Here are lists of problems for 1st and 2nd year students from New York University in Abu Dhabi during their preparation to IMC2018.
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0 |
Start test |
May 15th |
10 problems with short answers |
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1 |
Example+Estimate |
May 15th |
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) |
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2 |
Quadratic polynomials |
May 15th |
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. |
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3 |
Sequences and Sums |
May 16th |
How to find sums using equations |
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4T |
Prime Factorization: Right or Wrong |
May 16th |
11 problems with short answers |
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4C |
Prime Factorization: Construct using primes |
May 16th |
An example can be built of elements. Here we build integers as a product of prime numbers. |
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5 |
Polynomials and Equations |
May 17th |
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. |
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6 |
Coding |
May 17th |
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. |
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7 |
Polynomials with Integer Coefficients |
May 18th |
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 m–n. |
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8 |
Right or Wrong: Examples in Calculus |
May 20th |
10 problems with short answers |
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9 |
Inequalities and derivatives |
May 20th |
You know how to find intervals where a function is increasing or decreasing. Then you can compare function values and prove inequalities! |
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10 |
Vectors. Dot Product |
May 20th |
Vectors in the plane and in space and dot product: geometric and algebraic view without coordinates. |
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11 |
Inequalities and Second Derivatives |
May 21st |
Jensen's inequality give us many inequalities with few variables |
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12 |
Remainders and Modular Arithmetic |
May 21st |
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. |
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13 |
Go-betweens in Inequalities |
May 22nd |
One can prove A>B by prooving A>P and P>B. The art is to choose an appropriate go-between P. |
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14 |
Vector Spaces. Dimension |
May 22nd |
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. |
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15 |
Functional Equations |
May 23d |
Functional equations: substitution method. |
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16 |
Dimension and Matrices |
May 23d |
Subspace and its dimension. Linear span. Matrix subspaces. |
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17 |
Theorems in Modular Arithmetic |
May 24th |
Remainder cancellation, Fermat's little theorem, Wilson's theorem, Chinese remainder theorem |
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18 |
Sturm's method for inequalities |
May 24th |
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. |
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19 |
Rank and Determinant |
May 25th |
Rank of matrix as dimension of a linear span of its columns or rows. Rank of a matrix product. |
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20 |
Finite Fields Zp |
May 25th |
Projection of integer polynomials and vectors to remainder polynomial and vectors helps to prove divisibility and nonsingularity. |
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21 |
Matrix Algebra |
May 27th |
Algebraic properties of matrices, examples of matrices. |
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22 |
Integral and Differential Inequalities |
May 27th |
The main idea: integral of a positive function is positive. |
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23 |
Integral and Differential Inequalities |
May 28th |
Eigenvalues of matrix and its characteristic polynomial. |
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24 |
Integral and Differential Inequalities |
May 28th |
An algorithm can check an infinite sequence of hypotheses one by one until find the right one. |
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25 |
Continuous Functions |
May 29th |
Arithmetic of continuous functions, intermediate value theorem and extreme value theorem. |
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26 |
Selected problems |
May 29th |
A dozen credit problems with interesting solutions. |