The Second Annual West Windsor-Plainsboro Mathematics Tournament

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1 The Second Annual West Windsor-Plainsboro Mathematics Tournament Saturday October 9th, 0 Grade 8 Test RULES The test consists of 0 multiple choice problems and 0 short answer problems to be done in 0 minutes. All multiple choice answers should be marked on the scantron. No marks on this paper will be graded. All short answers you write should be in exact, simplified form, unless otherwise stated in the question, following the rules below: All fractions must be simplified. All fractional answers must be expressed in the form a b where a and b share no common factors other than. For example, if the answer is 9, then 8 (mixed fraction) or 0 is NOT ACCEPTABL Any question that involves the number π in the answer must be written using the symbol π. approximations of π are NOT ACCEPTABL In particular, decimal All radicals must be simplified. Radicals in the denominator of a final answer, a perfect n th power under a n th root, or fractions/decimals under a radical are NOT ACCEPTABL If you get up, you have to hand in your test, even if you need to go to the bathroom. Plan to go to the bathroom before the test starts. You can ask your proctor a question, but if they think answering your question will give you an unfair advantage on the test over other contestants, they don t have to respond. After the testing has completed, please go to your selected mini-event rotation (directions will be given for this after the main contest is over). DO NOT FLIP THIS PAGE TO BEGIN THE TEST UNTIL INSTRUCTED TO DO SO BY YOUR PROCTOR(S).

2 Multiple-Choice Questions Answers to this section of the test should go on the scantron multiple-choice form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORE. If a = 7 and b = 8, What is the value of a + ab + b? Bharath makes a bet with Dhruva about the upcoming Nets-Caveliers game. If the Cavaliers lose, Bharath has to pay Dhruva x dollars, where x is the difference in points between the two teams. The Nets beat the Cavaliers 0 to 9. How much does Bharath owe Dhruva? Which of the following numbers is divisible by? A double-double in basketball is when a player records double digit stats in two of the three categories: points, rebounds, and assists. Lebron James puts up a double-double in 8 of all his games. LaMarcus Aldridge records a double-double in half his games. If each player plays 80 games in a season, how many more double-doubles can we expect from Aldridge than from Lebron? If the average of a set of numbers is 6, how many 0 s should be added to the set in order to make the average of the numbers in the set 8? 6. We have rectangle ABCD with point E on side C What is the difference between the maximum possible area of ABE and the minimum possible area of ABE? 0 Not enough information 7. Find the area of a square inscribed in a circle of radius Vincent goes to the art museum twice this weekend. On Saturday, the museum randomly showcases three out of the ten paintings in their inventory. On Sunday, the museum chooses a random painting from the ten paintings and puts it on display. What is the probability that Vincent sees the same painting on both days? Sally has bags of marbles. The first bag has red and blue, and the second bag has red and blue, and the third bag has red and blue. Sally will choose one marble from each bag. What is the probability that all three marbles are blue? Julia s phone has 00% battery at 8 : 00 M. By : 00 P.M. her battery has dropped to 6%. If she uses her phone s battery at a constant rate, what time will her battery drop to 7%? : 00 P.M. : 0 P.M. : 00 P.M. : 0 P.M. : 00 P.M.. Am equilateral triangle and a square have equal perimeters. If a side of the triangle is 8, what is the area of the square?

3 . The diagonal of a rectangle has length 0. What is the rectangle s maximum possible area? Find the remainder when is divided by. 0. Given a circle centered at point (, 0) with radius 8. What is the length of the shortest chord of the circle that passes through the origin? Positive integers a, b, c, d exist that are relatively prime to each other. If a b = c d and d = 7b, what is a + b + c + d? The point (, ) is rotated clockwise around the origin. What are the new coordinates of the point? (0, ) (, 0) (0, ) (, 0) (, ) 7. Find the number of pairs of positive integers (a, b) such that ab + a + b = The sum of the height and width of a rectangle is. If the diagonal has length 7, what is the area of the rectangle? Ishan and Jason both flip coins times. Each time they get a head, they get a point. What is the probability that Ishan beats Jason? How many zeros are at the end of 0 factorial? 0 7. Jacques uses a calculator to find He then adds the digits of the product. He then adds the digits of the new result. He keeps on adding the digits of the results until he gets a single digit number. What is the single digit number? 6 9. Alex and Ethan are taking turns shooting a basketball. Alex shoots first and Ethan shoots second. The first person to make their shot wins. They keep shooting in this order until someone wins. Given that Alex makes / of his shots and Ethan makes / of his shots. What is the probability that Ethan wins. 6. DJ has integers x, y, a, and b. Instead of multiplying the sum of x and a with the sum of y and b, he multiplies the sum of x and b with the sum of y and a. If his result is 7 more than it should have been if he did it correctly, and knowing (b a)(y + x) = 7, compute y x

4 . The product of two numbers minus their sum is 7. What is their sum if both of them are less than 0? 6 6. A pyramid is sliced four times parallel to its base at equally spaced intervals (so that the line from the vertex of the pyramid perpendicular to the base is sliced into fifths). What fraction of the original volume is the third largest slice? Find the smallest positive integer such that n + divides n A circle of radius 6 is centered at point O. Three other points, M, P and B are chosen such that M, P, and B are collinear. Furthermore, M and B are on the circumference of the circle centered at O with radius 6. Given that the length of P O is, what is MP BP? A group of people sit around a round table. There are several different Angela s within the group. If the probability that no two Angela s are seated next to each other is 7 99, how many Angela s are there? 9. Let f(x) be a rational function such that when f(x) is in reduced form, the numerator and denominator both have degree. Which of the following is most likely to be the types of asymptotes that f(x) has? Note: if P (x) is a polynomial function and Q(x) is another polynomial function, then the function P (x) Q(x) Three vertical asymptotes only One vertical asyptote and one oblique asymptote One horizontal asyptote and either one or three vertical asymptotes One horizontal asymptote and one oblique asymptote One vertical asymptote and either one or three horizontal asymptotes is a rational function. 0. The incircle of a triangle ABC with AB =, BC =, and AC = is tangent to sides AB, BC, CA at D, E, F, respectively. Squares are drawn outside of triangle ABC with side lengths AD, BD, BE, CE, CF, and AF. Find the sum of the areas of these squares

5 Short Answer Questions Answers to this section of the test should go on the short answer questions answer form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORE. What is the probability that when you flip coins, you get at least 8 heads?. What are the digits of 000 in base 8??. Find the value of:. Find all the solutions of x log 0 x = x Saleem chooses random real numbers between - and. Find the probability that x + y + z. 6. Trapezoid ABCD is constructed, with legs BC and AD congruent. Diagonals AC and BD are drawn, intersecting at point E inside the trapezoid. If DE = 0, DC = 6, and AE =., what is the area of the trapezoid 7. You can purchase yellow balls and green balls, costing $ and $7 respectively. Find the largest amount of money you can have such that you cannot spend all of your money buying balls (you can only purchase integral numbers of balls). 8. Triangle ABC is on a coordinate plane, with A at (, ), B at (, 8), and C at ( 6, ). Make point X the midpoint of side AB and Y the midpoint of B Mark the intersection of lines CX and AY as M. What are the coordinates of M? 9. How many ways can you rearrange yellow balls, green balls, pink balls, and purple ball in a line where balls of the same color are indistinguishable? 0. What is the radius of the incircle of a triangle with side lengths, 6, 7? Note: the incircle of the triangle is the circle inside a triangle that is tangent to all three sides of the triangle.

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