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Gauss School and Gauss Math Circle
2016 Gauss Math Tournament
Grade 5-6 (Sprint Round 50 minutes)
1. What is (1003)(997)-(1008)(992)?
2. Solve the system of equations to get (x, y)
a. 3x+7y=38
b. x+10y=44
3. a/b=2, b/c=3. Find (a+c)/b.
4. Bob has a picture of dimensions 6 inch by 10 inch to which he has a ½ inch
border made. What is the ratio between the area of the border and the area
of the picture?
5. A plane makes a round trip from the East Pole to the West Pole. However,
the whole time, there is a constant strong wind going from east to west,
making the trip from east to west take 10 hours, and the trip west to east
take 12 hours. If the West Pole is 21,000 miles away from the East Pole and
the plane’s speed is constant, find the speed of the plane in mph.
6. What is x/y if (9x-5y)/(3x+11y) = ⅓
7. The sum of the squares of two numbers is 205 and their product is 78. Find
the larger of the 2 numbers.
8. Jane starts with 720 cookies and then gives ⅕ of them to Tiffany. Jane then
gives ⅓ of the remaining cookies to Lily and splits the rest with her sister.
How many cookies does Jane end up with?
9. In the school of West Pageborough, there are 253 students, each of which
takes orchestra, ba(n)d, both or none. If there are 163 orchestra students
and 111 band students, and 15 take none, how many students take both
orchestra and band?
10. In circle O with radius 5, minor arc EF measures 2pi, what is the angle
measure of <EOF?
11. Points E and F are the midpoints of AB and CD in rectangle ABCD,
respectively. DAEF is similar to ABCD. If AB =16, what is the area of ABCD?
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12. What is the side length of a cube whose volume is numerically 3 times its
surface area?
13. What is the sum of the external angles of a regular pentagon?
14. A rubber band is wrapped tightly around the outside of three tangent circles
with radius 2. What is the length of the rubber band?
15. In triangle ABC, AB=BC=1, and angle B is a right angle. Isosceles right
triangles are constructed on the hypotenuse of the previous triangle. Shown
are the first 3 triangles. How many times larger is the area of the 10th
triangle than the 1st triangle.
16. An icecream cone of radius 6 cm and height 20 cm is filled with liquefied
strawberries until the depth of the liquefied strawberries is ½ of the height
of the cone. What is the volume of the liquefied strawberries?
17. How many ways are there to rearrange the letters of the word GAUSS?
18. David is on the coordinate plane, and can only move 1 unit at a time in the
positive x or y directions. How many ways are there for David to get from (0,
0) to (3, 5)?
19. How many triangles are in the following diagram?
20. What is the chance of drawing two red marbles from a bag of 8 marbles with
5 red marbles?
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21. How many 3 digit numbers satisfy the condition that the tens digit is the
largest, and the hundreds digit is smaller than the ones digit?
22. License plates are made by 3 digits (0 through 9) followed by 2 letters, but
the letters cannot both be consonants (letters other than “a, e, i, o, or u”).
How many different license plates can be made?
23. A circle with is inscribed inside an equilateral triangle with side length
6in. What is the probability that a random point chosen inside the triangle is
within the circle?
24. What is the probability that when you roll 2 dice, the sum of the numbers you
roll is 8?
25. From a gumball machine, there is a 20% chance you will roll a green
gumball. If you roll the gumball machine 5 times, what is the chance you will
get exactly 2 green gumballs?
26. In Matthew’s wardrobe, there are 5 different T shirts, 3 different pairs of
pants and 4 different pairs of socks, and 2 types of Gauss makeup. How many
outfits consisting of one T shirt, two pairs of pants and one pair of socks, and
one application of makeup can Matthew make?
27. Two points C and D are selected from a line segment AB with C between A
and D, and D between C and B. What is the probability that AC+DB>CD.
28. March 6th, 2019 is a Wednesday. What day of the week will it be 192 days
from then?
29. Find the number of divisors of 2016.
30. How many integers between 1 and 50 inclusive are divisible by 2 or 3 (but
not both)?
31. How many two-digit numbers with 6 divisors are there?
32. How many different ordered pairs (A, B) are there such that A and B are
digits and make 12A73B divisible by 11?
33. What is the last digit of 13332016 ?
34. A number with 34 factors is squared. How many factors does this new
number have?
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35. A man who loves chocolate buys 100lbs of it. Every day, he eats 2/7 of the
chocolate. However, when there is less than 3 pounds of chocolate
remaining, he will eat all of it. How many days will it take him to finish the
chocolate?
36. A ball is dropped vertically from 40 feet. When the ball bounces, it only
bounces up to ½ the height of the previous bounce. How many feet would the
ball have traveled after hitting the ground for the 3rd time?
37. 4 lines and a circle are drawn on a plane. What is the maximum number of
intersections possible?
38. in triangle ABC with B=90. AD=4, DC=9, the altitude from B to AC intersects
AC at D. What is length of BD?
39. Let r and s be the roots of the equation
40. If x and y are integers such that
solutions are there for x?
Sprint Round Ends
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. Find r+s?
, how many distinct
Gauss School and Gauss Math Circle
2016 Gauss Math Tournament
Grade 5-6 (Target Round 20 minutes)
1. In triangle ABC with B=90. The altitude from B to AC intersects AC at D AD=4,
DC=9. What is length of BD?
2. David can eat 6 hot dogs in 3 minutes. Lawrence can eat 2 hot dogs in 4
minutes. Rachel can eat 4 hot dogs in 6 minutes. Finally, William can eat 1 hot
dog in 9 minutes. Working together, how many minutes will it take for the
four people to eat 20 hot dogs? Express your answer to the nearest tenth.
3. A classroom has 10 boys and 12 girls. How many ways are there to pick a
group of 4 students such that not all of them are the same gender?
4. How many ways are there to put a red ball, a blue ball, a green ball, and a
yellow ball into 2 indistinguishable boxes?
5. Three faces of a rectangular prism have areas 16, 20, and 45. What is the
volume of this prism?
6. Circles O and P are congruent circles with radius 3 and intersect at points A
and B. If AB = 3, what is the area of the intersection? Express your answer to
the nearest hundredth.
7. How many 8 digit integers are not palindromes? A palindrome is a number
that is the same when reading forwards to backwards as when it is read
backwards to forwards. For example, 14541, 108801, 5, and 104929401 are
all palindromes.
8. A sev-peating number is an integer consisting of only the digit 7. What is the
smallest integer n such that a sev-peating number with n digits is divisible by
133?
Target Round Ends
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Name: ______________________________
Grade:___________________________________
Sprint Round Answers:
1
21
2
22
3
23
4
24
5
25
6
26
7
27
8
28
9
29
10
30
11
31
12
32
13
33
14
34
15
35
16
36
17
37
18
38
19
39
20
40
Target Round Answers:
1
5
2
6
3
7
4
8
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G5-6 Answer Keys:
Sprint Round:
1. 55
2. (10/3, 4)
3. 7/3
4. 1/3
5. 1925
6. 13/12
7. 13
8. 192
9. 36
10. 72
11. 256 root(2)
12. 18
13. 360
14. 12+4pi
15. 1024
16. 30pi
17. 60
18. 56
19. 16
20. 5/14
21. 84
22. 23500
23. pi/3root(3)
24. 5/36
25. 128/536
26. 120
27. ¾
28. Saturday
29. 24
30. 33
31. 16
32. 5
33. 1
34. 99
35. 12
36. 100
37. 14
38. 6
39. -3
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40. 8
Target Keys:
1.
2.
3.
4.
5.
6.
7.
8.
6
5.3
6610
8
120
1.63
89991000
18
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