Elite 2008 March Competition USACO Contest
http://cerberus.delos.com:794/MAR08
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BRONZE PROBLEMS
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Three problems numbered 11 through 13
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Problem 11: Statistics [Traditional, 2008]
The cows have enrolled in elementary statistics and are having a tough time
with their homework. Please help them.
Given a set of N (1 <= N <= 500) numbers, X_i (-5,000 <= X_i <= 5000),
calculate two statistics:
* The mean (sum of the numbers divided by N)
* The median (if N is odd, the middle number when the N numbers
are sorted; if N is even, the mean of the two middle numbers
when the N numbers are sorted)
Your answer will be considered correct if it is within 0.002 of the
official answer.
PROBLEM NAME: stats
INPUT FORMAT:
* Line 1: A single integer: N
* Lines 2..N+1: Line i+1 contains a single integer: X_i
SAMPLE INPUT (file stats.in):
5
12
4
6
8
2
OUTPUT FORMAT:
* Line 1: The mean of the input set
* Line 2: The median of the input set
SAMPLE OUTPUT (file stats.out):
6.400000
6.000000
OUTPUT DETAILS:
(12+4+6+8+2)/5=6.4
Middle number of the set {2, 4, 6, 8, 12} is 6
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Problem 12: Lake Making [Rob Kolstad, 2008]
Farmer John wants his cows to help him make a lake. He has mapped
the pasture where he wants to build the lake by creating a R (3 <=
R <= 100) row by C (3 <= C <= 100) column grid of six foot by six
foot squares and then by determining the average elevation (10 <=
elev_rc <= 5000) in inches for each square.
Additionally, he has trained the cows in "stomp digging". The burly
bovines travel in a herd that just exactly covers a 3x3 grid of
squares to a grid whose upper left coordinate is R_s,C_s (1 <= R_s
<= R-2; 1 <= C_s <= C-2). The cows then stomp the ground to push
it down D_s (1 <= D_s <= 40) inches. The cows are quite meticulous;
the cows at lower elevations will not commence stomping until the
rest of the herd has joined them. Thus, not all the 3x3 grid is
necessarily stomped (or perhaps some part is stomped less than some
other part).
Given an initial set of elevations, an ordered set of N (1 <= N <=
20000) stomp digging instructions, and an elevation E (0 <= E <=
5000) for the lake's final water level, determine the volume of
water (in cubic inches) that the lake will hold. It is guaranteed
that the answer will not exceed 2,000,000,000. Presume that the
edge of the lake contains barriers so that water can not spill over
the border.
Consider a small 4 x 6 pasture to be turned into a lake. Its initial
elevations (annotated with row/column numbers) are:
column
1 2 3 4 5 6
row 1: 28 25 20 32 34 36
row 2: 27 25 20 20 30 34
row 3: 24 20 20 20 20 30
row 4: 20 20 14 14 20 20
Interpreting the map, we see a hill in the upper right corner that
rises to elevation 36; a small hill also rises to elevation 28 in
the upper left corner. The middle of row 4 has a slight depression.
After the cow-stomping instruction "1 4 4", the pasture has these
elevations:
1 2 3 4 5 6
row 1: 28 25 20 32 32 32
row 2: 27 25 20 20 30 32
row 3: 24 20 20 20 20 30
row 4: 20 20 14 14 20 20
Note that only three squares were stomped down. The other six sets
of cows were waiting for the stompers to get to their level, which
never happened. After stomping down the upper left corner with
this instruction "1 1 10", the pasture looks like this:
1 2 3 4 5 6
row 1: 18 18 18 32 32 32
row 2: 18 18 18 20 30 32
row 3: 18 18 18 20 20 30
row 4: 20 20 14 14 20 20
If the final elevation of the lake is to be 22 inches, the pasture
has these depths:
1 2 3 4 5 6
row 1: 4 4 4 -- -- --
row 2: 4 4 4 2 -- --
row 3: 4 4 4 2 2 --
row 4: 2 2 8 8 2 2
for a total aggregated depth of 66. Calculate the volume by multiplying
by 6 feet x 6 feet = 66 x 72 inches x 72 inches = 342,144 cubic
inches.
Write a program to automate this calculation.
PROBLEM NAME: makelake
INPUT FORMAT:
* Line 1: Four space-separated integers: R, C, E, N
* Lines 2..R+1: Line i+1 describes row of squares i with C
space-separated integers
* Lines R+2..R+N+1: Line i+R+1 describes stomp-digging instruction i
with three integers: R_s, C_s, and D_s
SAMPLE INPUT (file makelake.in):
4 6 22 2
28 25 20 32 34 36
27 25 20 20 30 34
24 20 20 20 20 30
20 20 14 14 20 20
1 4 4
1 1 10
INPUT DETAILS:
As per the example from the text.
OUTPUT FORMAT:
* Line 1: A single integer that is the volume of the new lake in cubic
inches
SAMPLE OUTPUT (file makelake.out):
342144
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Problem 13: Oh Those Rollers [Rob Kolstad, 2008]
Farmer John has installed a new winch that gives him mechanical
advantage when lifting bales of hay into the barn. The winch was
manufactured by the Rube Goldberg Winch Company and has way too
many rollers to make any sense at all. The winch has a huge steel
plate with a number of rollers whose ultimate source of power is
the drive-roller whose location FJ has denoted as the origin (0,0).
This roller drives a roller that drives another roller, etc. etc.
until the final roller is driven. FJ is trying to find that final
roller and wants to know which one it is.
FJ has recorded the x_i,y_i (-5,000 <= x_i <= 5,000; -5,000 <= y_i
<= 5,000) coordinates and the radius r_i (3 <= r_i <= 1024) of each
of the N (2 <= N <= 1080) rollers. Tell him the x,y coordinate of
the last roller in the chain (the roller that is driven but drives
no other roller). Every roller except the drive-roller is driven
by exactly one other roller.
PROBLEM NAME: rollers
INPUT FORMAT:
* Line 1: A single integer: N
* Lines 2..N+1: Line i+1 describes roller i with three space separated
integers: x_i, y_i, and r_i
SAMPLE INPUT (file rollers.in):
3
0 0 30
30 40 20
-15 100 55
INPUT DETAILS:
Three rollers. First is at the origin with radius 30. It drives the roller
at (30,40) whose radius is 20. That roller in turn drives the third roller
located at (-15,100) with radius 55.
OUTPUT FORMAT:
* Line 1: A single line with two space-separated integers that are
respectively the x,y coordinates of the last roller in the
chain of driven rollers.
SAMPLE OUTPUT (file rollers.out):
-15 100
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