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1、2016 AMC 12BProblem 1What is the value ofwhen？Solution By： DragonflyWe find thatis the same as， since a number to the power ofis just the reciprocal of that number. We then get the equation to beWe can then simplify the equation to getProblem 2The harmonic mean of two numbers can be calculated as tw
2、ice their product divided by their sum. The harmonic mean ofandis closest to which integer?SolutionBy： dragonflySince the harmonic mean istimes their product divided by their sum, we get the equationwhich is thenwhich is finally closest to。Problem 3Let。 What is the value of?SolutionBy： dragonflyFirs
3、t of all, lets plug in all of thes into the equation。Then we simplify to getwhich simplifies intoand finally we getProblem 4The ratio of the measures of two acute angles is, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree
4、measures of the two angles?SolutionBy： dragonflyWe set up equations to find each angle. The larger angle will be represented asand the larger angle will we represented as， in degrees. This implies thatandsince the larger the original angle， the smaller the complement.We then find thatand, and their
5、sum isProblem 5The War ofstarted with a declaration of war on Thursday, June，. The peace treaty to end the war was signeddays later， on December，. On what day of the week was the treaty signed?SolutionBy: dragonflyTo find what day of the week it is indays， we have to dividebyto see the remainder， an
6、d then add the remainder to the current day. We get thathas a remainder of 2, so we increase the current day byto getProblem 6All three vertices oflie on the parabola defined by， withat the origin andparallel to the-axis。 The area of the triangle is. What is the length of？SolutionBy: Albert471Plotti
7、ng pointsandon the graph shows that they are atand, which is isosceles。 By setting up the triangle area formula you get:Making x=4， and the length ofis， so the answer is.Problem 7Josh writes the numbers。 He marks out， skips the next number， marks out， and continues skipping and marking out the next
8、number to the end of the list. Then he goes back to the start of his list， marks out the first remaining number, skips the next number， marks out， skips, marks out， and so on to the end。 Josh continues in this manner until only one number remains。 What is that number?SolutionBy Albert471Following th
9、e pattern， you are crossing out.。Time 1: Every non-multiple ofTime 2: Every nonmultiple ofTime 3： Every non-multiple ofFollowing this pattern， you are left with every multiple ofwhich is only .Problem 8A thin piece of wood of uniform density in the shape of an equilateral triangle with side lengthin
10、ches weighsounces。 A second piece of the same type of wood, with the same thickness， also in the shape of an equilateral triangle, has side length ofinches. Which of the following is closest to the weight， in ounces， of the second piece?Solution 1By： dragonflyWe can solve this problem by using simil
11、ar triangles, since two equilateral triangles are always similar. We can then use。We can then solve the equation to getwhich is closest toSolution 2Another approach to this problem， very similar to the previous one but perhaps explained more thoroughly, is to use proportions. First, since the thickn
12、ess and density are the same, we can set up a proportion based on the principle that， thus.However， since density and thickness are the same and(recognizing that the area of an equilateral triangle is）, we can say that。Then, by increasing s by a factor of，is increased by a factor of, thusor.Problem
13、9Carl decided to fence in his rectangular garden. He boughtfence posts， placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactlyyards between neighboring posts. The longer side of his garden， including the corners, has twice as many posts
14、as the shorter side， including the corners. What is the area, in square yards， of Carls garden?SolutionBy Albert471To start， use algebra to determine the number of posts on each side. You have (the long sides count forbecause there are twice as many）(each corner is double counted so you must add) Ma
15、king the shorter end have， and the longer end have。 Therefore， the answer isProblem 10A quadrilateral has vertices, and， whereandare integers with. The area ofis. What is?Solution 1By distance formula we have. SImplifying we get. Thusandhave to be a factor of 8。 The only way for them to be factors ofand remain integers is ifand。 So the answer isSolution by I_Dont_Do_MathSolution 2Solution by e_power_pi_times_iBy the Shoelace Theorem， the area of the quad
