2000 H2 Math Paper 1

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Azalee Freas

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Aug 5, 2024, 5:44:43 AM8/5/24

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TheWilliam Lowell Putnam Mathematics Competition is a North American mathcontest for college students, organized by the Mathematical Association of America (MAA). Each year on the first Saturday in December,several thousand US and Canadian students spend 6 hours (in two sittings) trying to solve 12problems. Individual and team winners (and their schools, in the lattercase) get some money and a few minutes of fame.

Problem statements given here are verbatim from the competition (except somediagrams are missing), and are copyrighted by the MAA. They appear here with permission, but further redistribution is subject to copyright restrictions which I am not authorized to waive; contact the MAA for assistance.

Solutions given here have been compiled (in some combination) by Manjul Bhargava, Kiran Kedlaya, and Lenhard Ng based on numerous sources (see below). Copyright is held by the named authors, who request that youlink to this page in lieu of reproducing these solutions elsewhere. Also, pleasedo not refer to these as "official solutions", as this describes the solutions issued by the MAA which reflect the intentof the problem setters. Those appear in the official competition summary (see below).

New problems and solutions will be posted here no sooner than the following Monday at 8pm Eastern (5pm Pacific); this embargo was setin consultation with competition director Daniel Ullman, in order to give exam supervisors time to mail scan and upload their exam papers. We suggest that others observe a similar delay before discussing the exam online.

Competition results up to 2016 were transcribed by hand from the printed summary of results, which was sent by mail (I have added scans of the original documents where I have them).As of the 2017 competition, both the original summary of results, taken from the MAA historical Putnam results page, and old-style HTML transcriptions are provided.

This page is maintained by Kiran Kedlaya;contact me with corrections and other comments about this material only (including additional scans of historical results summaries).I cannot assist with exam administration. If you are a student, see your school's Putnam coordinator; if you are a coordinator, see the MAA's Putnam home page for instructions.

The Mathematics SubjectClassification (MSC) is an alphanumericalclassification scheme formulated by the American Mathematical Societybased on the coverage of two major reviewing databases MathematicalReviews and Zentralblatt MATH. It is used by many mathematics journals,which ask authors of research papers and expository articles to listsubject codes from the Mathematics Subject Classification in theirpapers.(Taken fromWikipedia.)

These Ten-Year-Series (TYS) worked solutions with video explanations for 2000 A Level H2 Mathematics Paper 1 Question 8 are suggested by Mr Gan. For any comments or suggestions please contact us at sup...@timganmath.edu.sg.

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collge de France. Three lectures were presented: Timothy Gowers spoke on The Importance of Mathematics; Michael Atiyah and John Tate spoke on the problems themselves.

The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years.

Following the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem.

It is of note that one of the seven Millennium Prize Problems, the Riemann hypothesis, formulated in 1859, also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.

The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

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This paper gives a review of cross-validation methods. The original applications in multiple linear regression are considered first. It is shown how predictive accuracy depends on sample size and the number of predictor variables. Both two-sample and single-sample cross-validation indices are investigated. The application of cross-validation methods to the analysis of moment structures is then justified. An equivalence of a single-sample cross-validation index and the Akaike information criterion is pointed out. It is seen that the optimal number of parameters suggested by both single-sample and two-sample cross-validation indices will depend on sample size. Copyright 2000 Academic Press.

Includes inequalities, functions and their graphs, polynomial and rational functions, exponential and logarithmic functions, systems of linear and nonlinear equations, matrices and determinants, arithmetic and geometric sequences, and the Binomial Theorem. May be delivered hybrid and/or online.

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Is an accelerated version of MATH 1050 and MATH 1060. Includes functions and their graphs including polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Covers inequalities, systems of linear and nonlinear equations, matrices, determinants, arithmetic and geometric sequences, the Binomial Theorem, the unit circle, right triangle trigonometry, trigonometric equations, trigonometric identities, the Law of Sines, the Law of Cosines, vectors, complex numbers, polar coordinates, and conic sections.

Uses linear, quadratic, power, polynomial, rational, exponential, logarithmic, and logistic functions to analyze business applications such as market equilibrium, rates of change, cost-benefit analysis, and inflation. Includes systems of linear and non-linear equations and inequalities, matrices and matrix equations, sequences and series, and financial mathematics. Canvas Course Mats $90/McGraw applies.

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Covers limits, continuity, differentiation, applications of differentiation, integration, and applications of integration, including derivatives and integrals of polynomial functions, rational functions, exponential functions, logarithmic functions, trigonometric functions, inverse trigonometric functions, and hyperbolic functions. Is a prerequisite for calculus-based sciences.

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