PAGE 2 - PART I HONORS MATH CONTEST 2008
1. There is a triangle garden T with sides 15, 14 and 14 feet. A curve C is to go around the garden. Each point of C is to be exactly 3.5 feet away from the nearest point of T. What is the approximate length of C in feet?
(A) 43
(B) 50
(C) 57
(D) 65
(E) 72


3. For any positive numbers a and b, consider the binary division operation a÷b. Which one of the following statements is true?
(A) If a,b, and c are positive numbers, then (a÷b)÷c=a÷(b÷c)
(B) If a,b, and c are positive numbers, then (ac)÷b=a÷(bc)
(C) If a and b are positive numbers, then a÷b=b÷a
(D) If a is a positive number, then there exists a positive number e such that a÷e=e÷a=a
(E) If a is a positive number, then there exists a positive number e such that a÷e=e÷a=1

PAGE 3 OF 12 - PART I HONORS MATH CONTEST 2008
4. A multiplication problem was done in a binary aritmetic. The digits have faded away, but the locations are still visible. The following is all that remains:
   ***
   ***
   ---
   ***
***
------
******
What was the arithmetic problem expressed in our usual base-10 (decimal) system?
(A) 7×7
(B) 7×6
(C) 7×5
(D) 6×6

回复：6楼
第四行***的地方最前边有一空格，被百度删掉了……

6. Consider the values of the following expressions for n, a positive integer. Which expression always gives a prime number?
(A)n²-n+41
(B)n²+n+17
(C)n²+n+91
(D)n²+n+41
(E)None of the above

PAGE 4 OF 12 - PART I HONORS MATH CONTEST 2008
7. Consider the following elementary sum of squares with alternating signs: 100²-99²+98²-97²+...+2²-1². What is the value of the sum?
(A) 10,100
(B) 5,050
(C) 5,000
(D) 4,950
(E) 2,475


PAGE 5 OF 12 - PART I HONORS MATH CONTEST 2008
10. There is a circle of radius 29 feet. In the same plane, there is a circle of radious 10 feet. Since the centers of the circles are 11 feet apart, the second circle is inside the first. Let R be the set of points between the circles together with the points on the two circles. If a straight-line interval is sufficiently short, it can be moved around the smaller circle while it stays in the region R. What is the largest number of feet long that the interval can be and still have the property that it can be so moved around the inner circle?

11. From a large set of positive integers, an integer is selected randomly. The probability that the integer a selected is odd is 1/3. The procedure is repeated two more times to obtain b and c. What is the probability that the product abc is odd?
(A) 1/27
(B) 7/18
(C) 2/3
(D) 4/9
(E) 13/27


12. A woman wants to tile the foyer with two types of tiles. One type has the shape of a square and the other type has the shape of a certain regular polygon. At each vertex of a square tile, there should be three tiles coming together: one square and two of the second type. Of course, the tiles are to cover the flat surface and they must not overlap. What is the number of sides of the polygonal boundary of the seond type of tile?
(A) 3
(B) 5
(C) 6
(D) 7
(E) 8


PAGE 6 OF 12 - PART I HONORS MATH CONTEST 2008
13. Consider a cube that is one unit on each edge. For a chosen vertex, there are three edges of the cube going out from that vertex. On each of those edges, there is a point half a unit from the chosen vertex. A single plane goes through those three points. Remove the part of the cube on the side of the plane containing the chosen vertex. Each vertex of the cube can be treated in similar fashion. The remaining solid is the solid S. The surface of S consists of squares and equilateral triangles. What are the numbers of each?
(A) 4 squares, 8 triangles
(B) 6 squares, 6 triangles
(C) 6 squares, 8 triangles
(D) 8 squares, 8 triangles
(E) 8 squares, 6 triangles


14. The following proposal is made for a game show:
Three identical boxes will be on stage. A randomly assigned box will have $300,000, and the other two will have no money. The contestant will make a tentative choice of a box. The host will then announce (truthfully) that he knows an empty unselected box, and he will then open such a box to demonstrate that is is empty. The contestant will then be allowed to make a final selection of a box. The contestant wins the $300,000 if the money is in the final selection. Assume that the contestants are honest and smart.
What will be the average payout for each game?
(A) $300,000
(B) $200,000
(C) $150,000
(D) $125,000
(E) $100,000


16. Each letter, A, B through G, represents a different digit. Consider the following exace calculation of a square root.
　　　Ａ　Ｂ　Ｃ
　　　－－－－－
　　√ＢＤＢＤＢ
　　　Ｃ
　　　－
ＣＢ／ＡＤＢ
　　　ＡＥＢ
　　　－－－
ＧＡＣ／ＡＦＤＢ
　　　　ＡＦＤＢ
　　　　－－－－
　　　　　　　０
What is the value of ABC?
(A)264
(B)426
(C)462
(D)624
(E)642


PAGE 7 OF 12 - PART I HONORS MATH CONTEST 2008
17. For this problem, model the earth as a sphere of radius 6,380 kilometers. Estimate the air pressure at sea level as 10,070 kilograms per square meter. What is the approximate weight of the earth's atmosphere in kilograms? [The surface area of a sphere of radius r is equal to 4πr²]
(A) 5.1×10^6
(B) 5.1×10^10
(C) 5.1×10^12
(D) 5.1×10^18
(E) 5.1×10^24

18. Let x=√(11+6√(2))+√(11-6√(2)). What is the strongest true statement that can be made about the number x?
(A) x is an integer
(B) x is rational
(C) x is 6
(D) x is irrational
(E) x satisfies a quadratic equation with integer coefficients