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Real Analysis 1-4 (semesters 1-4)
    Basic notions. Axioms of the real numbers (semester 1, weeks 1-2; 0 problems)
        Fundaments of Logic (semester 1, week 1; 16 problems)
        Proving Techniques: Proof by Contradiction, Induction (semester 1, week 1; 20 problems)
            Fibonacci Numbers (semester 1, week 1; 6 problems)
        Solving Inequalities and Optimization Problems by Inequalities between Means (semester 1, week 1; 19 problems)
        Sets, Functions, Combinatorics (semester 1, week 2; 22 problems)
Problem 81


     (1)  How many word of length \(\displaystyle k\) can be created using the letters \(\displaystyle A, B, C, D, E, F, G\)?
     (2)  How many such word of length \(\displaystyle 7\) can be created without repeating a letter?
     (3)  How many such word of length \(\displaystyle 7\) can be created with the property that \(\displaystyle A\) and \(\displaystyle B\) are neighbors (no repetition)?

    Difficulty: 1.


Problem 77

    Solve: \(\displaystyle |2x-1|<|x^2-4|\).

    Difficulty: 2.


Problem 79

    What are the solutions of the following equation?

    \(\displaystyle \left( \frac{x+|x|}{2}\right)^2+\left(\frac{x-|x|}{2}\right)^2=x^2\)

    Difficulty: 2.


Problem 82

    Show that

    \(\displaystyle \binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}.\)

    Difficulty: 2.


Problem 87

    Let \(\displaystyle A=\{ 1,2,...,n \}\) and \(\displaystyle B=\{ 1,...,k \}\).


     (1)  How many different functions \(\displaystyle f:A\to B\) do exist?
     (2)  How many different injective functions \(\displaystyle f:A\to B\) do exist?
     (3)  How many different functions \(\displaystyle f:A_{0}\to B\) do exist, where \(\displaystyle A_0\subset A\) is arbitrary?

    Difficulty: 2. Answer (final result) is provided for this problem.


Problem 90

    How many lines are determined by \(\displaystyle n\) points in the plane? And how many planes are determined by \(\displaystyle n\) points in the space?

    Difficulty: 2.


Problem 78

    Find the parallelogram with greatest area with given perimeter.

    Difficulty: 3.


Problem 84

    Which one is bigger? \(\displaystyle 639^9\) or \(\displaystyle 638^9+9\cdot 638^8\)?

    Difficulty: 3.


Problem 85

    Prove the De Morgan identities, i.e., \(\displaystyle \overline{A\cup B}=\overline A\cap \overline B,\) and \(\displaystyle \overline{A\cap B}=\overline A\cup \overline B.\)

    Difficulty: 3.


Problem 86

    Prove that \(\displaystyle A\cup(B\cap C) = (A\cup B)\cap (A\cup C)\)

    Difficulty: 3.


Problem 89

    Let \(\displaystyle A\Delta B=(A\setminus B)\cup (B\setminus A)\) denote the symmetric difference of the sets \(\displaystyle A\) and \(\displaystyle B\). Show that for any sets \(\displaystyle A,B,C\):

     (1)  \(\displaystyle A\Delta \emptyset=A\),     (2)  \(\displaystyle A\Delta A=\emptyset\),     (3)  \(\displaystyle (A\Delta B)\Delta C=A\Delta(B\Delta C)\).

    Difficulty: 3.


Problem 93

    How many ways can one put on the chessboard:
     (1)  2 white rooks,    
     (2)  2 white rooks such that they cannot capture each other,
     (3)  1 white rook and 1 black rook,    
     (4)  1 white rook and 1 black rook such that they cannot capture each other? EredmEn
     (1)  \(\displaystyle \binom{64}2,\)
     (2)  \(\displaystyle \frac12\cdot64\cdot49\),
     (3)  \(\displaystyle 64\cdot63\),
     (4)  \(\displaystyle 64\cdot49\).

    Difficulty: 3.


Problem 95

    Is it true for all triples \(\displaystyle A,B,C\) of sets that

    (a) \(\displaystyle (A\triangle B)\triangle C = A\triangle(B\triangle C)\);

    (b) \(\displaystyle (A\triangle B)\cap C = (A\cap C)\triangle(B\cap C)\);

    (c) \(\displaystyle (A\triangle B)\cup C = (A\cup C)\triangle(B\cup C)\)?

    Difficulty: 3. Answer (final result) is provided for this problem.


Problem 83

    Prove the so called binomial theorem:

    \(\displaystyle (a+b)^{n}=\binom{n}{ 0}a^{n}+ \binom{n}{1} a^{n-1}b+\cdots+\binom{n}{n}b^{n}. \)

    Difficulty: 4.


Problem 88

    Prove that \(\displaystyle x\in A_{1}\Delta A_{2}\Delta \cdots \Delta A_{n}\) if and only if \(\displaystyle x\) is an element of an odd number of \(\displaystyle A_{i}\)'s.

    Difficulty: 4.


Problem 94

    How many different rectangles can be seen on the chessboard?

    Difficulty: 4.


Problem 96

    Is it true that the subsets of a set \(\displaystyle H\) form a ring with identity using the symmetric difference and a) the intersection b) the union?

    Difficulty: 4.


Problem 97

    Let \(\displaystyle f:A\to B\). For any set \(\displaystyle X\subset A\) let \(\displaystyle f(X)=\{f(x): ~ x\in X\}\) (the image of the set \(\displaystyle X\)), and for any set \(\displaystyle Y\subset B\) let \(\displaystyle f^{-1}(Y)=\{x\in A: ~ f(x)\in Y\}\) (the preimage of the set \(\displaystyle Y\)). Is it true that

    (a) \(\displaystyle \forall X,Y\in \mathcal P(A) ~ f(X)\cup f(Y) = f(X\cup Y)\) ?

    (b) \(\displaystyle \forall X,Y\in \mathcal P(B) ~ f^{-1}(X)\cup f^{-1}(Y) = f^{-1}(X\cup Y)\) ?

    Difficulty: 4.


Problem 98

    Let \(\displaystyle f:A\to B\). Is it true that

    (a) \(\displaystyle \forall X,Y\in \mathcal P(A) ~ f(X)\cap f(Y) = f(X\cap Y)\) ?

    (b) \(\displaystyle \forall X,Y\in \mathcal P(B) ~ f^{-1}(X)\cap f^{-1}(Y) = f^{-1}(X\cap Y)\) ?

    Difficulty: 4.


Problem 103

    Show an example of an associative operation \(\displaystyle \circ:\mathcal P(\R)\times\mathcal P(\R)\to\mathcal P(\R)\) for which the union operation is left distributive but not right distributive. (Here \(\displaystyle \mathcal P(\R)\) denotes the set of all subsets of the real line \(\displaystyle \R\).)

    Difficulty: 7.


Problem 100

    Let \(\displaystyle A_1,A_2,\ldots\) be non-empty finite sets, and for all positive integer \(\displaystyle n\) let \(\displaystyle f_n\) be a map from \(\displaystyle A_{n+1}\) to \(\displaystyle A_n\)Prove that there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\) sorozat, such that for all \(\displaystyle n\) the conditions \(\displaystyle x_n\in A_n\) and \(\displaystyle f_n(x_{n+_1})=x_n\) hold. (König's lemma)

    Difficulty: 8.


Problem 101

    Using König's lemma verify that if all finite subgraph of a countable graph can be embedded into the plane then the whole graph can be embedded into the plane as well.

    Difficulty: 8.


    Axioms of the real numbers (semester 1, weeks 2-4; 0 problems)
        Field Axioms (semester 1, week 2; 5 problems)
        Ordering Axioms (semester 1, week 2; 6 problems)
        The Archimedean Axiom (semester 1, week 2; 3 problems)
        Cantor Axiom (semester 1, week 2; 7 problems)
        The Real Line, Intervals (semester 1, week 3; 15 problems)
        Completeness Theorem, Connectivity, Topology of the Real Line. (semester 1, week 3; 4 problems)
        Powers (semester 1, week 4; 3 problems)
    Convergence of Sequences (semester 1, weeks 4-6; 0 problems)
        Theoretical Exercises (semester 1, week 4; 61 problems)
        Order of Sequences, Threshold Index (semester 1, week 4; 22 problems)
        Limit Points, liminf, limsup (semester 1, week 5; 15 problems)
        Calculating the Limit of Sequences (semester 1, week 5; 33 problems)
        Recursively Defined Sequences (semester 1, week 5; 21 problems)
        The Number $e$ (semester 1, week 5; 13 problems)
        Bolzano–Weierstrass Theorem and Cauchy Criterion (semester 1, week 6; 5 problems)
        Infinite Sums: Introduction (semester 1, week 6; 21 problems)
    Cardinalities of Sets (semester 1, week 7; 0 problems)
        Countable and not countable sets (semester 1, week 7; 6 problems)
        Not countable Sets (semester 1, week 7; 0 problems)
    Limit and Continuity of Real Functions (semester 1, weeks 7-10; 0 problems)
        Global Properties of Real Functions (semester 1, week 7; 20 problems)
        Continuity and Limits of Functions (semester 1, week 8; 33 problems)
        Calculating Limits of Functions (semester 1, week 8; 29 problems)
        Continuity and Convergent Sequences (semester 1, week 9; 0 problems)
        Continuous Functions on a Closed Bounded Interval (semester 1, week 9; 9 problems)
        Uniformly Continuous Functions (semester 1, week 10; 5 problems)
        Monotonity and Continuity (semester 1, week 10; 2 problems)
        Convexity and Continuity (semester 1, week 10; 7 problems)
    Elementary functions (semester 1, weeks 11-12; 0 problems)
        Arclength of the Graph of the Function (semester 1, week 11; 0 problems)
        Exponential, Logarithm, and Power Functions (semester 1, week 11; 17 problems)
        Inequalities (semester 1, week 12; 1 problems)
        Trigonometric Functions and their Inverses (semester 1, week 12; 3 problems)
    Differential Calculus and its Applications (semester 2, weeks 0-3; 0 problems)
        The Notion of Differentiation (semester 2, week 0; 47 problems)
        Tangents (semester 2, week 1; 11 problems)
        Higher Order Derivatives (semester 2, week 1; 13 problems)
        Local Properties and the Derivative (semester 2, week 1; 4 problems)
        Mean Value Theorems (semester 2, week 1; 3 problems)
        Number of Roots (semester 2, week 1; 5 problems)
        Exercises for Extremal Values (semester 2, week 2; 2 problems)
            Inequalities, Estimates (semester 2, week 2; 14 problems)
        The L'Hospital Rule (semester 2, week 2; 14 problems)
        Polynomial Approximation, Taylor Polynomial (semester 2, week 3; 20 problems)
        Convexity (semester 2, week 3; 5 problems)
        Analysis of Differentiable Functions (semester 2, week 3; 6 problems)
    Riemann Integral (semester 2, weeks 4-11; 0 problems)
        Definite Integral (semester 2, week 4; 11 problems)
        Indefinite Integral (semester 2, weeks 5-6; 13 problems)
        Properties of the Derivative (semester 2, week 6; 2 problems)
        Newton-Leibniz formula (semester 2, week 6; 2 problems)
        Integral Calculus (semester 2, week 7; 5 problems)
        Applications of the Integral Calculus (semester 2, week 8; 4 problems)
            Calculating the Area and the Volume (semester 2, week 8; 0 problems)
            Calculating the Arclength (semester 2, week 8; 3 problems)
            Surface Area of Surfaces of Revolution (semester 2, week 8; 0 problems)
        Integral and Inequalities (semester 2, week 8; 5 problems)
        Improper Integral (semester 2, week 9; 9 problems)
        Liouville Theorem (semester 2, week 10; 0 problems)
        Functions of Bounded Variation (semester 2, week 11; 2 problems)
        Riemann-Stieltjes integral (semester 2, week 11; 2 problems)
    Infinite Series (semester 2, weeks 12-13; 38 problems)
    Sequences and Series of Functions (semester 3, weeks 1-2; 0 problems)
        Convergence of Dequences of Functions (semester 3, week 1; 16 problems)
        Convergence of Series of Functions (semester 3, week 1; 17 problems)
        Taylor and Power Series (semester 3, week 2; 12 problems)
    Differentiability in Higher Dimensions (semester 3, weeks 3-6; 0 problems)
        Topology of the $n$-dimensional Space (semester 3, week 3; 29 problems)
        Real Valued Functions of Several Variables (semester 3, weeks 3-4; 0 problems)
            Limits and Continuity in $R^n$ (semester 3, week 3; 16 problems)
            Differentiation in $R^n$ (semester 3, week 4; 62 problems)
        Vector Valued Functions of Several Variables (semester 3, weeks 5-6; 0 problems)
            Limit and Continuity (semester 3, week 5; 3 problems)
            Differentiation (semester 3, week 5; 11 problems)
            Implicite functions (semester 3, week 6; 0 problems)
    Jordan Measure and Riemann Integral in Higher Dimensions (semester 3, weeks 7-9; 60 problems)
    Integral Theorems of Vector Calculus (semester 3, week 10 -- semester 4, week 1; 0 problems)
        The Line Integral (semester 3, week 10; 11 problems)
        Newton-Leibniz Formula (semester 3, week 11; 6 problems)
        Existence of the Primitive Function (semester 3, week 12; 13 problems)
        Integral Theorems in 2D (semester 4, week 1; 2 problems)
        Integral Theorems in 3D (semester 4, week 1; 12 problems)
    Measure Theory (semester 4, weeks 3-99; 0 problems)
        Set Algebras (semester 4, week 3; 9 problems)
        Measures and Outer Measures (semester 4, week 4; 8 problems)
        Measurable Functions. Integral (semester 4, week 5; 10 problems)
        Integrating Sequences and Series of Functions (semester 4, weeks 7-8; 11 problems)
        Fubini Theorem (semester 4, week 9; 1 problems)
        Differentiation (semester 4, weeks 11-12; 7 problems)
Complex Analysis (semester 5)
    Complex differentiability (semester 5, week 0; 0 problems)
        Complex numbers (semester 5, week 0; 21 problems)
            The Riemann sphere (semester 5, week 0; 1 problems)
    Regular functions (semester 5, weeks 1-2; 0 problems)
        Complex differentiability (semester 5, week 1; 7 problems)
        The Cauchy-Riemann equations (semester 5, week 1; 3 problems)
        Power series (semester 5, weeks 1-2; 0 problems)
            Domain of convergence (semester 5, week 2; 9 problems)
            Regularity of power series (semester 5, week 2; 2 problems)
            Taylor series (semester 5, week 2; 1 problems)
        Elementary functions (semester 5, week 2; 0 problems)
            The complex exponential and trigonometric functions (semester 5, week 2; 8 problems)
            Complex logarithm (semester 5, week 2; 12 problems)
    Complex Line Integral and Applications (semester 5, weeks 3-5; 0 problems)
        The complex line integral (semester 5, week 3; 9 problems)
        Cauchy's theorem (semester 5, week 3; 6 problems)
        The Cauchy formula (semester 5, week 4; 12 problems)
        Power and Laurent series expansions (semester 5, week 5; 0 problems)
            Power series expansion (semester 5, week 5; 2 problems)
            Liouville's Theorem (semester 5, week 5; 7 problems)
        Local properties of holomorphic functions (semester 5, week 5; 0 problems)
        Consequences of analyticity (semester 5, week 5; 8 problems)
            The maximum principle (semester 5, week 5; 4 problems)
        Laurent series (semester 5, week 5; 9 problems)
    Isolated singularities (semester 5, weeks 5-8; 0 problems)
        Singularities (semester 5, week 5; 4 problems)
        Cauchy's theorem on residues (semester 5, weeks 7-8; 15 problems)
            Residue calculus (semester 5, week 7; 6 problems)
            Applications (semester 5, week 7; 0 problems)
                Evaluation of series (semester 5, week 7; 7 problems)
                Evaluation of integrals (semester 5, week 7; 26 problems)
            The argument principle and Rouche's theorem (semester 5, week 8; 7 problems)
    Conformal maps (semester 5, weeks 9-10; 0 problems)
        Fractional linear transformations (semester 5, week 9; 20 problems)
        Riemann mapping theorem (semester 5, week 9; 11 problems)
        Schwarz lemma (semester 5, week 9; 12 problems)
        Caratheodory's theorem (semester 5, week 9; 2 problems)
        Schwarz reflection principle (semester 5, week 10; 2 problems)
    Harmonic functions (semester 5, weeks 11-12; 8 problems)
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