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\centerline{\bf �vod: matematick� indukce}
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P��klad 1:

Doka�te matematickou indukc�, �e pro v�echna $n \in N$ plat�:

(i) $\sum_{k=1}^n k = \frac{n(n+1)}{2},$

(ii) $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6},$

(iii) $\sum_{k=1}^n k^3 = (\frac{n(n+1)}{2})^2,$

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P��klad 2:

Doka�te matematickou indukc�, �e pro v�echna $n \in N, a,b in R$ plat�:

 $(a + b)^n = \sum_{k=0}^n \pmatrix n\cr k\cr\endpmatrix a^k b^{n-k}.$
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P��klad 3:

Doka�te, �e ka�d� celo��seln� obnos v�t�� nejv�� rovn� 8 lze vyplatit 
t��korunami a p�tikorunami.
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P��klad 4:

Doka�te, �e ka�d� p�irozen� ��slo v�t�� ne� jedna je d�liteln� prvo��slem.
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P��klad 5: 

Doka�te matematickou indukc�:

(i)  pro v�echna p�irozen� $n \neq 3$ plat� $n^2 \leq 2^n$;

(ii)  pro v�echna p�irozen� $n \neq 3$ plat� $n+1)^n \leq n^{n+1};$

(iii)   pro v�echna p�irozen� $n \neq 3$ plat� $n! \leq (\frac{n+1}{2})^n$.

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P��klad 5 (Bernoulliova nerovnost): 

Doka�te, �e pro v�echna $n \in N$ a v�echna re�ln� $h$ takov�, �e $h > -1$ plat�
$$
(1 + h)^n \geq 1 + nh.
$$
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P��klad 6 (A-G nerovnost): 

Doka�te, �e pro v�echna $n \in N$ a v�echny n-tice nez�porn�ch re�ln�ch ��sel $a_1,...,a_n$ plat�
$$
(a_1.a_2...a_n)^{\frac1n} \leq \frac{a_1 + a_2 + ... + a_n}{n}.
$$
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P��klad 7: 

Doka�te de Morganova pravidla: Bu�te $A, B_i$  podmno�iny $X$. Pak
$$A\setminus \bigcap_{i\in I} B_i = \bigcup_{i\in I}
(A\setminus B_i))$$
$$A\setminus \bigcup_{i\in I} B_i = \bigcap_{i\in I}
(A\setminus B_i).$$

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Opakov�n�:
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I. Nalezn�te v�echna re�ln� $x$, spl�uj�c� nerovnosti

(i) $|2x + 3 |> x$;

(ii) $\frac{x-2}{2x + 4} \geq 1$;

(iii) $\frac{x+3}{x-1} \geq \frac{x+1}{x-5} $;

(iv) $\sqrt{x^2 +1} \geq 2x - 1.$


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II. Doka�te, �e pro v�echna re�ln� ��sla $a,b$ plat�

(i) $|a+b| \leq |a| + |b|$,

(ii) $|a-b|\geq ||a| - |b||$.

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III. Vyj�d�ete $\cos 5x$ pomoc� $\cos x$ a $\sin x$.

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IV. Rozhodn�te o pravdivosti a negujte v�roky

(i)$\forall x\in  N \exists y\in  N \forall z\in N (z > x \Rightarrow y < z);$

(ii)$\exists y \in N \forall x \in N \forall z\in N (z > x \Rightarrow y < z);$

(iii)$\exists y \in R \forall x \in R \forall z\in R (z > x \Rightarrow 
y < z)$.

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