-
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 t
he 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 (h
ollowing 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?
MCM 2010
A
题：解释棒球棒上的
“
最佳击球点<
/p>
”
每一个棒球手都知道在棒球棒比较粗
的
部分有一个击球点，这里可以把打击球的
力量最大程度地转移
到球上。为什么这个
点不在棒球棒的最末端？基于力矩的解释
或
许可以解释确定棒球棒的最末端就是最
佳的击球点，但是实际当中并不是这样
的。
构建一个模型帮助解释实际当中的这个
发现。有一些棒球手相信在最佳击球点添
充上软木塞可以提高打击效
果（在球棒头
部挖一个圆柱状槽，
填充上软木塞或者橡
皮。进一步扩展模型确认或者否定该结
论。
这个解释是否可以解释为什么棒球联
盟否定这种做法。
球棒是否和材质有关系，模型是否可以
预测木头和金属球棒的不同打击
效果？这
是否是联盟禁止金属球棒的原因？
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 seria
l 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.
MCM 2010
B
题：犯罪学问题
1981
年
Peter Sutcli
ffe(
萨克利夫被判刑因为他参与了十三起谋杀和对其他人的恶毒攻击。缩小搜索
Sutcliffe
的方法之一是发现一个攻击位置的
“
质心
”.
最终犯罪嫌疑人
恰好生活在该方法预测的同一个小镇。从那时
起，已经发展出一系列更加复杂的技术用来
预测基于犯罪地点的具有地理效应（地理轮廓）的系列犯罪行为。
你的团队被一个当地警察局要求发展出一种方法用来帮助他们的系列犯罪调查。你们的方法应该至少需要利用
两种不同的情景以生成地理效应（地理轮廓），进而根据不同情况下的分析结果对执法人
员提供有效的预测。基于
以往犯罪的时间和位置，预测信息应该提供一些估计或指导下次
可能的犯罪地点。如果在预测中用到了其它的信
息，必须提供特别的细节说明告诉我们这
些信息是如何被整合的。你们的方法中也应该包括在给定条件下（包括适
当警告信息）下
预测的可靠性估计。
除了必要的一页小结，你们的报告应该包
括两页额外的总结。这两页总结应该提供潜在问题的概述，它应该提
供什么情况下，你们
提出的方法是一个恰当的工具，在哪些情况下它不是。执行摘要将宣读了警察局长，并应包括
适当的目标受众的技术细节。
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 s
nowboarder.
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?
MCM 2011 A
题：滑雪场
<
/p>
请设计一个单板滑雪场（现为
“
半管
”
或
“U
型池
”
）的形状，以便能使熟练的单板滑雪选手最大限度地产生垂直腾<
/p>
空。
“
垂直腾
空
“
是超出
“
半管
”
边缘以上的最大的垂直距离。
定制形状时要优化其他可能的要求，如：在空中产生最大的身体扭曲。
< br>
在制定一个
“
实用
”
的场地时哪些权衡因素可能需要？
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
squelc
h 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.
MCM 2011
B
题：中继站的协调
特高频无线电频
谱包含信号的发送和接
受。这种限制可以被中继站所克服。中继
站可以捕捉到微弱的信号，然后把它放
大，再用不同的频率重新发送。这样，低
功耗的用户，例如移动电话用户，在不能<
/p>
直接与其他用户联系的地方可以通过中继
站来保持联系。然而
,
中继站之间会互相
影响
< br>,
除非彼此之间有足够远的距离或通
过充分分离的频率来
传送。
除了地理的分离、
“
连续编码音调控制
系统
” (CTCSS,
有时被称为
“
私人专线
”
(PL
、通过这项技术可以减轻干扰问题。
该系统连接每个中继站，靠的是所有通过
同一个中继站连接的用户发送的独
立的亚
音频音调来连接。中继站只回应接收到的
具有特殊
PL
的语调的信号。通过这个系
统
,
两个附近的中继站可以共享相同的频
率对
(
包括接收和发送
;
对于更多的中继
站
(
并且更多的用户
可以安置在一个特
定的区域。
在一个半径
40
英里的圆形区域，
请你
设计一个方案，用最少量的中继站来容纳
1000
同时在线用户。假设频谱范围是
145
到
148
兆赫
,
在中继站中的发射机<
/p>
的频率要么是
600
千赫以上，要么低于
接收机频率
600
千赫、并且这里有
54
个不同的
PL
可用。
如果这里有
10,000
个用户，如何改
变你的解决方案。
<
/p>
在由于山区引起信号传播的阻碍的地
区，讨论这样的情形。
2012 MCM Problems
PROBLEM A: The Leaves of a
Tree
tree
weigh?
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: