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Gauss School and Gauss Math Circle
2014 Gauss Math Tournament
Grade 5-6 (Sprint Round 50 minutes)
1. What is the smallest possible product of three distinct members of the set
(-1, -2, -3, 4, 5)?
10𝑥−3𝑦
3
2. What is the ratio of x to y if: 13𝑥−2𝑦 = 5 ? Express your answer as a common
fraction.
3. Compute 1-2+3-4+5-6+….+99-100?
4. A rectangle has an area of 100 square units. Its dimensions are whole units.
What is the number of units in the greatest possible perimeter?
5. Warren buys limes at a cost of three for 50 cents. He then sells them at a
price of five for 90 cents. How many limes must he sell to make a total profit
of $2?
6. Solve for x: 2𝑥−3 = 4𝑥+1
7. The digits 2,3,5,7 are arranged randomly to form a four-digit number. What
is the probability that the number is odd? Express your answer as a common
fraction.
8. How many factors of 25 × 36 are perfect squares?
9. School uniform parts are on sale. The $25 pair of slacks can be purchased at a
15% discount, and the $18 shirt can be purchased at a 20% discount. What is
the total number of dollars spent to purchase 1 pair of slacks and 1 shirt
during the sale? Express your answer to the nearest hundredth.
10. What is the least perfect square with 3 different prime factors?
11. Two integers have a sum of -21 and a product of 54. What is the smaller of
these two integers?
12. A number is defined as a falling number if each digit in the number is less
than the digit to its left. What is the least falling number that is greater than
97543?
13. Hillary has 11 coins, all dimes and nickels. In total, the coins are worth 75
cents. How many nickels does she have?
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14. What is the probability t hat a randomly-selected two-digit number is a
multiple of 14? Express your answer as a common fraction.
15. Three congruent squares are placed side-by-side to form a rectangle with a
perimeter of 104 inches. What is the number of square inches in the area of
one of the original squares?
16. A jar has 8 red, 9 blue and 10 green marbles. What is the least number of
marbles you can remove from the jar to guarantee that you have three of the
same color?
17. In Mathland, the area codes consist of three digits such that the first is a digit
from 2 to 9, inclusive. The second is either 0 or 1, and the third is a digit other
than 0. How many area codes are possible?
18. The average age of a group of doctors and lawyers is 40. The average age of
the doctors is 35, and the average age of the lawyers is 50. What is the ratio
of the number of lawyers to the number of doctors? Express your answer as a
common fraction.
19. The length of a diagonal of a rectangle is 10cm. What is the number of square
centimeters in the maximum possible area of this rectangle?
20. When n is divided by 3, the remainder is 2. What is the remainder when 5n is
divided by 3?
21. The perfect squares 1, 4, 9, 16, 25, …. are put into a string of digits 1491625….
What is the 40th digit in the string?
22. Let 2x+3y=72. How many ordered pairs (x, y) where x and y are whole
numbers, will satisfy the equation?
23. The hour hand of a clock is 6 inches long and the minute hand is 8 inches
long. What is the ratio of the distance in inches traveled by the tip of the hour
hand to the distance in inches traveled y the tip of the minute hand from
noon to 3 p.m.? Express your answer as a common fraction.
24. What is the ratio of the area of a square inscribed in a semicircle with radius
r to the area of a square inscribed in a circle with radius r? Express your
answer as a common fraction.
25. Two distinct circles and a square lie in the plane. What is the maximum
number of points of intersection of two or more of these three figures?
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26. A cube is painted so that one side is blue, two sides are red and three sides
are green. How many different such cubes can be painted? Two cubes are
considered the same if one cube can be rotated in any way to match the
second cube.
27. Forty-three percent of Americans have Type A molecules in their blood, 15%
have type B molecules and 46% have neither Type A nor Type B molecules.
What percent of Americans have both Type A and B molecules in their blood?
28. Each successive term of the sequence 1, 3, 7, 15, 31, … is obtained by adding
one to twice the previous term. The 10th term is 1023. What is the value of
the 9th term?
29. The side length of a triangle are 15 cm, 47 cm and x/2 cm. How many integer
values of x are possible?
30. A rectangular block of candy 10 inches by 10 inches by 5 inches is coated on
all faces with a very thin layer of chocolate. The block of candy is then cut
into cubes measuring 1 inch by 1 inch by 1 inch. What percent of the cubes
have no chocolate on them? Express your answer to the nearest hundredth.
31. The time right now is 9 o’clock. On a 12-hour clock, what time will it be 2004
hours from now?
32. How many liters of water must be added to 40 liters of a 40% salt solution in
order to produce a 25% salt solution?
33. Matt has exactly $1.60 in his pocket. He only has nickels and dimes. There are
twice as many nickels as dimes. What is the number of coins in Matt’s
pocket?
34. A square is divided into five congruent rectangles. If the perimeter of each of
these five rectangles is 36 inches, what is the perimeter of the square, in
inches?
35. The four-digit number “25AB” is divisible by 9, with A the tens digit and B the
units digit. How many different such four-digit numbers could “25AB”
represent?
36. In a bag with 20 marbles, five of them are blue. How many blue marbles must
be added to the bag so that the probability of selecting a blue marble at
random is ½?
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37. Brass is an alloy created using 80% cooper and 20% zinc. If Henri’s brass
trumpet contains 48 ounces of cooper, how many ounces of zinc are in the
trumpet?
38. When the two-digit integer MM with equal digits, is multiplied by the onedigit integer M, the result is the three-digit integer NPM. What is the greatest
possible value of NPM?
39. A triangle has a side of length 6cm, a side of length 8cm and a right angle.
What is the shortest possible length of the remaining side of the triangle?
40. A set of four consecutive integers has a sum of 22. If each integer of the set is
increased by 2 and then multiplied by 20, what is the sum of the new set of
integers?
Sprint Round Ends
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Gauss School and Gauss Math Circle
2014 Gauss Math Tournament
Grade 5-6 (Target Round 20 minutes)
1. In how many zeros does the decimal representation of the number 2002!
end? (2002! = 2002× 2001 × 2000 × …… × 3 × 2 × 1)
2. Quadrilateral ABCD is a trapezoid with AB parallel to CD. We know AB=20
and CD=12. What is the ratio of the area of triangle ACB to the area of the
trapezoid ABCD? Express as a common fraction.
3. In a particular sequence, each term after the first two terms is the sum of the
two preceding terms. If the first term is 20 and the sixth term is 220, what is
the value of the seventh term?
4. One interior angle of a convex polygon is 160 degrees. The rest of the interior
angles of the polygon are each 112 degrees. How many sides does the
polygon have?
5. Kristin has 5 different pairs of earrings. What is the probability that two
randomly-selected earrings will be a matching pair? Express your answer as
a common fraction.
6. What is the positive integer N for which 222 × 552 = 102 × 𝑁 2 ?
7. A running track is the ring formed by two concentric circles. If the
circumferences of the two circles differ by 10𝜋 feet, how wide is the track, in
feet?
8. One complete lap around a particular circular track is 400 meters. Jun and
Quan each start running at the starting line and run around the track; Jun
runs clockwise at 3 meters per second, and Quan runs counterclockwise at 5
meters per second. When they meet for the sixth time after starting, they
stop and both walk back together along the track to the starting line. What is
the shortest distance they could walk back on the track together?
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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Answer Keys:
Sprint Round:
1. -60
2. 9/11
3. -50
4. 202
5. 150
6. -5
7. ¾
8. 12
9. 35.65
10. 900
11. -18
12. 97610
13. 7
14. 7/90
15. 169
16. 7
17. 144
18. ½
19. 50
20. 1
21. 3
22. 13
23. 1/16
24. 2/5
25. 18
26. 3
27. 4%
28. 511
29. 59
30. 38.4%
31. 9
32. 24
33. 24
34. 60
35. 11
36. 10
37. 12
38. 396
39. 2√7
40. 600
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Target Round:
1.
2.
3.
4.
5.
6.
7.
8.
499
5/8
356
6
1/9
121
5
100
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