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The Contest Problem Book VIII: American Mathematics Competitions (AMC 10) (MAA Problem Book Series) [J. Douglas Faires, David Wells] on. The Contest Problem Book VIII is the first collection of problems from that competition covering the years – J. Douglas Faires and David Wells were. The Contest Problem Book VIII is the first collection of problems from that competition, covering the years J. Douglas Faires and David Wells were.

The Contest Problem Book Viii

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There are problems from the first 14 contests included in this collection. A Problem Index at the back of the book classifies the problems in the following. Dec 31, The Contest Problem Book VIII is the first collection of problems from that competition covering the years – J. Douglas Faires and. Compiled by J. Douglas Faires, Youngstown State University, David Wells, Pennsylvania State University at New Kensington; Publisher: Mathematical.

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The Contest Problem Book VIII

They could not do so because those relations at that time were not yet so plainly developed as they have been since from history. It is just those very campaigns of , , , and following ones, which have made it easier for us to form a conception of modern absolute war in its destroying energy. Theory demands, therefore, that at the commencement of every war its character and main outline shall be defined according to what the political conditions and relations lead us to anticipate as probable.

The more, that according to this probability its character approaches the form of absolute war, the more its outline embraces the mass of the belligerent states and draws them into the vortex, so much the more complete will be the relation of events to one another and the whole, but so much the more necessary it will also be not to take the first step without thinking what may be the last.

The compulsion which we must use towards our enemy will be regulated by the proportions of our own and his political demands. In so far as these are mutually known they will give the measure of the mutual efforts; but they are not always quite so evident, and this may be a first ground of a difference in the means adopted by each.

The situation and relations of the states are not like each other; this may become a second cause.

The strength of will, the character and capabilities of the governments are as little like; this is a third cause. These three elements cause an uncertainty in the calculation of the amount of resistance to be expected, consequently an uncertainty as to the amount of means to be applied and the object to be chosen. As in war the want of sufficient exertion may result not only in failure but in positive harm, therefore, the two sides respectively seek to outstrip each other, which produces a reciprocal action.

This might lead to the utmost extremity of exertion, if it was possible to define such a point. But then regard for the amount of the political demands would be lost, the means would lose all relation to the end, and in most cases this aim at an extreme effort would be wrecked by the opposing weight of forces within itself.

In this manner, he who undertakes war is brought back again into a middle course, in which he acts to a certain extent upon the principle of only applying so much force and aiming at such an object in war as are just sufficient for the attainment of its political object. To make this principle practicable he must renounce every absolute necessity of a result, and throw out of the calculation remote contingencies.

Here, therefore, the action of the mind leaves the province of science, strictly speaking, of logic and mathematics, and becomes, in the widest sense of the term, an art, that is, skill in discriminating, by the tact of judgment among an infinite multitude of objects and relations, that which is the most important and decisive. This tact of judgment consists unquestionably more or less in some intuitive comparison of things and relations by which the remote and unimportant are more quickly set aside, and the more immediate and important are sooner discovered than they could be by strictly logical deduction.

In order to ascertain the real scale of the means which we must put forth for war, we must think over the political object both on our own side and on the enemy's side; we must consider the power and position of the enemy's state as well as of our own, the character of his government and of his people, and the capacities of both, and all that again on our own side, and the political connections of other states, and the effect which the war will produce on those States.

That the determination of these diverse circumstances and their diverse connections with each other is an immense problem, that it is the true flash of genius which discovers here in a moment what is right, and that it would be quite out of the question to become master of the complexity merely by a methodical study, this it is easy to conceive.

In this sense Buonaparte was quite right when he said that it would be a problem in algebra before which a Newton might stand aghast. If the diversity and magnitude of the circumstances and the uncertainty as to the right measure augment in a high degree the difficulty of obtaining a right result, we must not overlook the fact that although the incomparable importance of the matter does not increase the complexity and difficulty of the problem, still it very much increases the merit of its solution.

In men of an ordinary stamp freedom and activity of mind are depressed not increased by the sense of danger and responsibility: but where these things give wings to strengthen the judgment, there undoubtedly must be unusual greatness of soul. First of all, therefore, we must admit that the judgment on an approaching war, on the end to which it should be directed, and on the means which are required, can only be formed after a full consideration of the whole of the circumstances in connection with it: with which therefore must also be combined the most individual traits of the moment; next, that this decision, like all in military life, cannot be purely objective but must be determined by the mental and moral qualities of princes, statesmen, and generals, whether they are united in the person of one man or not.

The subject becomes general and more fit to be treated of in the abstract if we look at the general relations in which States have been placed by circumstances at different times.

We must allow ourselves here a passing glance at history. Half-civilised Tartars, the Republics of ancient times, the feudal lords and commercial cities of the Middle Ages, kings of the eighteenth century, and, lastly, princes and people of the nineteenth century, all carry on war in their own way, carry it on differently, with different means, and for a different object. The Tartars seek new abodes. They march out as a nation with their wives and children, they are, therefore, greater than any other army in point of numbers, and their object is to make the enemy submit or expel him altogether.

By these means they would soon overthrow everything before them if a high degree of civilisation could be made compatible with such a condition. The old Republics with the exception of Rome were of small extent; still smaller their armies, for they excluded the great mass of the populace: they were too numerous and lay too close together not to find an obstacle to great enterprises in the natural equilibrium in which small separate parts always place themselves according to the general law of nature: therefore their wars were confined to devastating the open country and taking some towns in order to ensure to themselves in these a certain degree of influence for the future.

Rome alone forms an exception, but not until the later period of its history. For a long time, by means of small bands, it carried on the usual warfare with its neighbours for booty and alliances. It became great more through the alliances which it formed, and through which neighbouring peoples by degrees became amalgamated with it into one whole, than through actual conquests.

It was only after having spread itself in this manner all over Southern Italy, that it began to advance as a really conquering power. Carthage fell, Spain and Gaul were conquered, Greece subdued, and its dominion extended to Egypt and Asia. At this period its military power was immense, without its efforts being in the same proportion. These forces were kept up by its riches; it no longer resembled the ancient republics, nor itself as it had been; it stands alone.

Just as peculiar in their way are the wars of Alexander. With a small army, but distinguished for its intrinsic perfection, he overthrew the decayed fabric of the Asiatic States; without rest, and regardless of risks, he traverses the breadth of Asia, and penetrates into India. No republics could do this. Only a king, in a certain measure his own condottiere, could get through so much so quickly. The great and small monarchies of the middle ages carried on their wars with feudal armies.

Everything was then restricted to a short period of time; whatever could not be done in that time was held to be impracticable. The feudal force itself was raised through an organisation of vassaldom; the bond which held it together was partly legal obligation, partly a voluntary contract; the whole formed a real confederation. The armament and tactics were based on the right of might, on single combat, and therefore little suited to large bodies.

In fact, at no period has the union of States been so weak, and the individual citizen so independent. All this influenced the character of the wars at that period in the most distinct manner. They were comparatively rapidly carried out, there was little time spent idly in camps, but the object was generally only punishing, not subduing, the enemy. They carried off his cattle, burnt his towns, and then returned home again. The great commercial towns and small republics brought forward the condottieri.

That was an expensive, and therefore, as far as visible strength, a very limited military force; as for its intensive strength, it was of still less value in that respect; so far from their showing anything like extreme energy or impetuosity in the field, their combats were generally only sham fights.

The Contest Problem Book VIII: American Mathematics Competitions (AMC 10)...

In a word, hatred and enmity no longer roused a state to personal activity, but had become articles of trade; war lost great part of its danger, altered completely its nature, and nothing we can say of the character it then assumed, would be applicable to it in its reality. The feudal system condensed itself by degrees into a decided territorial supremacy; the ties binding the State together became closer; obligations which concerned the person were made subject of composition; by degrees gold became the substitute in most cases, and the feudal armies were turned into mercenaries.

The condottieri formed the connecting link in the change, and were therefore, for a time, the instrument of the more powerful States; but this had not lasted long, when the soldier, hired for a limited term, was turned into a standing mercenary, and the military force of States now became an army, having its base in the public treasury. It is only natural that the slow advance to this stage caused a diversified interweaving of all three kinds of military force.

Under Henry IV. The condottieri carried on their existence up to the period of the Thirty Years' War, indeed there are slight traces of them even in the eighteenth century.

The other relations of the States of Europe at these different periods were quite as peculiar as their military forces. Upon the whole, this part of the world had split up into a mass of petty States, partly republics in a state of internal dissension, partly small monarchies in which the power of the government was very limited and insecure.

A State in either of these cases could not be considered as a real unity; it was rather an agglomeration of loosely connected forces. The owner equivalent has written maybe was. William Shakespeare, Romeo and Juliet browser is dispatched comparable by a theory of ia Kant, First Critique, B ; Catholic F motivates used minor by a defense of g.

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