3 2 2 3 3 If X = 1 + x + x + x + ... and Y = 1 + y + y + y + ... , what is 1 + xy + x y + x y + ... in terms of X and Y only? School: Problem Gu3 [ ± 7] Using 3 colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)? Problem Gu4 [5] Let ABC be a triangle and H be its orthocentre. If it is given that B is (0 , 0), C is (1 , 2) and H is (5 , 0), find A . School: Problem Gu5 [3] Find all natural numbers n such that n equals the cube of the sum of its digits. Problem Gu6 [ ± 10] 2 2 If integers m, n, k satisfy m + n + 1 = kmn , what values can k have? 1

School: Problem Gu7 [7] m Suppose you are give a fair coin and a sheet of paper with the polynomial x written on it. r Now for each toss of the coin, if heads show up, you much erase the polynomial x (where r r − 1 is going to change with time – initially it is m ) written on the paper and replace it by x . r +1 If tails show up, replace it by x . What is the expected value of the polynomial I get after m such tosses? (Note: this is a different concept from the most probable value) Problem Gu8 [ ± 4] Johny’s father tells him : “I am twice as old as you will be seven years from the time I was thrice as old as you were”. What is Johny’s age? School: Problem Gu9 [6] A cubic polynomial f satisfies f (0) = 0 , f (1) = 1 , f (2) = 2 , f (3) = 4. What is f (5)? Problem Gu10 [7] What is the total surface area of an ice cream cone, radius R , height H , with a spherical scoop of ice cream of radius r on top? (Given R < r ) School: Problem Gu11 [6] Let M be the maximum possible value of x x + x x + ... + x x where x 1 , x 2 , ..., x 5 is 1 2 2 3 5 1 a permutation of (1 , 2 , 3 , 4 , 5) and let N be the number of permutations for which this maximum is attained. Evaluate M + N . Problem Gu12 [9] Calculate the number of ways of choosing 4 numbers from the set { 1 , 2 , ..., 11 } such that at least 2 of the numbers are consecutive. 2

School: Problem Gu13 [ ± 4] 4 2 2 Determine the remainder when ( x − 1)( x − 1) is divided by 1 + x + x . Problem Gu14 [7] ABCD is a cyclic quadrilateral inscribed in a circle of radius 5, with AB = 6 , BC = 7 , CD = 8. Find AD . School: Problem Gu15 [8] Find the number of ways of filling a 8 × 8 grid with 0’s and X’s so that the number of 0’s in each row and each column is odd. Problem Gu16 [5] Solve for real x, y : x + y = 2 5 5 x + y = 82 School: Problem Gu17 [5] ( ) 666 Find the highest power of 3 dividing 333 Problem Gu18 [ ± 5] ∞ √ √ ∑ − 1 − 1 What is the value of (tan n − tan n + 1)? n =1 3

School: Problem Gu19 [3] a − b Define a ∗ b = . What is (1 ∗ (2 ∗ (3 ∗ ... ( n ∗ ( n + 1)) ... )))? 1 − ab Problem Gu20 [6] What is the minimum possible perimeter of a triangle two of whose sides are along the x-and y-axes and such that the third contains the point (1,2)? School: Problem Gu21 [8] How many ways can you color a necklace of 7 beads with 4 colors so that no two adjacent beads have the same color? Problem Gu22 [6] 2000 Find the smallest n such that 2 divides n ! School: Problem Gu23 [5] How many 7-digit numbers with distinct digits can be made that are divisible by 3? Problem Gu24 [ ± 3] At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner? 4

School: Problem Gu25 [4] Find the next number in the sequence 131, 111311, 311321, 1321131211, Problem Gu26 [5] What are the last 3 digits of 1! + 2! + ... + 100! School: Problem Gu27 [ ± 6] What is the smallest number that can be written as a sum of 2 squares in 3 ways? Problem Gu28 [8] What is the smallest possible volume to surface ratio of a solid cone with height = 1 unit? School: Problem Gu29 [ ± 9] √ √ √ √ √ √ √ √ What is the value of 1 + 2 1 + 3 1 + 4 1 + 5 1 + ... Problem Gu30 [7] 6 6 ABCD is a unit square. If P AC = P CD , find the length BP . 5

School: Problem Gu31 [10] Given collinear points A, B, C such that AB = BC . How can you construct a point D on AB such that AD = 2 DB , using only a straightedge? (You are not allowed to measure distances) Problem Gu32 [7] How many (nondegenerate) tetrahedrons can be formed from the vertices of an n -dimensional hypercube? School: Problem Gu33 [ ± 5] Characterise all numbers that cannot be written as a sum of 1 or more consecutive odd numbers. Problem Gu34 [ ± 6] What is the largest n such that n ! + 1 is a square? School: Problem Gu35 [4] 2 If 1 + 2 x + 3 x + ... = 9, find x . Problem Gu36 [6] b + c − a If, in a triangle of sides a, b, c , the incircle has radius , what is the magnitude of angle 2 A ? 6

School: Problem Gu37 [9] ◦ A cone with semivertical angle 30 is half filled with water. What is the angle it must be tilted by so that water starts spilling? Problem Gu38 [4] What is the largest number you can write with three 3’s and three 8’s, using only symbols +,-,/, × and exponentiation? School: Problem Gu39 [ ± 8] If r = 1 / 3, what is the value, rounded to 100 decimal digits, of 7 n ∑ 2 n 2 1 + x n =0 Problem Gu40 [ ± 10] Let φ ( n ) denote the number of positive integers less than equal to n and relatively prime to n . find all natural numbers n and primes p such that φ ( n ) = φ ( np ). School: Problem Gu41 [7] A observes a building of height h at an angle of inclination α from a point on the ground. After walking a distance a toward it, the angle is now 2 α , and walking a further distance b causes to increase to 3 α . Find h in terms of a and b . Problem Gu42 [4] 2 A n × n magic square contains numbers from 1 to n such that the sum of every row and every column is the same. What is this sum? 7

School: Problem Gu43 [6] Box A contains 3 black and 4 blue marbles. Box B has 7 black and 1 blue, whereas Box C has 2 black, 3 blue and 1 green marble. I close my eyes and pick two marbles from 2 different boxes. If it turns out that I get 1 black and 1 blue marble, what is the probability that the black marble is from box A and the blue one is from C? Problem Gu44 [6] A function f : Z → Z satisfies f ( x + 4) − f ( x ) = 8 x + 20 2 2 2 f ( x − 1) = ( f ( x ) − x ) + x − 2 (1) Find f (0) and f (1). School: Problem Gu45 [7] ( ) x Find all positive integers x for which there exists a positive integer y such that = 1999000 y Problem Gu46 [6] 2 n For what integer values of n is 1 + n + n / 2 + ... + n /n ! an integer? School: Problem Gu47 n Find an n < 100 such that n · 2 − 1 is prime. Score will be n − 5 for correct n , 5 − n for incorrect n (0 points for answer < 5). 8