Mathematical Olympiad Challenges

Andreescu, Titu; Gelca, Răzvan

doi:10.1007/978-1-4612-2138-8

Cited by 21 publications

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“…Una vez que llega al punto E, a Jennifer no le queda otra alternativa que dirigirse al punto F, pero Steven podría moverse hacia el punto E o hacia el punto G, por lo que la probabilidad de que Steven y Jennifer se encuentren en el segmento de la calle que une E y F es entonces: 1 8 • 1 8 • 1 2 = 1 2 7 . Caso 2: El caso en el que Jennifer llega al punto Q y Steven al punto T es análogo al anterior, y por lo tanto, la probabilidad de que Steven y Jennifer se encuentren en el segmento de la calle que une T con Q es también: 1 8 • 1 8 • 1 2 = 1 2 7 . Caso 3: El caso en el que Jennifer llega al punto Q y Steven al punto P. La probabilidad de que Jennifer llegue a Q es de 1 2 • 1 2 • 1 2 = 1 8 .…”

Section: 111unclassified

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Problemas de exámenes y soluciones XXX Olimpiada Costarricense de Matemática, 2018

Páez

Abarca

Chaves

et al. 2020

RDMEI

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OLCOMA es la Comisión de Olimpiadas Costarricenses de Matemáticas. Esta comisión está conformada por académicos en el área de Matemática de la Universidad Nacional, del Instituto Tecnológico de Costa Rica, de la Universidad de Costa Rica, de la Universidad Estatal a Distancia, y por representantes del Ministerio de Educación Pública y del Ministerio de Ciencia, Tecnología y Telecomunicaciones. Dentro de la estructura de las Olimpiadas Costarricenses de Matemáticas, se tienen tres niveles de competencia. En Nivel I participan estudiantes de sétimo año, entre 10 y 13 años, aproximadamente (también pueden optar por participar en este nivel los estudiantes de primaria que cursan sexto grado). En Nivel II participan estudiantes de octavo y noveno años, entre 14 y 15 aõs, aproximadamente. En Nivel III participan estudiantes de décimo, undécimo y duodécimo años, más de 15 años. Durante el año 2017, se trabajó con tres etapas. En la I Eliminatoria los estudiantes resolvieron un examen de selección única con 25 ítems. En la II Eliminatoria, los clasificados realizaron un examen con 12 preguntas de selección única y tres problemas de desarrollo. En la Etapa Final, los estudiantes clasificados realizaron dos pruebas en días consecutivos, cada prueba con tres problemas de desarrollo. El objetivo de este libro es que los futuros participantes en Olimpiadas Costarricenses de Matemáticas posean material de consulta. Los problemas que en este libro se enuncian representan un reto para quienes gustan resolver ejercicios matemáticos. Cada uno de los problemas tiene la solución respectiva, así que una vez que intenten resolver cada problema, pueden comparar el procedimiento realizado con la solución propuesta. Se han incorporado todos los problemas de I Eliminatoria para cada uno de los tres niveles, los problemas de II Eliminatoria para cada uno de los tres niveles, los problemas propuestos para conformar los exámenes de la Etapa Final y los problemas que conformaron los exámenes de la Etapa Final. Abstract OLCOMA is the Costa Rican Mathematics Olympics Commission. This commission is made up of academics in the area of Mathematics from the Universidad Nacional, the Instituto Tecnológico de Costa Rica, the Universidad de Costa Rica, the Universidad Estatal a Distancia, and the Universidad Autónoma de Costa Rica and by representatives of the Ministry of Public Education and of the Ministry of Science, Technology, Culture and TelecomuncationsWithin the structure of the Costa Rican Mathematical Olympiads, there are three levels of competition. In Level I there are students of seventh year, between 10 and 13 years old, approximately (also the students of primary school that are studying in this level can choose to participate in this level sixth grade). In Level II, students of eighth and ninth years, between 14 and 15 years, approximately, participate. In Level III there are students of tenth, eleventh and twelfth years, plus 15 years old. During the year 2017, we worked with three stages. In the First Elimination, the students resolved toa single selection test with 25 items. In the II Eliminatory, the classified ones carried out aexam with 12 single-choice questions and three developmental problems. In the Final Stage, the classified students took two tests on consecutive days, each test with three tests developmental problems. The objective of this book is that future participants in Costa Rican Mathematics Olympiads possess reference material. The problems listed in this book represent a challenge for those who like to solve mathematical exercises. Each of the problems has the respective solution, so once who try to solve each problem, can compare the procedure carried out with the proposed solution. All the problems of I Eliminatory for each one of the three levels, the problems of II Eliminatory for each one of the three levels, the problems proposed to conform the examinations of the Final Stage and the problems that conformed the examinations of the Final Stage have been incorporated.

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Section: 111unclassified

“…a. Steven está en el punto F: Steven llega al punto F con una probabilidad de 1 8 . En ese caso se cruzarían en el segmento de la calle que une F con G. Steven en ese caso tendría que escoger una de dos opciones y de igual manera Jennifer.…”

Section: 111unclassified

Problemas de exámenes y soluciones XXX Olimpiada Costarricense de Matemática, 2018

Páez

Abarca

Chaves

et al. 2020

RDMEI

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“…As shown in [4] and [5] such a tetrahedron has congruent (acute-angled) faces and is called a disphenoid. Also shown in Figure 2 is a net that folds up to produce a disphenoid.…”

Section: Agure 2: Disphenoid Illustrating Viviani In 3dmentioning

confidence: 99%

3D Generalisations of Viviani's theorem

Villiers

2013

Math. Gaz.

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“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

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“…This non-obvious and interesting fact is attributed to the Romanian mathematician Dimitrie Pompeiu (1873-1954), and now carries his name. It, too, appeared, with different proofs quite frequently; see, for example, [2], [3], [11], [1], [26], [20], [4], and [31].…”

Section: Introductionmentioning

confidence: 99%

Pre-kites: simplices having a regular facet

2017

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Abstract. The investigation of the relation among the distances of an arbitrary point in the Euclidean space R n to the vertices of a regular n-simplex in that space has led us to the study of simplices having a regular facet. Calling an n-simplex with a regular facet an n-pre-kite, we investigate, in the spirit of [14], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if n ≥ 3, then the intersection of the family of n-pre-kites with any of the four known special families is the family of n-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.Mathematics Subject Classification (2010). Primary 52B11; Secondary 52B12, 52B15, 51M20, 52B10.

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Mathematical Olympiad Challenges

Cited by 21 publications

References 0 publications

Problemas de exámenes y soluciones XXX Olimpiada Costarricense de Matemática, 2018

Problemas de exámenes y soluciones XXX Olimpiada Costarricense de Matemática, 2018

3D Generalisations of Viviani's theorem

Pre-kites: simplices having a regular facet

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