Gauss School and Gauss Math Circle 2014 Gauss Math Tournament Grade 7-8

1. 
   A full bottle of milk weighs 32 ounces and a half-full bottle of milk weighs 19 ounces. What is the weight, in ounces, of an empty bottle?
   

2. 
   Dena and Tom each took three math tests. The median of Dena’s three scores is the same as Tom’s. Their lowest scores are also the same. However, Dena’s mean is 2 points higher than Tom’s mean. How many points higher is Dena’s best test score than Tom’s?
   

3. 
   If 4 daps are equivalent to 3 dops, and 2 dops are equivalent to 7 dips, how many daps are equivalent to 42 dips?
   

4. 
   A student added up the first 100 positive integers. However, instead of getting 5050 like he was supposed to, he accidentally switched the two digits of one of the integers and got 5068 instead. What is the largest possible integer whose digits could have been switched to get this erroneous result?
   

5. 
   A group of 20 students planned to share equally the total expense for renting a bus. When 4 students drop out, each remaining student’s share of the expense increased by m%. What is the value of m?
   

6. 
   The equation 4x + 6y = 128 has many solutions. If x and y are positive integers, what value of x is associated with the smallest positive integer value of y-x?
   

7. 
   What is the simplified numerical value of (a+11b)/(a-b) if (4a+3b)/(a-2b) = 5?
   

8. 
   Running at a constant speed, Andy can run around a circular track once in 40 seconds. Bob, running in the opposite direction, slaps hands with Andy every time they run past each other. If they only slap hands every 15 seconds, how many seconds does it take Bob to run around the track once?
   

9. 
   ABCD is a rectangle, Z is a point on AD and W is a point on BC. ZW and BD intersect at Q. AZ=WC=6, AB=12 and the area of trapezoid ZWCD is 120. What is the area of triangle BQW?
   

10. 
    A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction.
    

11. 
    How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4?
    

12. 
    The marching band has more than 100 members but fewer than 200 members. When they line up in rows of 4 there is one extra person; when they line up in rows of 5 there are two extra people; and when they line up in rows of 7 there are three extra people. How many members are in the marching band?
    

13. 
    Yesterday Teddy played a game repeatedly on his computer and won exactly 85% of the games he played. Today, he played the same game repeatedly and won every game he played until his over-all winning percentage for yesterday and today was exactly 94%. What is the minimum number of games that he played today?
    

14. 
    In the coordinate plane, points A (-4, 3), B (0, 6) and C (2, -2) make a triangle. What is the area of the triangle ABC?
    

15. 
    A car travels the 120 miles from A to B at 60 miles per hour, and then returns to A on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car travelling back from B to A?
    

16. 
    A group of juniors and seniors took a test. Exactly 3/5 of the juniors and exactly 6/7 of the seniors passed the test. If the number of juniors who passed the test was 2/3 of the number of seniors who passed the test, what fraction of the entire group passed the test?
    

17. 
    How many even divisors does 7! have?
    

18. 
    Jack bought a new cape. The cape was reduced from the original price by 20% and then a 5% sales tax was added. If he received $14.72 in change from a $50 bill, what was the original price of the cape?
    

19. 
    The volume of a cube is equal to six times the sum of the lengths of its edges. What is the volume of the cube?
    

20. 
    In numbering the pages of a book from 1 to 576, how many times is the digit 3 used?
    

21. 
    What is the value of x + y + z when 6x + 5y – 2z = -4 and 4x + 5y + 12z = 54?
    

22. 
    The value of the numerator of a particular fraction is one-half the value of its denominator. If the numerator is increased by 2 and the denominator is decreased by 2, the value of the resulting fraction is ¾. What is the denominator of the original fraction?
    

23. 
    What is the probability of rolling six standard, six-sided dice and getting six distinct numbers? Express as a common fraction.
    

24. 
    The product of three consecutive, odd positive integers is seven times their sum. What is the sum of the three integers?
    

25. 
    How many positive, three-digit integers contain at least one 3 as a digit but do not contain a 5 as a digit?
    

26. 
    The GCF (72, x) = 24 and the GCF (64, x) = 32. What is the smallest possible positive value of x?
    

27. 
    How many distinct equilateral triangles with integral side lengths have a perimeter less than 200?
    

28. 
    The positive integers A, B and C form an arithmetic sequence while the integers B, C and D form a geometric sequence. If C/B = 5/3, what is the smallest possible value of A + B + C + D?
    

29. 
    For how many positive values of n are both n/3 and 3n four-digit integers?
    

30. 
    The mean of the seven numbers, 2x-4, 4x-3, -13, 9, 5x+2 and x-2 is 4. What is the integer value of the mode of this set of numbers?
    

31. 
    The 80^th term of an arithmetic sequence is twice the 30^th term. If the first term of the sequence is 7, what is the 40^th term?
    

32. 
    The numbers a, b, c and d form a geometric sequence, in that order. If b is three more than a, and c is nine more than b, what is the value of d?
    

33. 
    The function f is defined by f(n) = f(n+1) + f(n+2) with f(1) = 0 and f(3) = 4. What is the value of f(4)?
    

34. 
    The surface area of a rectangular prism is 32 square inches. The volume of the prism is 12 cubic inches. The sum of all edge lengths is 28 inches. If the length, width and height of the prism each are increased by 1 inch, what is the volume of the new prism?
    

35. 
    If the degree measure of an arc of a circle in increased by 20% and the radius of the circle in increased by 25%, by what percent does the length of the arc increased?
    

36. 
    A rectangle is divided into two congruent trapezoids. If both trapezoids have legs of lengths 4 and 5, and both trapezoids have a shorter base of length 3, what is the area of the rectangle?
    

37. 
    What is the only negative integer x for which I x + 2 I = I x + 4 I?
    

38. 
    How many integers n > 0 satisfy the inequality 3/65 < 1/n < 9/100?
    

39. 
    The length of each side of rectangle R is an integer, and the area of R is 2009. What is the largest possible perimeter of R?
    

40. 
    For what integer n are the roots of x^2 – 7x + n = 0 consecutive integers?
    

1. 
   1. A 30-60-90 triangle is drawn on the exterior of an equilateral triangle so that the hypotenuse of the right triangle is one side of the equilateral triangle. If the shorter leg of the triangle is 6 units, what is the distance between the two vertices that the triangles do not have in common? Express your answer in simplest radical form.
   

2. 
   Points X and Y lie on sides AB and AC respectively of triangle ABC. If AB=7, AC=10, AX=4 and AY=6, what is the ratio of the area of triangle AXY to the area of triangle ABC? Express as a common fraction.
   

3. 
   How many positive three-digit integers are there in which each digit is exactly one more or one less than the digits next to it?
   

4. 
   Let A be a vertex of a unit cube and let B, C, D be the vertices adjacent to A. The tetrahedron is cut off the cube. What is the surface area of the remaining solid?
   

5. 
   A circle centered at (x, y) passes through the points (-2, 0), (0, 2), and (8, 0). What are the coordinates of the center of the circle? Express as an ordered pair.
   

6. 
   What is the number of square units in the region satisfying IxI + IyI 9 and -5 x 4?
   

7. 
   A point P is randomly placed in the interior of the right triangle ABC (C is the right angle). What is the probability that the area of triangle PBC is less than half of the area of triangle ABC?
   

8. 
   There are two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the region between two circles?
   

Name: ______________________________ Grade:___________________________________

1. 
   6
   
2. 
   6
   
3. 
   16
   
4. 
   79
   
5. 
   25%
   
6. 
   11
   
7. 
   2
   
8. 
   24
   
9. 
   42
   
10. 
    4/3
    
11. 
    42
    
12. 
    157
    
13. 
    30
    
14. 
    19
    
15. 
    36
    
16. 
    30/41
    
17. 
    48
    
18. 
    42
    
19. 
    432
    
20. 
    218
    
21. 
    5
    
22. 
    14
    
23. 
    5/324
    
24. 
    15
    
25. 
    200
    
26. 
    96
    
27. 
    66
    
28. 
    52
    
29. 
    112
    
30. 
    9
    
31. 
    20
    
32. 
    81/2
    
33. 
    -8
    
34. 
    36
    
35. 
    50%
    
36. 
    36
    
37. 
    -3
    
38. 
    10
    
39. 
    4020
    
40. 
    12
    

Target Round:

1. 
   6
   
2. 
   12/35
   
3. 
   32
   
4. 
   (9+)/2
   
5. 
   (3, -3)
   
6. 
   121
   
7. 
   ¾
   
8. 
   1600