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 
5 problems from Tournament of Towns 

2 
10 problems with a short answer 
April 27th 
10 problems to which students may give only short answer (e.g. just a number) 

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). 

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. 

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. 
6 
Sums 
May 14th 
10 sums to train yourself. Give just an answer. 

7 
Sequnces and Sums 
May 21st 
How to find sums using equations. 

8 
Right or Wrong: Examples in Calculus 
May 25th 
How to construct an example and find out, if an assumption is true or false. 

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! 

10 
Right or Wrong: Prime Factors 
May 26th 
A complex construction can be made of simple bricks. In number theory these are prime numbers. 

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. 

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. 

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. 

13 
Finite Fields 
May 28th 
Z_{p} is like numbers but finite. A polynomial with integer coefficients can be replaced with the polynomial with Z_{p} coefficients. 

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. 

15 
Gobetweens in Inequalities 
May 30th 
One can prove A>B by prooving A>P and P>B. The art is to choose an appropriate gobetween P. 

16 
Test 2 
June 6th 
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 
Vectors in the plane and in space and dot product: geometric and algebraic view without coordinates. 

18 
Functional Equations 
June 11th 
Functional equations: substitution method. 

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. 

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. 

21 
Dimension and Matrices 
June 13th 
The dimension of a linear span is connected with the rank of a matrix. 

22 
Functional and Differential Inequalities 
June 14th 
To solve an inequality start with the corresponding equation. 

24 
Inequalities: Stepwise Improvement 
June 15th 
Some problens can by solved via step by step simplification. 

25 
Matrices: Convenient Basis 
June 15th and 16th 
There are some bases when eigenvalues reside on the main diagonal of the matrix. 