HMMT 二月 2000 · 冲刺赛

英文原题

Guts
HMMT 2000
School:
Problem Gu 1 [4]
The sum of 3 real numbers is known to be zero. If the sum of their cubes is π , what is theireproduct equal to?
Problem Gu 2 [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 Gu 3 [ ± 7]
Using 3 colors, red, blue and yellow, how many different ways can you color a cube (modulorigid rotations)?
Problem Gu 4 [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 Gu 5 [3]
Find all natural numbers n such that n equals the cube of the sum of its digits.
Problem Gu 6 [ ± 10]
2 2
If integers m, n, k satisfy m + n + 1 = kmn , what values can k have?
School:1
Problem Gu 7 [7]
Suppose you are give a fair coin and a sheet of paper with the polynomial x written on it.mr
Now for each toss of the coin, if heads show up, you much erase the polynomial x (where rr − 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 afterm such tosses? (Note: this is a different concept from the most probable value)
Problem Gu 8 [ ± 4]
Johny’s father tells him : “I am twice as old as you will be seven years from the time I wasthrice as old as you were”. What is Johny’s age?
School:
Problem Gu 9 [6]
A cubic polynomial f satisfies f (0) = 0 , f (1) = 1 , f (2) = 2 , f (3) = 4. What is f (5)?
Problem Gu 10 [7]
What is the total surface area of an ice cream cone, radius R , height H , with a sphericalscoop of ice cream of radius r on top? (Given R < r )
School:
Problem Gu 11 [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 thismaximum is attained. Evaluate M + N .
Problem Gu 12 [9]
Calculate the number of ways of choosing 4 numbers from the set { 1 , 2 , ..., 11 } such that atleast 2 of the numbers are consecutive.
School:2
Problem Gu 13 [ ± 4]
4 2 2
Determine the remainder when ( x − 1)( x − 1) is divided by 1 + x + x .
Problem Gu 14 [7]
ABCD is a cyclic quadrilateral inscribed in a circle of radius 5, with AB = 6 , BC = 7 , CD =
8. Find AD .
School:
Problem Gu 15 [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 ineach row and each column is odd.
Problem Gu 16 [5]
Solve for real x, y :
x + y = 2
5 5
x + y = 82
School:
Problem Gu 17 [5]
( )
Find the highest power of 3 dividing 666
Problem Gu 18 [ ± 5]333
∞
√
√
∑
− 1 − 1
What is the value of (tan n − tan n + 1)?
n =1
School:3
Problem Gu 19 [3]
a − b
Define a ∗ b = . What is (1 ∗ (2 ∗ (3 ∗ ... ( n ∗ ( n + 1)) ... )))?
1 − ab
Problem Gu 20 [6]
What is the minimum possible perimeter of a triangle two of whose sides are along the x-andy-axes and such that the third contains the point (1,2)?
School:
Problem Gu 21 [8]
How many ways can you color a necklace of 7 beads with 4 colors so that no two adjacentbeads have the same color?
Problem Gu 22 [6]
2000
Find the smallest n such that 2 divides n !
School:
Problem Gu 23 [5]
How many 7-digit numbers with distinct digits can be made that are divisible by 3?
Problem Gu 24 [ ± 3]
At least how many moves must a knight make to get from one corner of a chessboard to theopposite corner?
School:4
Problem Gu 25 [4]
Find the next number in the sequence 131, 111311, 311321, 1321131211,
Problem Gu 26 [5]
What are the last 3 digits of 1! + 2! + ... + 100!
School:
Problem Gu 27 [ ± 6]
What is the smallest number that can be written as a sum of 2 squares in 3 ways?
Problem Gu 28 [8]
What is the smallest possible volume to surface ratio of a solid cone with height = 1 unit?
School:
Problem Gu 29 [ ± 9]
√
√
√
√
√
√
√
√
What is the value of 1 + 2 1 + 3 1 + 4 1 + 5 1 + ...
Problem Gu 30 [7]
6 6
ABCD is a unit square. If P AC = P CD , find the length BP .
School:5
Problem Gu 31 [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 measuredistances)
Problem Gu 32 [7]
How many (nondegenerate) tetrahedrons can be formed from the vertices of an n -dimensionalhypercube?
School:
Problem Gu 33 [ ± 5]
Characterise all numbers that cannot be written as a sum of 1 or more consecutive oddnumbers.
Problem Gu 34 [ ± 6]
What is the largest n such that n ! + 1 is a square?
School:
Problem Gu 35 [4]
If 1 + 2 x + 3 x + ... = 9, find x .2
Problem Gu 36 [6]
b + c − a
If, in a triangle of sides a, b, c , the incircle has radius , what is the magnitude of angle
A ?2
School:6
Problem Gu 37 [9]
°
A cone with semivertical angle 30 is half filled with water. What is the angle it must betilted by so that water starts spilling?
Problem Gu 38 [4]
What is the largest number you can write with three 3’s and three 8’s, using only symbols +,-,/, × and exponentiation?
School:
Problem Gu 39 [ ± 8]
If r = 1 / 3, what is the value, rounded to 100 decimal digits, ofn 7
∑
n 2
1 + x 2
n =0
Problem Gu 40 [ ± 10]
Let φ ( n ) denote the number of positive integers less than equal to n and relatively prime ton . find all natural numbers n and primes p such that φ ( n ) = φ ( np ).
School:
Problem Gu 41 [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 bcauses to increase to 3 α . Find h in terms of a and b .
Problem Gu 42 [4]
A n × n magic square contains numbers from 1 to n such that the sum of every row and 2
every column is the same. What is this sum?
School:7
Problem Gu 43 [6]
Box A contains 3 black and 4 blue marbles. Box B has 7 black and 1 blue, whereas Box Chas 2 black, 3 blue and 1 green marble. I close my eyes and pick two marbles from 2 differentboxes. If it turns out that I get 1 black and 1 blue marble, what is the probability that theblack marble is from box A and the blue one is from C?
Problem Gu 44 [6]
A function f : Z → Z satisfiesf ( 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 Gu 45 [7]
( )
Find all positive integers x for which there exists a positive integer y such that = 1999000 xy
Problem Gu 46 [6]
2 n
For what integer values of n is 1 + n + n / 2 + ... + n /n ! an integer?
School:
Problem Gu 47
Find an n < 100 such that n · 2 − 1 is prime. Score will be n − 5 for correct n , 5 − n fornincorrect n (0 points for answer < 5). 8