Problem 1Problem 1 What is the value of Problem 2Problem 2 Two of the three sides of a triangle are 20 and 15 Which of the following numbers is not a possible perimeter of the triangle Problem 3Problem 3 Mr Patrick teaches math to 15 students He was grading tests and found that when he graded everyone s test except Payton s the average grade for the class was 80 after he graded Payton s test the class average became 81 What was Payton s score on the test Problem 4 The sum of two positive numbers is 5 times their difference What is the ratio of the larger number to the smaller Problem 5Problem 5 Amelia needs to estimate the quantity whereandare large positive integers She rounds each of the integers so that the calculation will be easier to do mentally In which of these situations will her answer necessarily be greater than the exact value of Problem 6Problem 6 Two years ago Pete was three times as old as his cousin Claire Two years before that Pete was four times as old as Claire In how many years will the ratio of their ages be Problem 7Problem 7 Two right circular cylinders have the same volume The radius of the second cylinder ismore than the radius of the first What is the relationship between the heights of the two cylinders Problem 8Problem 8 The ratio of the length to the width of a rectangle is If the rectangle has diagonal of length then the area may be expressed asfor some constant What is Problem 9Problem 9 A box contains 2 red marbles 2 green marbles and 2 yellow marbles Carol takes 2 marbles from the box at random then Claudia takes 2 of the remaining marbles at random and then Cheryl takes the last 2 marbles What is the probability that Cheryl gets 2 marbles of the same color Problem 10Problem 10 Integersandwithsatisfy What is Problem 11Problem 11 On a sheet of paper Isabella draws a circle of radius a circle of radius and all possible lines simultaneously tangent to both circles Isabella notices that she has drawn exactlylines How many different values ofare possible Problem 12Problem 12 The parabolasandintersect the coordinate axes in exactly four points and these four points are the vertices of a kite of area What is Problem 13Problem 13 A league with 12 teams holds a round robin tournament with each team playing every other team exactly once Games either end with one team victorious or else end in a draw A team scores 2 points for every game it wins and 1 point for every game it draws Which of the following is NOT a true statement about the list of 12 scores Problem 14Problem 14 What is the value offor which Problem 15Problem 15 What is the minimum number of digits to the right of the decimal point needed to express the fractionas a decimal Problem 16Problem 16 Tetrahedronhasand What is the volume of the tetrahedron Problem 17Problem 17 Eight people are sitting around a circular table each holding a fair coin All eight people flip their coins and those who flip heads stand while those who flip tails remain seated What is the probability that no two adjacent people will stand Problem 18Problem 18 The zeros of the functionare integers What is the sum of the possible values of Problem 19Problem 19 For some positive integers there is a quadrilateralwith positive integer side lengths perimeter right angles atand and How many different values ofare possible Problem 20Problem 20 Isosceles trianglesandare not congruent but have the same area and the same perimeter The sides ofhave lengths ofand while those ofhave lengths ofand Which of the following numbers is closest to Problem 21Problem 21 A circle of radiuspasses through both foci of and exactly four points on the ellipse with equation The set of all possible values ofis an interval What is Problem 22Problem 22 For each positive integer n letbe the number of sequences of length n consisting solely of the lettersand with no more than threes in a row and no more than threes in a row What is the remainder whenis divided by 12 Problem 23Problem 23 Letbe a square of side length 1 Two points are chosen independently at random on the sides of The probability that the straight line distance between the points is at leastis whereandare positive integers and What is Problem 24Problem 24 Rational numbersandare chosen at random among all rational numbers in the intervalthat can be written as fractionswhereandare integers with What is the probability thatis a real number Problem 25Problem 25 A collection of circles in the upper half plane all tangent to the axis is constructed in layers as follows Layerconsists of two circles of radiiandthat are externally tangent For the circles inare ordered according to their points of tangency with the axis For every pair of consecutive circles in this order a new circle is constructed externally tangent to each of the two circles in the pair Layerconsists of thecircles constructed in this way Let and for every circledenote byits radius What is DIAGRAM NEEDED 。