2010 AMC 10B1 . What is ?SolutionWe first expand the first term, simplify, and then compute to get an answer of .2 、Makarla attended two meetings during her -hour work day. The firstmeeting took minutes and the second meeting took twice as long.What percent of her work day was spent attending meetings?SolutionThe percentage of her time spent in meetings is the total amount of time spent in meetings divided by the length of her workday.The total time spent in meetings isTherefore, the percentage is3、A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?SolutionAfter you draw socks, you can have one of each color, so (according tothe pigeonhole principle), if you pull then you will be guaranteed a matching pair.4 、For a real number , define to be the average of and . Whatis ?SolutionThe average of two numbers, and , is defined as . Thus theaverage of and would be . With that said, we need to find thesum when we plug, , and into that equation. So:.5 、A month with days has the same number of Mondays andWednesdays.How many of the seven days of the week could be the first day of this month?Solution（B）. 36 、A circle is centered at , is a diameter and is a point on thecircle with . What is the degree measure of ?SolutionAssuming the reader is not readily capable to understand how willalways be right, the I will continue with an easily understandable solution.Since is the center, are all radii, they are congruent.Thus, and are isosceles triangles. Also, note thatand are supplementary, then . Sinceis isosceles, then . They also sum to , so eachangle is .7 、A triangle has side lengths , , and . A rectangle has width andarea equal to the area of the rectangle. What is the perimeter of this rectangle?SolutionThe triangle is isosceles. The height of the triangle is therefore given byNow, the area of the triangle isWe have that the area of the rectangle is the same as the area of the triangle, namely 48. We also have the width of the rectangle: 4.The length of the rectangle therefor is:The perimeter of the rectangle then becomes:The answer is:8 、A ticket to a school play cost dollars, where is a whole number. A group of 9th graders buys tickets costing a total of $, and a group of 10th graders buys tickets costing a total of $. How many values for are possible?SolutionWe see how many common integer factors 48 and 64 share. Of the factors of 48 - 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48; only 1, 2, 4, 8, and 16are factors of 64. So there are possibilities for the ticket price.9 、Lucky Larry's teacher asked him to substitute numbers for , , ,, and in the expression and evaluate the result.Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry sustitued for , , , and were , , , and , respectively. What number didLarry substitude for ?SolutionSimplify the expression . I recommend to start with the innermost parenthesis and work your way out.So you get:Henry substituted with respectively.We have to find the value of , such that(the same expression withoutparenthesis).Substituting and simplifying we get:So Henry must have used the value for .Our answer is:10、Shelby drives her scooter at a speed of miles per hour if it is notraining, and miles per hour if it is raining. Today she drove in the sunin the morning and in the rain in the evening, for a total of miles inminutes. How many minutes did she drive in the rain?SolutionWe know thatSince we know that she drove both when it was raining and when it was not and that her total distance traveled is miles.We also know that she drove a total of minutes which is of an hour.We get the following system of equations, where is the time traveled when it was not raining and is the time traveled when it was raining:Solving the above equations by multiplying the second equation by 30 and subtracting the second equation from the first we get:We know now that the time traveled in rain was of an hour, which is minutesSo, our answer is:11 、A shopper plans to purchase an item that has a listed price greater than $and can use any one of the three coupns. Coupon A gives off the listed price, Coupon B gives $off the listed price, andCoupon C gives off the amount by which the listed price exceeds $.Let and be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is −SolutionLet the listed price be , whereCoupon A saves us:Coupon B saves us:Coupon C saves us:Now, the condition is that A has to be greater than or equal to either B or C which give us the following inequalities:We see here that the greatest possible value for p is and the smallest isThe difference between and isOur answer is:12 、At the beginning of the school year, of all students in Mr. Wells'math class answered "Yes" to the question "Do you love math", and answered "No." At the end of the school year, answered "Yes"and answerws "No." Altogether, of the students gave a differentanswer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values ofSolutionThe minimum possible value occurs when of the students whooriginally answered "No." answer "Yes." In this case,The maximum possible value occurs when of the students whooriginally answered "Yes." answer "No." and the of the students whooriginally answered "No." answer "Yes." In this case,Subtract to obtain an answer of13 、What is the sum of all the solutions of ?SolutionCase 1:Case 1a:Case 1b:Case 2:Case 2a:Case 2b:Since an absolute value cannot be negative, we exclude . Theanswer is14 、The average of the numbers and is . What isSolutionWe must find the average of the numbers from to and in terms of. The sum of all these terms is . We must divide this by the total number of terms, which is . We get: . Thisis equal to , as stated in the problem. We have: .We can now cross multiply. This gives:This gives us our answer.15 、On a -question multiple choice math contest, students receivepoints for a correct answer, points for an answer left blank, andpoint for an incorrect answer. Jesse’s total score on the contest was .What is the maximum number of questions that Jesse could have answered correctly?SolutionLet be the amount of questions Jesse answered correctly, be theamount of questions Jesse left blank, and be the amount of questions Jesse answered incorrectly. Since there were questions on the contest,. Since his total score was , . Also,. We can substitute this inequality into the previous equation to obtain another inequality:. Since is an integer, themaximum value for is .16 、A square of side length and a circle of radius share the samecenter. What is the area inside the circle, but outside the square?Solution(B)17 、Every high school in the city of Euclid sent a team of students to amath contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed th and th, respectively. How many schools are in the city?SolutionLet the be the number of schools, be the number of contestants, andbe Andrea's score. Since the number of participants divided by three isthe number of schools, . Andrea received a higher scorethan her teammates, so . Since is the maximum possiblemedian, then is the maximum possible number ofparticipants. Therefore, . This yields the compound inequality: . Since a set with an even number of elements has a median that is the average of the two middle terms, anoccurrence that cannot happen in this situation, cannot be even.is the only other option.18 、Positive integers , , and are randomly and independentlyselected with replacement from the set . What is the probability that is divisible by ?Solution（E）13/2719 、A circle with center has area . Triangle is equilateral,is a chord on the circle, , and point is outside .What is the side length of ?Solution(B)620 、Two circles lie outside regular hexagon . The first istangent to , and the second is tangent to . Both are tangent tolines and . What is the ratio of the area of the second circle to thatof the first circle?SolutionA good diagram here is very helpful.The first circle is in red, the second in blue. With this diagram, we can see that the first circle is inscribed in equilateral triangle while thesecond circle is inscribed in . From this, it's evident that the ratio ofthe red area to the blue area is equal to the ratio of the areas of triangles toSince the ratio of areas is equal to the square of the ratio of lengths, we know our final answer is From the diagram, we can see that thisisThe letter answer is D21 、A palindrome between and is chosen at random. What isthe probability that it is divisible by ?SolutionView the palindrome as some number with form (decimalrepresentation): . But because the number is a palindrome, . Recombining this yields .1001 is divisible by 7, which means that as long as , thepalindrome will be divisible by 7. This yields 9 palindromes out of 90() possibilities for palindromes. However, if , then this givesanother case in which the palindrome is divisible by 7. This adds another 9 palindromes to the list, bringing our total to22 、Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?SolutionWe can count the total number of ways to distribute the candies (ignoring the restrictions), and then subtract the overcount to get the answer.Each candy has three choices; it can go in any of the three bags.Since there are seven candies, that makes the total distributionsTo find the overcount, we calculate the number of invalid distributions: the red or blue bag is empty.The number of distributions such that the red bag is empty is equal to ,since it's equivalent to distributing the 7 candies into 2 bags.We know that the number of distributions with the blue bag is empty will be the same number because of the symmetry, so it's also .The case where both the red and the blue bags are empty (all 7 candies are in the white bag) are included in both of the above calculations, and this case has only distribution.The total overcount isThe final answer will beThat makes the letter choice C23 、The entries in a array include all the digits from through ,arranged so that the entries in every row and column are in increasing order. How many such arrays are there?Solution（D）6024 、A high school basketball game between the Raiders and Wildcatswas tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than points. What was the total number of points scoredby the two teams in the first half?SolutionRepresent the teams' scores as: andWe have Manipulating this, we can get, orSince both are increasing sequences, . We can check cases up tobecause when , we get . When▪▪▪Checking each of these cases individually back into the equation, we see that only when a=5 and n=2, we get an integer value for m, which is 9. The original question asks for the first half scores summed, so we must find25 、Let , and let be a polynomial with integer coefficients suchthat, and.What is the smallest possible value of ?SolutionThere must be some polynomial such thatThen, plugging in values of we getThus, the least value ofmust be the . Solving, we receive , so our answer is.。
