# 2012 AMC 10B PROBLEMS PDF

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Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?

Solution If the side lengths of the dart board and the side lengths of the center square are all then the side length of the legs of the triangles are.

Use Geometric probability by putting the area of the desired region over the area of the entire region. What is the degree measure of Solution We can let be and be because they are in the ratio.

When an inscribed anglecontains the diameter, the inscribed angle is a right angle. Therefore by triangle sum theorem, and. Point is chosen on side? What is the degree measure of Solution It is given that interior angles and ,. Since and are alternate. Use the Base Angle Theorem to show rectangle, so it follows that triangle, and then.

We know that. We notice that. Solve However, this is not the final answer. Plug it back into the original equation to ensure it still works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways.

What is the area of? Then and let if and only if be is on the same be. The line divides the plane into two half-planes; let. Let us define similarly and the half-plane containing Then is equal to. We can make it easier to find the area of this region by dividing it into four triangles: Since equilateral, contains , contains and and , and are contains. Then with and so and has area. The pairwise. What is the sum of the Solution The largest difference, The smallest difference, must be between and must be directly between two integers.

This also means The only However, those the differences directly between the other two should add up to remaining differences that would make this possible are and two differences can't be right next to each other because they would make a difference of This means must be the difference between and We can express the possible configurations as the lines.

Solution Since we know To compute this, write it as and use the binomial theorem. What is the maximum are integers. The graph of for all? For , are the points of tangency of the incircle and respectively, then is a triangle with side lengths triangle in the sequence if it exists. What is the perimeter of the last Solution By constructing the bisectors of each angle and the perpendicular radii of the incircle the triangle consists of 3 kites.

Hence Let and and and. Subbing in gives that has sides of. All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinately until the side lengths no longer form a triangle. Subbing in gives has sides with sides.

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## 35 Sets of Previous Official AMC 10 Tests with Answer Keys (PDF files)

Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is What is the area of the rectangle? The point in the xy-plane with coordinates is reflected across the line.

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## Slope of straight line – AMC 10B, 2012 Problem 3

The AMC 10 and AMC 12 are both question, minute, multiple choice examinations in high school mathematics designed to promote the development and enhancement of problem-solving skills. The AMC 10 is for students in 10th grade and below, and covers the high school curriculum up to 10th grade. Students in grade 10 or below and under The AMC 12 covers the entire high school curriculum including trigonometry, advanced algebra, and advanced geometry, but excluding calculus. Students in grade 12 or below and under These competitions are administered around the country on Thursday, January 30, and Wednesday, February 5,

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