Problem 81
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 \(\displaystyle B=\{ 1,...,k \}\).
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: 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. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |