2023年12月7日发(作者：罗庄区数学试卷九年级)
2014 MCM Problems
PROBLEM A: The Keep-Right-Except-To-Pass Rule
In countries where driving automobiles on the right is the rule (that is, USA,
China and most other countries except for Great Britain, Australia, and some
former British colonies), multi-lane freeways often employ
a rule that
requires drivers to drive in the right-most lane unless they are
passing another vehicle, in which case they move one lane to the left,
pass, and return to their former travel lane.
Build and analyze a mathematical model to analyze the performance of this rule
in light and heavy traffic. You may wish to examine tradeoffs between traffic
flow and safety, the role of under- or over-posted speed limits (that is, speed
limits that are too low or too high), and/or other factors that may not be
explicitly called out in this problem statement. Is this rule effective in promoting
better traffic flow? If not, suggest and analyze alternatives (to include possibly
no rule of this kind at all) that might promote greater traffic flow, safety, and/or
other factors that you deem important.
In countries where driving automobiles on the left is the norm, argue whether
or not your solution can be carried over with a simple change of orientation, or
would additional requirements be needed.
Lastly, the rule as stated above relies upon human judgment for compliance. If
vehicle transportation on the same roadway was fully under the control of an
intelligent system – either part of the road network or imbedded in the design
of all vehicles using the roadway – to what extent would this change the results
of your earlier analysis?
PROBLEM B: College Coaching Legends
Sports Illustrated, a magazine for sports enthusiasts, is looking for the ―best all
time college coach‖ male or female for the previous century. Build a
mathematical model to choose the
best college coach or coaches (past or
present) from among either male or female coaches in such sports as college
hockey or field hockey, football, baseball or softball, basketball, or soccer. Does
it make a difference which time line horizon that you use in your analysis, i.e.,
does coaching in 1913 differ from coaching in 2013? Clearly articulate your
metrics for assessment. Discuss how your model can be applied in general
across both genders and all possible sports. Present your model’s top 5 coaches
in each of 3 different sports.
In addition to the MCM format and requirements, prepare a 1-2 page article
for
Sports Illustrated that explains your results and includes a non-technical
explanation of your mathematical model that sports fans will understand. 2013 MCM Problems
PROBLEM A: The Ultimate Brownie Pan
When baking in a rectangular pan heat is concentrated in the 4 corners and the
product gets overcooked at the corners (and to a lesser extent at the edges). In
a round pan the heat is distributed evenly over the entire outer edge and the
product is not overcooked at the edges. However, since most ovens are
rectangular in shape using round pans is not efficient with respect to using the
space in an oven. Develop a model to show the distribution of heat across the
outer edge of a pan for pans of different shapes - rectangular to circular and
other shapes in between.
Assume
1. A width to length ratio of
W/L for the oven which is rectangular in shape.
2. Each pan must have an area of
A.
3. Initially two racks in the oven, evenly spaced.
Develop a model that can be used to select the best type of pan (shape) under
the following conditions:
1. Maximize number of pans that can fit in the oven (N)
2. Maximize even distribution of heat (H) for the pan
3. Optimize a combination of conditions (1) and (2) where weights p and (1-
p)
are assigned to illustrate how the results vary with different values
of
W/L and
p.
In addition to your MCM formatted solution, prepare a one to two page
advertising sheet for the new Brownie Gourmet Magazine highlighting your
design and results.
PROBLEM B: Water, Water, Everywhere
Fresh water is the limiting constraint for development in much of the world.
Build a mathematical model for determining an effective, feasible, and
cost-efficient water strategy for 2013 to meet the projected water needs of
[pick one country from the list below] in 2025, and identify the best water
strategy. In particular, your mathematical model must address storage and
movement; de-salinization; and conservation. If possible, use your model to
discuss the economic, physical, and environmental implications of your strategy.
Provide a non-technical position paper to governmental leadership outlining
your approach, its feasibility and costs, and why it is the ―best water strategy
choice.‖
Countries: United States, China, Russia, Egypt, or Saudi Arabia 2012 MCM Problems
PROBLEM A: The Leaves of a Tree
\"How much do the leaves on a tree weigh?\" How might one estimate the actual
weight of the leaves (or for that matter any other parts of the tree)? How might
one classify leaves? Build a mathematical model to describe and classify leaves.
Consider and answer the following:
• Why do leaves have the various shapes that they have?
• Do the shapes ―minimize‖ overlapping individual shadows that are cast, so as
to maximize exposure? Does the distribution of leaves within the ―volume‖ of
the tree and its branches effect the shape?
• Speaking of profiles, is leaf shape (general characteristics) related to tree
profile/branching structure?
• How would you estimate the leaf mass of a tree? Is there a correlation
between the leaf mass and the size characteristics of the tree (height, mass,
volume defined by the profile)?
In addition to your one page summary sheet prepare a one page letter to an
editor of a scientific journal outlining your key findings.
PROBLEM B: Camping along the Big Long River
Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting
white water rapids. The river is inaccessible to hikers, so the only way to enjoy
it is to take a river trip that requires several days of camping. River trips all start
at First Launch and exit the river at Final Exit, 225 miles downstream.
Passengers take either oar- powered rubber rafts, which travel on average 4
mph or motorized boats, which travel on average 8 mph. The trips range from 6
to 18 nights of camping on the river, start to finish.. The government agency
responsible for managing this river wants every trip to enjoy a wilderness
experience, with minimal contact with other groups of boats on the river.
Currently,
X trips travel down the Big Long River each year during a six month
period (the rest of the year it is too cold for river trips). There are
Y camp sites
on the Big Long River, distributed fairly uniformly throughout the river corridor.
Given the rise in popularity of river rafting, the park managers have been asked
to allow more trips to travel down the river. They want to determine how they
might schedule an optimal mix of trips, of varying duration (measured in nights
on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to
the Big Long River’s rafting season? The river managers have hired you to
advise them on ways in which to develop the best schedule and on ways in
which to determine the carrying capacity of the river, remembering that no two
sets of campers can occupy the same site at the same time. In addition to your
one page summary sheet, prepare a one page memo to the managers of the
river describing your key findings.
2011 MCM Problems
PROBLEM A: Snowboard Course
Determine the shape of a snowboard course (currently known as a ―halfpipe‖)
to maximize the production of ―vertical air‖ by a skilled snowboarder.
\"Vertical air\" is the maximum vertical distance above the edge of the halfpipe.
Tailor the shape to optimize other possible requirements, such as maximum
twist in the air.
What tradeoffs may be required to develop a ―practical‖ course?
PROBLEM B: Repeater Coordination
The VHF radio spectrum involves line-of-sight transmission and reception. This
limitation can be overcome by ―repeaters,‖ which pick up weak signals, amplify
them, and retransmit them on a different frequency. Thus, using a repeater,
low-power users (such as mobile stations) can communicate with one another
in situations where direct user-to-user contact would not be possible. However,
repeaters can interfere with one another unless they are far enough apart or
transmit on sufficiently separated frequencies.
In addition to geographical separation, the ―continuous tone-coded squelch
system‖ (CTCSS), sometimes nicknamed ―private line‖ (PL), technology can be
used to mitigate interference problems. This system associates to each
repeater a separate subaudible tone that is transmitted by all users who wish to
communicate through that repeater. The repeater responds only to received
signals with its specific PL tone. With this system, two nearby repeaters can
share the same frequency pair (for receive and transmit); so more repeaters
(and hence more users) can be accommodated in a particular area.
For a circular flat area of radius 40 miles radius, determine the minimum
number of repeaters necessary to accommodate 1,000 simultaneous users.
Assume that the spectrum available is 145 to 148 MHz, the transmitter
frequency in a repeater is either 600 kHz above or 600 kHz below the receiver
frequency, and there are 54 different PL tones available.
How does your solution change if there are 10,000 users?
Discuss the case where there might be defects in line-of-sight propagation
caused by mountainous areas.
2010 MCM Problems
PROBLEM A: The Sweet Spot
Explain the ―sweet spot‖ on a baseball bat.
Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power
is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple
explanation based on torque might seem to identify the end of the bat as the sweet spot, but
this is known to be empirically incorrect. Develop a model that helps explain this empirical
finding.
Some players believe that ―corking‖ a bat (hollowing out a cylinder in the head of the bat and
filling it with cork or rubber, then replacing a wood cap) enhances the ―sweet spot‖ effect.
Augment your model to confirm or deny this effect. Does this explain why Major League
Baseball prohibits ―corking‖?
Does the material out of which the bat is constructed matter? That is, does this model predict
different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major
League Baseball prohibits metal bats?
PROBLEM B: Criminology
In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other
people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to
find a ―center of mass‖ of the locations of the attacks. In the end, the suspect happened to live
in the same town predicted by this technique. Since that time, a number of more sophisticated
techniques have been developed to determine the ―geographical profile‖ of a suspected serial criminal based on the locations of the crimes.
Your team has been asked by a local police agency to develop a method to aid in their
investigations of serial criminals. The approach that you develop should make use of at least
two different schemes to generate a geographical profile. You should develop a technique to
combine the results of the different schemes and generate a useful prediction for law
enforcement officers. The prediction should provide some kind of estimate or guidance about
possible locations of the next crime based on the time and locations of the past crime scenes. If
you make use of any other evidence in your estimate, you must provide specific details about
how you incorporate the extra information. Your method should also provide some kind of
estimate about how reliable the estimate will be in a given situation, including appropriate
warnings.
In addition to the required one-page summary, your report should include an additional
two-page executive summary. The executive summary should provide a broad overview of the
potential issues. It should provide an overview of your approach and describe situations when it
is an appropriate tool and situations in which it is not an appropriate tool. The executive
summary will be read by a chief of police and should include technical details appropriate to the
intended audience
2009 MCM Problems
PROBLEM A: Designing a Traffic Circle
Many cities and communities have traffic circles—from large ones with many
lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory
Monument in Bangkok) to small ones with one or two lanes in the circle. Some
of these traffic circles position a stop sign or a yield sign on every incoming road
that gives priority to traffic already in the circle; some position a yield sign in
the circle at each incoming road to give priority to incoming traffic; and some
position a traffic light on each incoming road (with no right turn allowed on a
red light). Other designs may also be possible.
The goal of this problem is to use a model to determine how best to control
traffic flow in, around, and out of a circle. State clearly the objective(s) you use
in your model for making the optimal choice as well as the factors that affect
this choice. Include a Technical Summary of not more than two double-spaced
pages that explains to a Traffic Engineer how to use your model to help choose
the appropriate flow-control method for any specific traffic circle. That is,
summarize the conditions under which each type of traffic-control method
should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may
vary according to the time of day and other factors). Illustrate how your model
works with specific examples.
PROBLEM B: Energy and the Cell Phone
This question involves the ―energy‖ consequences of the cell phone revolution. Cell phone
usage is mushrooming, and many people are using cell phones and giving up their landline
telephones. What is the consequence of this in terms of electricity use? Every cell phone comes
with a battery and a recharger.
Requirement 1
Consider the current US, a country of about 300 million people. Estimate from available data
the number
H of households, with
m members each, that in the past were serviced by landlines.
Now, suppose that all the landlines are replaced by cell phones; that is, each of the
m members
of the household has a cell phone. Model the consequences of this change for electricity
utilization in the current US, both during the transition and during the steady state. The analysis
should take into account the need for charging the batteries of the cell phones, as well as the
fact that cell phones do not last as long as landline phones (for example, the cell phones get
lost and break).
Requirement 2
Consider a second ―Pseudo US‖—a country of about 300 million people with about the same
economic status as the current US. However, this emerging country has neither landlines nor
cell phones. What is the optimal way of providing phone service to this country from an energy
perspective? Of course, cell phones have many social consequences and uses that landline
phones do not allow. A discussion of the broad and hidden consequences of having only
landlines, only cell phones, or a mixture of the two is welcomed.
Requirement 3
Cell phones periodically need to be recharged. However, many people always keep their
recharger plugged in. Additionally, many people charge their phones every night, whether they
need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US
based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity
from oil. Interpret your results in terms of barrels of oil.
Requirement 4
Estimates vary on the amount of energy that is used by various recharger types (TV, DVR,
computer peripherals, and so forth) when left plugged in but not charging the device. Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day.
Requirement 5
Now consider population and economic growth over the next 50 years. How might a typical
Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for
providing phone service based upon your analysis in the first three requirements. Again,
assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.
2008 MCM Problems
PROBLEM A: Take a Bath
Consider the effects on land from the melting of the north polar ice cap due to the predicted
increase in global temperatures. Specifically, model the effects on the coast of Florida every ten
years for the next 50 years due to the melting, with particular attention given to large
metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the
data used is an important part of the answer.
PROBLEM B: Creating Sudoku Puzzles
Develop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to
define a difficulty level. The algorithm and metrics should be extensible to a varying number of
difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your
algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your
objective should be to minimize the complexity of the algorithm and meet the above
requirements.
2007 MCM Problems
PROBLEM A: Gerrymandering
The United States Constitution provides that the House of Representatives shall be composed
of some number (currently 435) of individuals who are elected from each state in proportion to
the state\'s population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the
district represented by a particular representative shall be determined geographically. This
oversight has led to egregious (at least some people think so, usually not the incumbent)
district shapes that look \"unnatural\" by some standards.
Hence the following question: Suppose you were given the opportunity to draw congressional
districts for a state. How would you do so as a purely \"baseline\" exercise to create the
\"simplest\" shapes for all the districts in a state? The rules include only that each district in the
state must contain the same population. The definition of \"simple\" is up to you; but you need to
make a convincing argument to voters in the state that your solution is fair. As an application of
your method, draw geographically simple congressional districts for the state of New York.
PROBLEM B: The Airplane Seating Problem
Airlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has
become customary to seat passengers with special needs first, followed by first-class
passengers (who sit at the front of the plane). Then coach and business-class passengers are
seated by groups of rows, beginning with the row at the back of the plane and proceeding
forward.
Apart from consideration of the passengers\' wait time, from the airline\'s point of view, time is
money, and boarding time is best minimized. The plane makes money for the airline only when
it is in motion, and long boarding times limit the number of trips that a plane can make in a day.
The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the
problem of minimizing boarding (and deboarding) time.
Devise and compare procedures for boarding and deboarding planes with varying numbers of
passengers: small (85-210), midsize (210-330), and large (450-800).
Prepare an executive summary, not to exceed two single-spaced pages, in which you set out
your conclusions to an audience of airline executives, gate agents, and flight crews.
Note: The 2 page executive summary is to be included IN ADDITION to the reports required by
the contest guidelines.
An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being
followed and the importance to the airline of finding better solutions. The article can be seen
at:/2006/11/14/business/
2006 MCM Problems
PROBLEM A: Positioning and Moving Sprinkler Systems for Irrigation
There are a wide variety of techniques available for irrigating a field. The technologies range
from advanced drip systems to periodic flooding. One of the systems that is used on smaller
ranches is the use of \"hand move\" irrigation systems. Lightweight aluminum pipes with
sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals
to insure that the whole field receives an adequate amount of water. This type of irrigation
system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use
on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time
and effort to move and set up the equipment at regular intervals.
Given that this type of irrigation system is to be used, how can it be configured to minimize the
amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are
asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the
amount of time required by a rancher to maintain the irrigation system. One pipe set is used in
the field. You should determine the number of sprinklers and the spacing between sprinklers,
and you should find a schedule to move the pipes, including where to move them.
A pipe set consists of a number of pipes that can be connected together in a straight line. Each
pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter.
When put together the resulting pipe is 20 meters long. At the water source, the pressure is
420 Kilo- Pascal\'s and has a flow rate of 150 liters per minute. No part of the field should
receive more than 0.75 cm per hour of water, and each part of the field should receive at least
2 centimeters of water every 4 days. The total amount of water should be applied as uniformly
as possible
PROBLEM B: Wheel Chair Access at Airports
One of the frustrations with air travel is the need to fly through multiple airports, and each stop
generally requires each traveler to change to a different airplane. This can be especially difficult
for people who are not able to easily walk to a different flight\'s waiting area. One of the ways
that an airline can make the transition easier is to provide a wheel chair and an escort to those
people who ask for help. It is generally known well in advance which passengers require help,
but it is not uncommon to receive notice when a passenger first registers at the airport. In rare
instances an airline may not receive notice from a passenger until just prior to landing.
Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are
expensive and require maintenance. There is also a cost for making the escorts available.
Moreover, wheel chairs and their escorts must be constantly moved around the airport so that
they are available to people when their flight lands. In some large airports the time required to move across the airport is nontrivial. The wheel chairs must be stored somewhere, but space is
expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas
represent a liability risk as people try to move around them. Finally, one of the biggest costs is
the cost of holding a plane if someone must wait for an escort and becomes late for their flight.
The latter cost is especially troubling because it can affect the airline\'s average flight delay
which can lead to fewer ticket sales as potential customers may choose to avoid an airline.
Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the
issues and costs of keeping and maintaining wheel chairs and escorts available for passengers.
The airline needs to find a way to schedule the movement of wheel chairs throughout each day
in a cost effective way. They also need to find and define the costs for budget planning in both
the short and long term.
Epsilon Airlines has asked your consultant group to put together a bid to help them solve their
problem. Your bid should include an overview and analysis of the situation to help them decide
if you fully understand their problem. They require a detailed description of an algorithm that
you would like to implement which can determine where the escorts and wheel chairs should be
and how they should move throughout each day. The goal is to keep the total costs as low as
possible. Your bid is one of many that the airline will consider. You must make a strong case as
to why your solution is the best and show that it will be able to handle a wide range of airports
under a variety of circumstances.
Your bid should also include examples of how the algorithm would work for a large (at least 4
concourses), a medium (at least two concourses), and a small airport (one concourse) under
high and low traffic loads. You should determine all potential costs and balance their respective
weights. Finally, as populations begin to include a higher percentage of older people who have
more time to travel but may require more aid, your report should include projections of
potential costs and needs in the future with recommendations to meet future needs.
2005 MCM Problems
PROBLEM A: Flood Planning
Lake Murray in central South Carolina is formed by a large earthen dam, which
was completed in 1930 for power production. Model the flooding downstream
in the event there is a catastrophic earthquake that breaches the dam.
Two particular questions:
Rawls Creek is a year-round stream that flows into the Saluda River a short
distance downriver from the dam. How much flooding will occur in Rawls Creek
from a dam failure, and how far back will it extend? Could the flood be so massive downstream that water would reach up to the
S.C. State Capitol Building, which is on a hill overlooking the Congaree River?
PROBLEM B: Tollbooths
Heavily-traveled toll roads such as the Garden State Parkway , Interstate 95,
and so forth, are multi-lane divided highways that are interrupted at intervals
by toll plazas. Because collecting tolls is usually unpopular, it is desirable to
minimize motorist annoyance by limiting the amount of traffic disruption
caused by the toll plazas. Commonly, a much larger number of tollbooths is
provided than the number of travel lanes entering the toll plaza. Upon entering
the toll plaza, the flow of vehicles fans out to the larger number of tollbooths,
and when leaving the toll plaza, the flow of vehicles is required to squeeze back
down to a number of travel lanes equal to the number of travel lanes before the
toll plaza. Consequently, when traffic is heavy, congestion increases upon
departure from the toll plaza. When traffic is very heavy, congestion also builds
at the entry to the toll plaza because of the time required for each vehicle to
pay the toll.
Make a model to help you determine the optimal number of tollbooths to deploy
in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one
tollbooth per incoming travel lane. Under what conditions is this more or less
effective than the current practice? Note that the definition of \"optimal\" is up
to you to determine.
以上是2005年——2014年美国大学生数学建模竞赛试题
更多试题详见/undergraduate/contests/matrix/
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