USACO Winter Championship
Wed-Mon, January 15-20, 1997
Introduction
WHAT: The USA Computer Olympiad Winter Championship is a computer programming
contest open to all pre-college students on the hs-computing@delos.com
mailing list. Four or five problems are presented for solution.
WHO: All pre-college students who are members of the hs-computing mailing
list are eligible. This is an individual contest, not a team contest.
WHEN: Problems will be posted to the mailing list between 0750 and 0800
a.m. US Mountain Standard Time (1450-1500 GMT) on Wednesday, January
15, 1997. All results are due by 0800 am US Mountain Standard Time
(1500 GMT) Monday, January 20, 1997. Late solutions will not be
accepted.
The extended solution interval is required this time since several
schools are conducting final examinations this week.
WHERE: Students can work on problem solutions anywhere they wish, including
school and home.
HOW: Problems and rules will be posted to the hs-computing mailing list.
Solutions must be written in Borland C/C++ or Turbo Pascal and must
be returned via e-mail by the contest completion deadline along with
answers to test data supplied with the problems. Choosing a different
compiler than the Borland compilers will probably cause a solution
to compile incorrectly on the Borland compilers on the judges' machines
(which will result in no points scored for that problem).
PRIZES: Winners will receive a handsome plaque, have their names immortalized
on the USACO Web Pages, and be the subject of a press release.
FEES: No entry fees are charged.
RULES: * No consultation with people is allowed.
* Submit questions to kolstad@bsdi.com who will try to fathom
an appropriate answer for you (and post it to the hs-computing
list, if appropriate).
* Any amount of consultation with written materials (such as
programming books) is allowed (but this doesn't mean you can ask
someone to write an article on how to solve a problem, of course).
* Spend no more than six hours solving problems.
* Spend no more than three sessions solving problems (i.e., don't work
one hour in a session, take a one hour break, work another hour,
take another break, ...).
* Submit source code for the problems and answers to the test data
to the contest coordinators via e-mail to win97@delos.com (see
below).
* Solutions will be judged for correctness by running them against
several (usually 10) different sets of data. Points are accrued
for correct solutions. Some datasets for a problem will be worth
more points than others. Solutions to some problems will be worth
more points. Judging will be performed expeditiously but
could easily take more than one week to complete.
* Unless otherwise stated, no one solution can run longer than 10
seconds on the judge's computer (a medium speed Pentium).
* Judges will not test all datasets against a program that crashes
the computer for initial datasets.
* Judges will not test all datasets against a program whose author
submits erroneous solutions for the supplied test data.
* Make sure your program has any special compiler options
listed near its first line so the graders can get them right.
* Decision of the judges is final.
* Do not cheat; it's no fun for anyone.
* Do not use inline assembler statements.
* Don't submit programs that use data files that aren't supplied by
the contest coordinators
* Keep in mind that this is neither a `tricky input' nor a
`tricky output' contest, in general.
* All problems are intended to be straightforward; no problems
are intended to have `tricks' in them.
* If you find a problem with wording or an ambiguity, document your
assumptions carefully and move on. Send e-mail; we'll reply if we
can.
NOTES: * We will mostly be using automated grading procedures.
* The registration information at the front of the solution
should be in precisely the same format as requested below.
* Submitters of programs that attempt to thwart grading procedures
will be forever disqualified from USACO contests.
* Rob Kolstad is this contest's director.
* During the contest, feel free to send questions (one per e-mail
please!) to kolstad@bsdi.com. He will reply by e-mail when he is
available. Questions judged to be germane to the entire hs-computing
list will be summarized and posted there, also. Be sure to check
your e-mail occasionally for contest updates, should there be any.
For fairness, sometimes your question will be answered with a null
answer (`sorry, I can't elaborate on that').
* Rob Kolstad judges Russ Cox's programs (at kolstad's site) before
any other program is judged.
* All programs read their input from file INPUT.DAT.
* All programs write their output to the console.
* DO NOT USE conio.h or its Pascal equivalent -- just write your
answer straight to the console as if it were a teletype-style
printer (using no screen manipulation or cursor positioning
commands).
ACKNOWLEDGMENTS:
* Thanks to Hal Burch for problem and solution creation and testing.
* Thanks to Don Piele for problem selection and creation.
* Thanks in advance to Russ Cox for creating grading procedures and
performing them for the contest entrants.
----------------------------------------------------------------------
No special registration form is required. Each problem solution has,
within it, an entry form to be completed (see below).
----------------------------------------------------------------------
Submit your answers to problems via e-mail to win97@delos.com. Please send
precisely one problem's solution (and test results) per e-mail (i.e., send
three separate e-mails for three problems, etc.). Submit only one solution
per problem. If you submit more than one solution, only the last one received
will be graded. Every e-mail is acknowledged almost instantly by an automated
mail reader. If you do not receive an acknowledgment, double-check the
address. Note that some networks connect to the Internet only occasionally.
Each solution e-mail submission should have the following format for the
e-mail message. PLEASE SEND ASCII MESSAGES. DO NOT USE ATTACHMENTS, MIME
ENCODING, fancy mail options, or anything else. The programs that read the
mail are looking for just plain old ASCII.
The submissions have special comments separating sections of the mail. These
begin with three `#'s and are followed by an informational comment. Those
comment lines are surrounded by a single blank line. Here is the format for
submitting solutions so the grading programs can read them:
### Program
... [text of program in C or pascal; see below for header format] ...
### Test Data 1
... [output of program for first run of test data] ...
### Test Data 2
... [output of program for second run of test data] ...
### Test Data 3
... [output of program for third run of test data] ...
### Test Data 4
... [output of program for fourth run of test data] ...
### Test Data 5
... [output of program for fifth run of test data] ...
### End
NOTES: * Put comments precisely like these at top of each solution.
* These are read by a program, do not change the format.
* Be sure your email address and name are always spelled precisely
the same way. We use the combination of email address and name as
a unique identifier for you.
* Data after `### End' is ignored (e.g., signatures)
Here is the C/C++ header filled out for Rob Kolstad (my comments on far
right). You should fill yours out for each problem with your own data (why
not save it in a file on your computer for easy inclusion on later
submissions):
/*
Compiler options: (any special options needed)
Problem: 1 (single digit problem number)
Name: Rob Kolstad (first name then last name)
Email: kolstad@bsdi.com (email address)
School: Norman High School
Grade: 12 (grade in school, numerical: 7, 8, 9, ...,
13)
Age: 18
Country: USA (three letters: CAN, COL, DEU, ROM, USA,
etc.)
*/
Pascal users substitute { for /* and } for */.
Here's a clarification for screen I/O. Don't even clear the screen.
* DO NOT USE conio.h (in C) or `uses dos;' in Pascal -- just write
your answer straight to the console as if it were a teletype-style
printer (using no screen manipulation or cursor positioning
commands).
Here's a hint about testing your program:
* Note that test data run by the judges will almost surely be
more challenging than the test data supplied with each problem.
Here's even broader clarification about how your program should get
its input:
* All programs read their input from file INPUT.DAT; do not
specify a path name in your `open' statement.
Watch your mailbox for a quick (automatically-generated) reply. If you don't
see one fairly soon, send another e-mail to kolstad@bsdi.com explaining the
situation (which problem you submitted, when you sent it).
Good luck!
Problems
PROBLEM 1: Milk Containers [Piele, 1997]
Farmer Paul has many milk containers of each of the following sizes:
Can 10 gallons
Pail 2 gallons
Gallon
Quart 1/4 gallon
Pint 1/8 gallon
Cup 1/16 gallon
Write a program that will compute the number of ways farmer Paul can store
X gallons of milk using any combination of these containers. For instance,
farmer Paul can store one Quart four ways:
1: 1 quart
2: 2 pints
3: 1 pint + 2 cups
4: 4 cups
One gallon can be stored 26 different ways.
In all data, X is a positive integer number and 1 <= X gallons <= 50. Your
program must compute the number of combinations for each separate input value
in less than ten seconds (which means that your program might run as long as
10*n seconds for n input values). Your program should read values from the
file INPUT.DAT one per line (and compute and print the number of combinations)
until encountering a value of 0.
----------------------------------------------------------------------
SAMPLE INPUT (file INPUT.DAT):
5
10
0
SAMPLE OUTPUT:
5 1308
10 12477
TEST DATA SET #1:
1
5
10
15
39
0
PROBLEM 2: Super Roman Numerals [Kolstad, 1997]
You've heard the story of Romans like Midas who had the `Golden Touch'.
Midas was no fool and sold his gold for lots of money. The traditional Roman
numerals proved inconvenient for expressing the value of his fortune, which
ranged into the millions. He invented Super Roman Numerals.
Super Roman Numerals follow the traditional rules for Roman numerals but have
many more single-character values. Consider the traditional Roman numeral
values, shown here with the single letter and the decimal number it represents:
I 1 L 50 M 1000
V 5 C 100
X 10 D 500
As many as three of the same marks that represent 10^n may be placed
consecutively:
III is 3
CCC is 300
Marks that are 5 * 10^n are never used consecutively.
Generally (with the exception of the next rule), marks are connected together
and written in descending order:
CCLXVIII = 100+100+50+10+1+1+1 = 263
Sometimes, a mark that represents 10^n is placed before a mark of one of the
two next higher values (I before V or X; X before L or C; etc.). In this
case, the value of the smaller mark is SUBTRACTED from the mark it precedes:
IV = 4
IX = 9
XL= 40
But compound marks like XD, IC, and XM are not legal, since the smaller mark
is too much smaller than the larger one. For XD (wrong for 490), one would
use CDXC; for IC (wrong for 99), one would use XCIX; for XM (wrong for 990),
one would use CMXC.
Regrettably, in standard Roman numerals, numbers like 10,000 are represented
as MMMMMMMMMM. In Super Roman Numerals, the table of marks is extended:
I 1 L 50 M 1,000 R 50,000 U 1,000,000 N 50,000,000
V 5 C 100 P 5,000 S 100,000 B 5,000,000 Y 100,000,000
X 10 D 500 Q 10,000 T 500,000 W 10,000,000 Z 500,000,000
Numbers greater than 100 million are now easily expressible.
Write a program that reads non-negative decimal numbers (one per line) from
the file INPUT.DAT and prints the Super Roman Numeral equivalent. It is
promised that the input data will require an answer that is representable
using the given rules. Stop your program when the input number is a 0.
----------------------------------------------------------------------
SAMPLE INPUT (file INPUT.DAT):
18
1997
12345678
0
SAMPLE OUTPUT:
18 XVIII
1997 MCMXCVII
12345678 WUUSSSQRPDCLXXVIII
TEST DATA SET #1:
18
1998
87654321
99999999
11111
0
PROBLEM 3: Babylonian Fractions [Traditional]
In ancient Babylon, fractions were poorly understood. The strange way they
represented fractions was as a sum of the minimal number of descending size
terms of unique fractions with numerators of 1. Consider 2/3:
2/3 = 1/2 + 1/6
Write a program that reads pairs of numbers, one per line, from the file
INPUT.DAT (stopping when either of the numbers is 0). It should print the
Babylonian equivalent of the quotient of that pair of numbers formatted
similarly to the examples below.
Your program must compute the number of combinations for each separate input
value in less than ten seconds (which means that your program might run as
long as 10*n seconds for n input values).
No solution will require a fraction whose denominator (bottom number) is
greater than 10,000.
----------------------------------------------------------------------
SAMPLE INPUT (file INPUT.DAT):
2 3
3 4
1 2
0 0
SAMPLE OUTPUT (note the format):
2/3 = 1/2 + 1/6
3/4 = 1/2 + 1/4
1/2 = 1/2
TEST DATA SET #1:
2 3
3 4
1 2
79 1200
26 27
0 0
PROBLEM 4: Electrical Engineering Lab [Kolstad, 1997]
A poorly funded high school in Elbonia wishes to construct a laboratory for
Electrical Engineering students. They are so poor that they wish to build
a software laboratory. They are concerned only with resistors.
Resistors are electrical devices with two `terminals' (wires going into the
resistor). Between these two terminals there is some amount of `resistance'
measured in `ohms'. A single resistor is diagrammed (in the USA) somewhat
like this:
----/\/\/\/----
10 ohms
There isn't a notion of an input or output terminal (wire) -- that all depends
on which way electricity flows through the device. Furthermore, the device
is perfectly symmetrical and bidirectional.
Hooking two resistors `in series' is diagrammed like this:
----/\/\/\/------/\/\/\/----
10 ohms 20 ohms
These two resistors in series behave precisely a single resistor whose value
is the sum of the two resistors (10+20 in this example):
----/\/\/\/----
30 ohms
Another way (not the only other way) to connect resistors is known as
connecting them `in parallel' and is diagrammed like this:
+----/\/\/\/----+
----+ 10 ohms +-------
+----/\/\/\/----+
20 ohms
The resistors' left terminals is connected together as are the right terminals.
These two resistors in series behave precisely a single resistor whose value
is calculated thusly (given two resistors whose values are R1 and R2 ohms):
R1*R2
R = -------
tot R1+R2
----/\/\/\/----
R ohms
tot
For the example above (10 and 20 ohms), the equivalent value for a pair
resistors of 10 and 20 ohms connected in parallel is:
R1*R2 10*20 200
R = ------- = ------ = ----- = 6.667 ohms
tot R1+R2 10+20 30
Other configurations of resistors require more complex analysis and formulae.
The Elbonians wish their software circuit simulator to be able to find
equivalent resistances for sets of resistors connected in parallel and series
(and combinations thereof, of course). For instance, they wish to calculate
the equivalent resistance for a slightly more complex circuit that looks like
this:
+----/\/\/\/----+
----+ 10 ohms +-------/\/\/\/----
+----/\/\/\/----+ 25 ohms
20 ohms
This circuit's equivalent resistance is found by finding the parallel
resistance and substituting an equivalent transistor
----/\/\/\/\/-------/\/\/\/----
6.667 ohms 25 ohms
then using the series formula to combine these two resistors:
----/\/\/\/\/----
31.667 ohms
More complex examples are easily designed:
+----/\/\/\/----+
+ 25 ohms +
+----/\/\/\/----+ +----/\/\/\/----+
----+ 10 ohms +-------/\/\/\/-----+ 22 ohms +-----
+----/\/\/\/----+ 25 ohms +----/\/\/\/----+
+ 20 ohms + 44 ohms
+----/\/\/\/----+
41 ohms
This circuit is reduced pair-by-pair: maybe starting with the 25 and 10 ohm
resistors in parallel, then the 41 and 20 in parallel, then those two
equivalents in parallel, then the series combination of that equivalent with
the 25 ohm resistor, and so on.
Write a program that accepts a set of resistor connections described by the
resistors' values and endpoints:
25 A B
100 B C
with a zero ohm resistor designating the endpoints (and designating the end
of the input):
0 A C
Consider the previous example, now augmented with arbitrarily chosen SINGLE
LETTER connection names:
+----/\/\/\/----+ B
A ----+ 10 ohms +-------/\/\/\/---- C
+----/\/\/\/----+ 25 ohms
20 ohms
whose input description is:
10 A B
20 A B
25 B C
0 A C
Connect names can be upper or lower case -- and node upper-case `A' is
different from node lower-case `a'.
There will be no more than 500 resistors. Calculate your answers in floating
point printing four decimal places of results. No formulae other than those
presented here will be required (though this notation enables potential test
data that is not solvable with these equations -- such test data will not be
presented to your program).
Read your input data from the file INPUT.DAT that has a single set of test
data with the 0 ohm connection data as its last entry.
Print your answer with four decimal places:
31.6667
----------------------------------------------------------------------
SAMPLE INPUT (file INPUT.DAT):
10 A B
20 A B
25 B C
0 A C
SAMPLE OUTPUT:
31.6667
TEST DATASET #1:
10 A B
20 A B
25 B C
0 A C
TEST DATASET #2:
10 A B
10 B C
10 C D
10 D E
10 E F
10 F G
10 G H
10 H I
10 I J
10 J K
10 K L
0 A L
TEST DATASET #3:
10 A B
10 B C
10 C D
10 D E
10 E F
10 F G
10 G H
10 H I
10 I J
10 J K
10 K L
10 A L
0 A L
TEST DATASET #4:
10 A B
10 A B
10 A B
10 A B
0 A B
TEST DATASET #5:
10 A B
10 B A
10 C D
10 D C
10 B C
10 A D
0 D A
PROBLEM 5: Planetary Timekeeping [Kolstad, 1997 after Heinlein]
Consider the Angusonian planetary system. It has N (1 <= N <= 9) planets
circling the Angus sun in perfect circles. The system is so perfect that
the planets trace circular, clockwise orbits that are all in the same plane.
Let's call the planets P1, P2, ..., PN and the respective radius of their
orbits R1, R2, ..., RN. Any planet with orbit of radius 100 mooters requires
exactly one year to orbit the sun. Planets closer to the sun require less
time to complete an orbit; planets farther from the sun require more time.
In fact, the length of a planet's year is precisely:
_ _ 2
| R_i |
Yearlen =| ------- | years
i |_ 100 _|
So a planet whose orbit has radius 200 mooters would require (200/100)^2 =
2^2 = 4 years to complete an orbit. A planet whose orbit has radius 50
mooters would require (1/2)^2 = 0.25 years to complete an orbit.
You will be given a set of planets. For each planet, you will be given the
size of its radius (in mooters) and its initial position for this observation.
Its initial position will be expressed in clock time. The clock time
represents the location of the planet in its orbit. At the specified time,
the `little hand' of an analog clock will point to the position of the planet
in its orbit. By way of example, a planet located at 12:15 is 180 degrees
around the sun from a planet positioned at 6:15.
Given the positions and orbital radii of a set of planets, you are to determine
the date (in years) when the planets will all line up at 3:00 (i.e., all
planets will be positioned in their orbits at the 3:00 angle).
Consider a single planet of orbital radius 100 mooters and located at 12:00.
In 0.25 years, it will be at 3:00.
Consider a pair of planets. Planet 1 has radius 100 mooters and is currently
located at 9:00. Planet 2 has radius 200 mooters and is located at 1:30.
After 0.5 years, they will both be aligned at 3:00.
Your input file is named INPUT.DAT and includes a set of lines each with a
radius and starting time. The last input line is all zeroes and is to be
ignored. The input file looks like this for the previous example:
100 9 00
200 1 30
0 0 0
You are to calculate the number of years until the planets align at 3:00 and
print the answer to four decimal places:
0.5000
It is guaranteed that the planets will eventually line up. The least common
multiple of the planets' radii will be less than 2,000,000,000.
NOTE: `Starting times' (initial angles) might have a floating point number
for the minutes. A planet of radius 100 mooters might have an input line
like this:
100 9 42.5
----------------------------------------------------------------------
SAMPLE INPUT (file INPUT.DAT):
100 9 00
200 1 30
0 0 0
SAMPLE OUTPUT:
0.5000
TEST DATA SET #1:
100 9 00
200 1 30
0 0 0
TEST DATA SET #2:
100 9 00
200 7 30
0 0 0
TEST DATA SET #3:
100 9 00
200 1 30
300 10 20
0 0 0
TEST DATA SET #4:
100 12 00
200 8 15
300 9 20
75 8 20
0 0 0
TEST DATA SET #5:
5000 3 0
100 3 0
200 3 0
0 0 0