Virginia Mathematics Teacher Fall 2016

So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems

Construct the third median BG. The three medians AD, CF and BG split ABC into six triangles (AOF, AOG, GOC, DOC, BOD, and BOF) whose areas are equal to one sixth of the ABC area. The area of the triangle BDF is one fourth of ABC area, as those triangles are similar. Thus the area DOF equals area of DOFB - area of BDF = 2 x one sixth - one fourth = one twelfth of ABC. The area of DOF equals the area of DOFB minus the area of BDF. The area of DOFB equals 2 times 1/6 th the area of ABC, as it is the sum of the areas of the trian­ gles BOD and BOF. As the area of BDF equals 1/4 th the area of ABC, the area of DOF equals of the area of triangle ABC. June Challenge: AB is the diameter of a semicircle. AC is the diameter of smaller semicircle which completely inside the first semicircle. The line FG is tangent to the smaller semicircle and parallel to AB. The length of FG is 10 units. Find the ar­ ea of the bigger semicircle that does not include the area of the smaller semicircle.

SOLUTION :

Let’s translate the smaller semicircle to the right, so the center points of both semicircles will overlap. The shaded area will remain the same, as the area of the smaller semicircle is not changing: Therefore:

Shaded area equals:

July Challenge Mr Saver has $ A, spends nothing and saves all money he has been paid by the clients. If a client pays any­ thing, Mr Saver estimates the "significance" of the payment by calculating the ratio of this payment to the total amount he has saved, including the last payment. During this month, Mr Saver has had N clients, who paid him $ B in total. Assume R - the maximum "significance" ratio. Find the minimum value of R.

SOLUTION :

Assume so far Mr. Saver has had N clients who have paid him the amounts x1, x2, x3, respectively. If R is the "significance" of their payments, then:

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Virginia Mathematics Teacher vol. 43, no. 1

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