* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download File
Survey
Document related concepts
Mathematics of radio engineering wikipedia , lookup
Law of large numbers wikipedia , lookup
Large numbers wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Positional notation wikipedia , lookup
Collatz conjecture wikipedia , lookup
Location arithmetic wikipedia , lookup
Weber problem wikipedia , lookup
Approximations of π wikipedia , lookup
Transcript
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? www.gaussmath.org 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? www.gaussmath.org 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? www.gaussmath.org 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 www.gaussmath.org 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 www.gaussmath.org 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 www.gaussmath.org 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 www.gaussmath.org 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 www.gaussmath.org