MathDB

2016 Purple Comet Problems

Part of Purple Comet Problems

Subcontests

(30)
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Purple Comet 2016 HS Problem 30

Some identically sized spheres are piled in nn layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of n2n^2 spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer mm such that as nn gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches mπ\frac{\sqrt{m}}{\pi}. Find mm.
https://snag.gy/bIwyl6.jpg
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Purple Comet HS 2016 Problem 13

In △\triangleABC shown below, AB = AC, AF = EF, and EH = CH = DH = GH = DG = BG. Also, ∠CHE = ∠F GH. Find the degree measure of ∠BAC. https://i.snag.gy/ZyxQVX.jpg

Purple Comet 2016 MS Problem 13

One afternoon Elizabeth noticed that twice as many cars on the expressway carried only a driver as compared to the number of cars that carried a driver and one passenger. She also noted that twice as many cars carried a driver and one passenger as those that carried a driver and two passengers. Only 10% of the cars carried a driver and three passengers, and no car carried more than four people. Any car containing at least three people was allowed to use the fast lane. Elizabeth calculated that mn\frac{m}{n} of the people in cars on the expressway were allowed to ride in the fast lane, where mm and nn are relatively prime positive integers. Find m+nm + n.
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Purple Comet 2016 HS Problem 10

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is mn\frac{m}{n}, where m and n are relatively prime positive integers. Find m+nm + n.

Purple Comet 2016 MS Problem 10

Mildred the cow is tied with a rope to the side of a square shed with side length 10 meters. The rope is attached to the shed at a point two meters from one corner of the shed. The rope is 14 meters long. The area of grass growing around the shed that Mildred can reach is given by nπn\pi square meters, where nn is a positive integer. Find nn.
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Purple Comet 2016 HS Problem 8

The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
https://i.snag.gy/Ik094i.jpg

Purple Comet 2016 MS Problem 8

The figure below has a 1 × 1 square, a 2 × 2 square, a 3 × 3 square, a 4 × 4 square, and a 5 × 5 square. Each of the larger squares shares a corner with the 1 × 1 square. Find the area of the region covered by these five squares.
https://snag.gy/1AfJWt.jpg
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