Undergraduate amth problems

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Undergraduate amth problems

  1. 1. Undergraduate Math Problems Júlia Turró Comas
  2. 2. O bjectives <ul><li>Learn new mathematic techniques </li></ul><ul><li>Solve problems of a certain difficulty </li></ul><ul><li>Try to find more than one solution </li></ul>
  3. 3. Structure <ul><li>Geometry </li></ul><ul><ul><ul><li>Theoretical compendium </li></ul></ul></ul><ul><ul><ul><li>Problems </li></ul></ul></ul><ul><li>Algebra </li></ul><ul><ul><ul><li>Theoretical compendium </li></ul></ul></ul><ul><ul><ul><li>Problems </li></ul></ul></ul><ul><li>Number Theory </li></ul><ul><ul><ul><li>Theoretical compendium </li></ul></ul></ul><ul><ul><ul><li>Problems </li></ul></ul></ul><ul><li>Miscellaneous </li></ul><ul><ul><ul><li>Theoretical compendium </li></ul></ul></ul><ul><ul><ul><li>Problems </li></ul></ul></ul>
  4. 4. Geometry <ul><li>Relation between angles, sides and area of plane triangles </li></ul><ul><li>Pythagoras' theorem </li></ul><ul><li>Thales theorem. Similarity of triangles </li></ul><ul><li>Apollonius’ theorem </li></ul><ul><li>Ptolemy’s theorem </li></ul><ul><li>Angles in a circumference </li></ul><ul><li>Trigonometric functions of multiples angles and other relations </li></ul><ul><li>Golden ratio </li></ul><ul><li>Outstanding points of a triangle </li></ul><ul><li>Power of a point with respect to a circumference </li></ul>
  5. 5. Problem 1 <ul><li>A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? </li></ul>
  6. 6. Algebra <ul><li>Fundamental theorem of Algebra </li></ul><ul><li>Cardano-Vieta formulas </li></ul><ul><li>Maximum and minimum of derivable functions </li></ul><ul><li>Binomial formula </li></ul><ul><li>Polynomial remainder theorem </li></ul><ul><li>Homogeneous systems </li></ul>
  7. 7. Problem 2 <ul><li>Show that the following system has no real solution. </li></ul>
  8. 8. Number theory <ul><li>Fermat’s little theorem </li></ul><ul><li>Congruence relation </li></ul><ul><li>Geometric series </li></ul>
  9. 9. Miscellaneous <ul><li>Pigeonhole principle </li></ul><ul><li>Principle of mathematical induction </li></ul><ul><li>Integer part </li></ul>
  10. 10. Conclusions <ul><li>Acquisition of new mathematic concepts </li></ul><ul><li>Discovered mathematics' ambiguity </li></ul><ul><li>Gained knowledge in English </li></ul><ul><li>Unknown mathematics’ world </li></ul>

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