【全集】1997 MCM A (疾走龙属)问题 特等奖论文 教程

raisor

谢谢分享～～～～～～～～～～～～～～～～～～

Ms浩

Very good~!Very good~!Very good~!

huang200832000

很好，很强大！！！！！！！！！！！！

sky22710

很好的东东呀！我顶顶~~~

一杯子

太感谢了……好人啊……………………………………………………

一杯子

看了楼主的帖子,不由得精神为之一振,自觉七经八脉为之一畅,七窍倒也开了六巧半,自古英雄出少年,楼主年纪轻轻,就有经天纬地之才,定国安邦之智,古人云,卧龙凤雏得一而安天下,而今,天佑我大中华,沧海桑田5000年,中华神州平地一声雷,飞沙走石,大雾迷天,朦胧中,只见顶天立地一金甲天神立于天地间,这人英雄手持双斧,二目如电,一斧下去,混沌初开,二斧下去,女娲造人,三斧下去,小生倾倒.得此大英雄,实耐之幸也,民之福也,怎不叫人喜极而泣.......古人有少年楼主说为证,少年之楼主如红日初升，其道大光；河出伏流，一泻汪洋；潜龙腾渊，鳞爪飞扬；乳虎啸谷，百兽震惶；鹰隼试翼，风尘吸张；奇花初胎，皇皇；干将发硎，有作其芒；天戴其苍，地履其黄；纵有千古，横有八荒；小生对楼主之仰慕如滔滔江水连绵不绝,海枯石烂,天崩地裂,永不变心.

yates

好东西!!!!
顶
% x/ ]# I7 D" A& Y% v$ l' l* G7 Z这是不是今年某个学校的的数模题目吖? 真的吗????

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AMCM97问题-A- z( Q1 d2 U; ?  O; F9 X5 P( a  c
Velociraptor(疾走龙属)问题




; }% k' ?7 P& xVelociraptor，Velociraptor mongo1iensis是生活在距今约7500万年前后白垩纪(译注：白垩纪为距今1.36-0.65亿年的地质年代，是中生代最后的纪)的一种食肉(捕食其他动物的)恐龙。
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* ^& M% L0 n" E6 u( ^    古生物学家认为这是一种非常顽强的猎食其他动物的野兽，而且可能是成对或成群地外出追猎。然而，不幸的是无法像观察现代哺乳食肉动物在野外是如何迫猎其食物的行为那样观察到Velociraptor在野外的追猎行为。一组古生物学家来到你们队请求你们在Velociraptor的追猎行为的建模方面给予帮助。他们希望把你们的结果与研究狮子、老虎及其他类似的食肉动物行为的生物学家的研究报告相比较。



2 H) c" p* S8 h2 W  ]    平均的成年Velociraptor长3米，髋高0.5米，重约45公斤。据估计，这种动物跑得非常快，速度可达60公里/小时，持续约15秒。在以这种速度开始助冲刺后．它要停下来在其肌肉中积聚乳酸以恢复体力。
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    假设Velociraptor摘食一种称为Thescelosaurus(太西龙届)neglectus的大小与Velociraptor差不多的双足食草动物。从Thescelosaurus化石的生物力学分析得知Thescelosaurus可以50公里/小时的速度长时间奔跑。
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* H: h: ^. t* g( |9 G第1部分
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( |' X- X7 {: s; i; {9 Y3 g2 S    假设Velociraptor是一只独居的猎食其他动物的野兽，试设计一个单个的Velociraptor潜近猎物并追猎一只单个的Thescelosaurus的策略以及被追捕物逃避迟捕的策略的数学模型。假设当Vclociraptor潜近15米内时，Thescelosaurus总能觉察到，根据栖息地及气候的条件不同，甚至在(多达50米的)更大的范围内觉察欲捕食它的动物的存在。此外，由于Vclociraptor的身体结构及体能，它在全速奔跑时的拐弯半径是受到**的。据估计，拐弯半径大约是其髋高的三倍。另一方面，Thescelosaurus却是极其灵活的．其拐弯半径只有0.5米。
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第2部分
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    更现实地假设Vclociraptor是成对外出追猎，试设计一个新的关于成对的Vclociraptor潜近猎物并追猎一只单个的Thescelosaurus的策略以及被追捕物逃避追捕的策略的数学模型。利用第1部分给出假定和**。

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Problem A: The Velociraptor Problem
The velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter and may that hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild, as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.
The average adult velociraptor was 3 m long with a hip height of 0.5 m and an approxiate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60km/hr, for about 15 sec. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.
Suppose that velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr for long periods of time.
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Part 1
: k) q4 q/ l% T6 [Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15m,but may detect the predator at even greater ranges (up to 50m) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 m.

, X1 v0 x* K* K5 l0 J! b+ NPart 2
Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.

3 [/ g; h' j0 \. ?Problem B: Mix Well For Fruitful Discussions
; r' c+ {( k0 _8 R% ]Small group meeting for the discussion of important issues, particularly long-range planning ,are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings, the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run the risk of control by a dominant personality. In an attempt to reduce this danger, it is common to schedule several sessions with a different mix of people in each group.
' r% z( D) T( L5 |* V' M) _5 A0 wA meeting of An Tostal Corporation will be attended by 29 board members of which nine are in-house members (i.e., corporate employees). The meeting is to be an ally-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A. M. To 4:00 P. M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by one of the corporation's six senior officers. None of these officers is a board member. Thus, each senior officer will lead three different discussion groups. The senior officers will not be involved in the afternoon sessions, and each of these sessions will consist of only four different discussion groups.
The president of the corporation wants a list of board-member assignments to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions. The assignments should also satisfy the following criteria:

1．For the morning sessions, no board member should be in the same senior officer's discussion group twice.
0 ~- X2 w, k8 J: i2．No discussion group should contain a disproportionate number of in-house members.
: r# t& n5 I; W6 zGive a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee.

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建模Problem A: The Velociraptor Problem
The velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter and may that hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild, as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.
The average adult velociraptor was 3 m long with a hip height of 0.5 m and an approxiate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60km/hr, for about 15 sec. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.
  G" C/ T5 C) C5 j4 t  RSuppose that velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr for long periods of time.

# k, x9 ~0 X) v' v- S  H4 I+ QPart 1
0 U2 [/ s0 F7 ]$ o* q/ r- PAssuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15m,but may detect the predator at even greater ranges (up to 50m) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 m.
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! o+ U0 v' {! }# Q% \7 a- kPart 2
Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.
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Problem B: Mix Well For Fruitful Discussions
Small group meeting for the discussion of important issues, particularly long-range planning ,are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings, the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run the risk of control by a dominant personality. In an attempt to reduce this danger, it is common to schedule several sessions with a different mix of people in each group.
  W! f( w2 K+ n7 ?7 P0 w! ?8 ^5 XA meeting of An Tostal Corporation will be attended by 29 board members of which nine are in-house members (i.e., corporate employees). The meeting is to be an ally-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A. M. To 4:00 P. M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by one of the corporation's six senior officers. None of these officers is a board member. Thus, each senior officer will lead three different discussion groups. The senior officers will not be involved in the afternoon sessions, and each of these sessions will consist of only four different discussion groups.
The president of the corporation wants a list of board-member assignments to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions. The assignments should also satisfy the following criteria:

! _9 o/ S+ Y% y' L5 q1．For the morning sessions, no board member should be in the same senior officer's discussion group twice.
2．No discussion group should contain a disproportionate number of in-house members.
5 t  x% g* Q5 C, ?$ G5 a$ i4 s6 i1 @Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee.