A.Shapovalov => Lessons

NYUAD, 2015

Here are lists of problems for 1st year students from New York University in Abu Dhabi during their preparation to IMC2015.

1

5 problems for the first acquaintance

April 20th

pdf    doc

 5 problems from Tournament of Towns

2

10 problems with a short answer

April 27th

pdf    doc

 10 problems to which students may give only short answer (e.g. just a number)

3

Example+Estimate

May 2nd

pdf    odt

 The problems of this kind consist of two parts to be treated separately. First, one should give an example with the optimal value. Second, one should prove that a better example does not exist (this can be usually done by giving proving an upper or a lower estimate for all possible examples).

4

Quadratic polynomials

May 5th

Problems: pdf    odt
Answers: pdf    odt

 A quadratic polynomial makes a good step towards a general polynomial. Train your intuition. Try to decide if the answer is yes or no before you find a complete solution of the problem. If you think there should be an example, try to find it. Otherwise try to find an explanation why any example is not possible.

5

Polynomials and Equations

May 9th

Theorems and Problems: pdf    odt
Answers and Solutions: pdf    odt

 Polynomials occur quite often in mathematics. One can look at them from different point of view – as expressions, as functions, as curves. Learn you to switch from one point of view to another at the right moment.

6

Sums

May 14th

pdf    odt

 10 sums to train yourself. Give just an answer.

7

Sequnces and Sums

May 21st

pdf    odt

 How to find sums using equations.

8

Right or Wrong: Examples in Calculus

May 25th

pdf    odt

 How to construct an example and find out, if an assumption is true or false.

9

Inequalities and Derivatives

May 25th

pdf    odt

 You know how to find intervals where a function is increasing, decreasing, convex, or concave. Then you know how to prove inequalities!

10

Right or Wrong: Prime Factors

May 26th

pdf    odt

  A complex construction can be made of simple bricks. In number theory these are prime numbers.

11

Polynomials with Integer Coefficients

May 26th

pdf    odt

  The main idea: if m, n are distinct integers, and P is a polynomial with integer coefficients, then P(m)P(n) is divisible by mn.

12

Remainders and Modular Arithmetic

May 27th

pdf    odt

  Remainders can be treated as "usual" numbers. The advantage is the list of such numbers is finite. Translation to the "language of remainders" (and back) proved to be very useful.

12a

Diophantine Equations and Factorization

May 27th

pdf    odt

  It is easy to prove that an equation has no solution modulo p if you choose an apropriate prime p. But that proves that the equation has no integer solution.

13

Finite Fields

May 28th

pdf    odt

  Zp is like numbers but finite. A polynomial with integer coefficients can be replaced with the polynomial with Zp coefficients.

14

Infinite Algorithms

May 29th

pdf    odt

  Inductive algorithms help to understand relation between infinite and finite. Cantors method as a way to get interesting examples like the functiion tht increases faster then any "elementary" function.

15

Go-betweens in Inequalities

May 30th

pdf    odt

  One can prove A>B by prooving A>P and P>B. The art is to choose an appropriate go-between P.

16

Test 2

June 6th

pdf    odt

  Train yourself. You have 4 problems and 3 hours to solve some of them and to write down solutions.

17

Vectors. Dot Product

June 11th

pdf    odt

 Vectors in the plane and in space and dot product: geometric and algebraic view without coordinates.

18

Functional Equations

June 11th

pdf    odt

 Functional equations: substitution method.

19

Vector Spaces. Dimension

June 12th

pdf    odt

 The notion of dimension helps us to use the pigeonhole principle for vector spaces: though a space usually consists of an infinite number of vectors, the number of vectors in a basis is finite.

20

Continuous Functional Equations

June 12th

pdf    odt

 It is easier to search for only continuous or monotonic solutions of functional equations. Sometimes the extra property of a solution can be established before we find it.

21

Dimension and Matrices

June 13th

pdf    odt

 The dimension of a linear span is connected with the rank of a matrix.

22

Functional and Differential Inequalities

June 14th

pdf    odt

  To solve an inequality start with the corresponding equation.

24

Inequalities: Stepwise Improvement

June 15th

pdf    odt

 Some problens can by solved via step by step simplification.

25

Matrices: Convenient Basis

June 15th and 16th

pdf    odt

 There are some bases when eigenvalues reside on the main diagonal of the matrix.