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Gauss School and Gauss Math Circle
2016 Gauss Math Tournament
Grade 7-8 (Sprint Round 50 minutes)
1. Euler solves 1/7 of a problem in 7 seconds. Euclid solves ⅕ of a problem in
10 seconds. How long in seconds does it take both of them working together
to solve 99 problems?
2. Construct medians AD, BE, and CF of triangle ABC and have their intersection
be G. Let the midpoint of AG be X, the midpoint of EG be Y, and the midpoint
of FG be Z. If the area of XYZ is 1, what is the area of ABC?
3. There are six blank fish drawn in a line on a piece of paper. Lucy wants to
color them all either red or blue, but she refuses to color two adjacent fish
red. How many ways can she do this?
4. Compute the number of ordered pairs (a,b) of positive integers less than or
equal to 100, such that 𝑎𝑏 − 1 is a multiple of 4.
5. What is the sum of the first 10 even integers?
6. There are currently 175 problems submitted for the Gauss Math
Tournament. Chris has submitted 51 of them. Given that nobody else submits
any more problems, how many more problems must Chris submit so that he
has submitted ⅓ of the problems?
7. A circle with center O and radius 1 intersects segment AB of square ABCD at
points E and F, such that the arc EF that goes through the interior of the
square measures 120°. If ABCD has side length 2, what is the area inside
ABCD but outside of the circle?
8. The planning committee at school has 10 members. Exactly four of these
members are teachers. A four-person subcommittee with at least one
member who is a teacher must be formed from the members of the planning
committee. How many distinct subcommittees are possible?
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9. How many different triangles have sides whose lengths are integers if the
longest length is 6?
10. Find the remainder when 20112011 is divided by 7.
11. What is the positive square root of the product 10 * 24 * 15?
12. A man born in the first half of the 1800s was x years old in the year x2. What
year was he born in?
13. Three rooks are arranged randomly on an 8x8 chessboard. What is the
probability that none of the rooks are attacking each other (two rooks are
attacking each other if they share the same row or column)?
14. What is the units digit of 2323?
15. Point P lies inside triangle ABC such that ∠PBC=30° and ∠PAC=20°. If angle
APB is a right angle, find the measure of ∠BCA in degrees.
16. A pool table is 2 units long and 1 unit wide. If the ball starts .4 units from the
left side on the bottom side, which is 1 unit in length, the ball must be hit at
an angle 𝜃 going counterclockwise relative to the horizontal so that the ball
ends up in the bottom left pocket after exactly two bounces. Express 𝑡𝑎𝑛𝜃 in
terms of p/q, where p and q are coprime, positive integers.
17. Find all prime numbers n such that 𝑛2 − 1 is also prime.
18. There are 30 mathletes in the Euclid Math League. How many ways are there
to choose 4 mathletes to make a team if James and Lucas hate each other and
refuse to be on the team together?
19. Positive integer n has remainder 1 when divided by 2, remainder 2 when
divided by 3, remainder 3 when divided by 4, remainder 4 when divided by
5, remainder 5 when divided by 6, remainder 6 when divided by 7, and
remainder 7 when divided by 8. What is the sum of the digits of n?
20. Joey writes down the numbers 1 through 10. He randomly crosses one
number out and sums the remaining numbers. What is the probability that
this sum is less than or equal to 47?
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21. For a positive integer n, let sn be the sum of the n smallest prime numbers.
Find the least n such that sn is a perfect square.
22. What is the minimum value of x2+8x for all real values of x?
23. A and B together can do a job in 2 days, B and C can do it in 4 days, and A and
C can do it in 2.4 days. How many days does A need to do the job alone?
24. Triangle ABC has BC=14, CA=15, and AB=13. The altitude from A is extended
to meet the circumcircle of ABC at P. What is the distance from P to BC?
25. The perimeter of an isosceles right triangle is 18. What is its area?
𝑏
26. If each letter in the expression (𝑎 + 𝑐 )(𝑑 + 𝑒) is replaced by a different digit
from 1 through 9, inclusive, what is the smallest possible integer value of the
expression?
27. What is the ratio of the area of a square inscribed in a semicircle of radius r
to the area of a square inscribed in a circle of radius r?
28. For how many different positive integers n does √𝑛 differ from √100 by less
than 1?
29. Convex polygons P1 and P2 are drawn in the same plane with n1 and n2 sides,
respectively, where n1≤n2. If P1 and P2 do not have any line segment in
common, then find the maximum number of intersections of P1 and P2 in
terms of n1 and n2
30. Let p(x) = x2 + bx + c, where b and c are integers. If p(x) is a factor of both x4 +
6x2 + 25 and 3x4 + 4x2 + 28x + 5, what is p(1)?
31. Two of the altitudes of the scalene triangle ABC have length 4 and 12. If the
length of the third altitude is also an integer, what is the biggest it can be?
32. What is the least positive six-digit integer with distinct digits that is divisible
by 11?
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33. The roots of the equation 𝑥 3 − 24𝑥 2 + 183𝑥 − 440 = 0 are the first three
terms of an increasing arithmetic sequence. Find the 10th term of this
sequence.
34. In triangle ABC, AB=5, BC=7, AC=9, and D is on line AC with BD=5. Find
AD/DC.
35. Positive integer K has 24 factors, 2K has 30 factors, and 3K has 32 factors.
How many factors does 𝐾 2 have?
36. If ⌊x⌋ is the greatest integer less than or equal to x, then find ∑1024
𝑁=1 ⌊𝑙𝑜𝑔 2 𝑁⌋.
37. Six blocks are stacked on top of each other to create a pyramid, as shown
below. Carl removes blocks one at a time until all the blocks are removed. He
never removes a block until all the blocks that rest on top of it have been
removed. In how many different orders can Carl remove the blocks?
38. In quadrilateral ABCD, it is given that ∠A=120o, angles B and D are right
angles, AB=13, and AD=46. Find AC.
39. In a certain sequence of numbers, the first number is 1, and for all n > 1, the
product of the first n numbers in the sequence is n2. Find the 20th number in
the sequence.
40. What is the maximum integer n such that 3n is a factor of the product of all
the odd integers between 1 and 200?
Sprint Round Ends
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Gauss School and Gauss Math Circle
2016 Gauss Math Tournament
Grade 7-8 (Target Round 20 minutes)
1. The perimeter of a sector of a circle is the sum of the two sides formed by the
radii and the length of the included arc. A sector of a particular circle has a
perimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this
sector in cm?
2. A classroom has 10 desks in two rows. Anne and Ben, Charlie and Darla, Eddie
and Fiona, Gina and Hank, and Isabella and Jack are five boyfriend/girlfriend
pairs. In how many ways can the 10 students be seated if the two people in each
boyfriend/girlfriend pair always sit in adjacent seats?
3. Find all ordered pairs (x,y) of real numbers such that −𝑥 2 + 3𝑦 2 − 5𝑥 + 7𝑦 +
4 = 0 and 2𝑥 2 − 2𝑦 2 − 𝑥 + 𝑦 + 21 = 0.
4. Let T be a positive integer whose only digits are 0s and 1s. If X=T/12 and X is an
integer, what is the smallest possible value of X?
5. What is the least prime number that has 9 digits in base 2, 4 digits in base 7, and 3
digits in base 9?
6. The hypotenuse of a right triangle is 10 inches and the radius of the inscribed
circle is 1 inch. What is the perimeter of the triangle, in inches?
1
1
7. Let 𝑓(𝑥) = √2𝑥 + 1 + 2√𝑥 2 + 𝑥. Determine the value of 𝑓(1) + 𝑓(2) +
1
𝑓(3)
1
+. . . + 𝑓(24).
8. How many ways are there to rearrange the letters in the word MATHEMATICS
such that no three consonants are adjacent?
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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G7-8 Answers Keys:
Sprint Round
1. 2450
2. 16
3. 21
4. 3750
5. 110
6. 11
𝜋
7. 4 − 3 +
√3
4
8. 195
9. 11
10. 2
11. 60
12. 1806
13. 14/31
14. 7
15. 40
16. 5/2
17. 2
18. 27027
19. 20
20. 3/10
21. 9
22. -16
23. 3
24. 15/4
25. 81(3 − 2√2) OR 243 − 162√2
26. 8
27. ⅖
28. 39
29. 2n1
30. 4
31. 5
32. 102465
33. 32
34. 19/8
35. 84
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36. 8204
37. 16
38. 62
39. 400/361
40. 49
Target Round
1. 14
2. 30720
3. (3, -4)
4. 925
5. 347
6. 22
7. 4
8. 453600
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