2010 Math Prize for Girls, Problems 6-10

Problem #6 If you know the standard formula for the area of a trapezoid, then you’ll know that the missing piece of information needed to complete this problem is the height of the trapezoid. If you draw in the height in strategic locations, right triangles emerge and so we have an application of the Pythagorean theorem. If you don’t remember the area of a trapezoid, a general strategy for computing areas is to dissect the shape into various rectangles and triangles or try to place the shape in a rectangle or triangle and subtract the complementary area within the larger shape from the area of the containing shape. In fact, such manipulations are how the formula for the area of a trapezoid is derived in the first place!

If you’re having trouble, you need to learn and develop tools pertaining to length and area. By solving many problems, you’ll begin to memorize the most useful formulas, and if you keep on solving problems, you’ll discover or learn about more exotic area and length formulas, like Heron’s formula or the law of cosines. And if you keep on solving problems, you’ll memorize even those formulas, and the process goes on and on!

A note about the solution at Math Prize for Girls: Their solution relies on the “look” of the picture, which is not really rigorous. For a rigorous solution, you would demonstrate algebraic inequalities that show that the 7 points obtained are the only ones.

Problem #7 This problem is similar to problem #3, only this time, instead of finding the lattice points on a sphere, you have to find the lattice points inside a heart shape. Just like problem #3, this problem also has a very small number of solutions, so you can find them by just trying a few cases.

If you’re having trouble, it might mean that psychologically, you were intimidated by the unusual shape of the region and the equation that defines it or that you resist examining cases. If you were intimidated, try to make headway by figuring out what you can about the situation no matter how insignificant the facts you find may at first seem. For example, maybe you can’t solve the problem in general, but perhaps you can find solutions when x = 0 or y = 0 (i.e. find out where the graph intersects the axes). As you work these out, you’ll probably begin to see how you can go about resolving the whole problem.

Problem #8 This problem exploits the fact that the number of divisors of a number can be obtained from the exponents that appear in the prime factorization. Precisely, if

n = p_1^{m_1}p_2^{m_2}p_3^{m_3}\cdots p_n^{m_n}

is the prime factorization of n, then the number of divisors of n is given by

(m_1 + 1)(m_2 + 1)(m_3 + 1) \cdots (m_n + 1).

This is true because any divisor has a prime factorization where the exponents corresponding to each prime must be between 0 and the corresponding exponent in the prime factorization of the original number. This fact sharply reduces the possibilities to a level that can be checked case by case.

If you’re having trouble, it probably means you were unaware of the formula for the number of divisors of a number. You can learn these kinds of facts in any introduction to Number Theory.

Problem #9 In this problem, you are given information about every set of four consecutive test scores and are asked to maximize the average of all 10 tests, or, what is the same, maximum the total score of all the tests. This problem is an example of a linear programming problem, though it can be solved without knowing the general theory of linear programming problems because there is a lot of symmetry in the given information.

If you’re having trouble, study systems of linear equations and inequalities. And although it’s a bit much for this problem, it’s still worthwhile to learn about linear programming (Google it!). Linear programming is very useful and important in many engineering applications.

Problem #10 You don’t need to know about vectors to solve this problem, but thinking about this problem in terms of vectors is very natural. You could also approach this problem using coordinate geometry to obtain a system of linear equations as was done in the posted solution.

Because the vector approach leads to quite a nice solution, here it is: Notice that vertex A completes a parallelogram with the 3 given midpoints. Therefore, point A can be reached from the midpoint of AB by translating along the same translation that carries the midpoint of BC to the midpoint of AC. Thus, point A is (-16, -63) + ((13, 50) – (6, -85)) = (-16, -63) + (7, 135) = (-9, 72). (You could also get to A via the midpoint of AC by translating via the translation that carries the midpoint of BC to the midpoint of AB.)

If you’re having trouble, the relevant topics to study are coordinate geometry (or “analytic geometry”) and vectors.

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