A.Shapovalov => Lessons

NYUAD, 2018

Here are lists of problems for 1st and 2nd year students from New York University in Abu Dhabi during their preparation to IMC2018.

0

Start test

May 15th

pdf    odt

 10 problems with short answers

1

Example+Estimate

May 15th

pdf    odt

 A problem of this kind consists 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)

2

Quadratic polynomials

May 15th

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.

3

Sequences and Sums

May 16th

Problems: pdf    odt

 How to find sums using equations

4T

Prime Factorization: Right or Wrong

May 16th

Test: pdf    odt

 11 problems with short answers

4C

Prime Factorization: Construct using primes

May 16th

pdf    odt

 An example can be built of elements. Here we build integers as a product of prime numbers.

5

Polynomials and Equations

May 17th

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

Coding

May 17th

pdf    odt

 A coding makes one-to-one corresponding between objects or situations from a the problem to some combinatorial set. For example, such a correspondence to sequences of 0 and 1 is called a binary coding.

7

Polynomials with Integer Coefficients

May 18th

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.

8

Right or Wrong: Examples in Calculus

May 20th

Test: pdf    odt

 10 problems with short answers

9

Inequalities and derivatives

May 20th

pdf    odt

 You know how to find intervals where a function is increasing or decreasing. Then you can compare function values and prove inequalities!

10

Vectors. Dot Product

May 20th

pdf    odt

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

11

Inequalities and Second Derivatives

May 21st

pdf    odt

 Jensen's inequality give us many inequalities with few variables

12

Remainders and Modular Arithmetic

May 21st

pdf    odt

 If the polynomial equation has an integer solution, it has a solution modulo any positive integer. And vice versa: if the equation has no solution module some positive integer it has no integer solution.

13

Go-betweens in Inequalities

May 22nd

pdf    odt

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

14

Vector Spaces. Dimension

May 22nd

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.

15

Functional Equations

May 23d

pdf    odt

  Functional equations: substitution method.

16

Dimension and Matrices

May 23d

pdf    odt

  Subspace and its dimension. Linear span. Matrix subspaces.

17

Theorems in Modular Arithmetic

May 24th

pdf    odt

  Remainder cancellation, Fermat's little theorem, Wilson's theorem, Chinese remainder theorem

18

Sturm's method for inequalities

May 24th

pdf    odt

  Let A and B be expressions on same variables, equal both to C when all variables has the same mean value. To prove Ai values equal to mean value.

19

Rank and Determinant

May 25th

pdf    odt

  Rank of matrix as dimension of a linear span of its columns or rows. Rank of a matrix product.

20

Finite Fields Zp

May 25th

pdf    odt

  Projection of integer polynomials and vectors to remainder polynomial and vectors helps to prove divisibility and nonsingularity.

21

Matrix Algebra

May 27th

pdf    odt

  Algebraic properties of matrices, examples of matrices.

22

Integral and Differential Inequalities

May 27th

pdf    odt

  The main idea: integral of a positive function is positive.

23

Integral and Differential Inequalities

May 28th

pdf    odt

  Eigenvalues of matrix and its characteristic polynomial.

24

Integral and Differential Inequalities

May 28th

pdf    odt

  An algorithm can check an infinite sequence of hypotheses one by one until find the right one.

25

Continuous Functions

May 29th

pdf    odt

  Arithmetic of continuous functions, intermediate value theorem and extreme value theorem.

26

Selected problems

May 29th

pdf    odt

  A dozen credit problems with interesting solutions.