Here are lists of problems for 1st year students from New York University in Abu Dhabi during their preparation to IMC2015.
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1 |
5 problems for the first acquaintance |
April 20th |
5 problems from Tournament of Towns |
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2 |
10 problems with a short answer |
April 27th |
10 problems to which students may give only short answer (e.g. just a number) |
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3 |
Example+Estimate |
May 2nd |
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). |
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4 |
Quadratic polynomials |
May 5th |
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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5 |
Polynomials and Equations |
May 9th |
Theorems and Problems: 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. |
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6 |
Sums |
May 14th |
10 sums to train yourself. Give just an answer. |
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7 |
Sequnces and Sums |
May 21st |
How to find sums using equations. |
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8 |
Right or Wrong: Examples in Calculus |
May 25th |
How to construct an example and find out, if an assumption is true or false. |
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9 |
Inequalities and Derivatives |
May 25th |
You know how to find intervals where a function is increasing, decreasing, convex, or concave. Then you know how to prove inequalities! |
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10 |
Right or Wrong: Prime Factors |
May 26th |
A complex construction can be made of simple bricks. In number theory these are prime numbers. |
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11 |
Polynomials with Integer Coefficients |
May 26th |
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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12 |
Remainders and Modular Arithmetic |
May 27th |
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. |
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12a |
Diophantine Equations and Factorization |
May 27th |
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. |
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13 |
Finite Fields |
May 28th |
Zp is like numbers but finite. A polynomial with integer coefficients can be replaced with the polynomial with Zp coefficients. |
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14 |
Infinite Algorithms |
May 29th |
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. |
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15 |
Go-betweens in Inequalities |
May 30th |
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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16 |
Test 2 |
June 6th |
Train yourself. You have 4 problems and 3 hours to solve some of them and to write down solutions. |
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17 |
Vectors. Dot Product |
June 11th |
Vectors in the plane and in space and dot product: geometric and algebraic view without coordinates. |
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18 |
Functional Equations |
June 11th |
Functional equations: substitution method. |
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19 |
Vector Spaces. Dimension |
June 12th |
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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20 |
Continuous Functional Equations |
June 12th |
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. |
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21 |
Dimension and Matrices |
June 13th |
The dimension of a linear span is connected with the rank of a matrix. |
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22 |
Functional and Differential Inequalities |
June 14th |
To solve an inequality start with the corresponding equation. |
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24 |
Inequalities: Stepwise Improvement |
June 15th |
Some problens can by solved via step by step simplification. |
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25 |
Matrices: Convenient Basis |
June 15th and 16th |
There are some bases when eigenvalues reside on the main diagonal of the matrix. |