Chapter 1

Let \(\boldsymbol{A}\) and \(\boldsymbol{B}\) be non-empty sets of real numbers. Put $$ -A=\\{-x: x \in A\\}, \quad A-B=\\{a-b:(a, b) \in A \times B\\} $$ Prowe that 1\. If \(A\) is bounded above, then \(-A\) is bounded below and \(\sup A=\) \(-\inf (-A)\) 2\. If \(\boldsymbol{A}\) and \(\boldsymbol{B}\) are bounded above then \(\boldsymbol{A} \cup \boldsymbol{B}\) is also bounded above and \(\sup (A \cup B)=\max (\sup A, \sup B)\). 3\. If \(\boldsymbol{A}\) is bounded above and \(\boldsymbol{B}\) is bounded below, then \(\boldsymbol{A}-\boldsymbol{B}\) is bounded above and \(\sup (A-B)=\sup A-\inf B\).

Let \((a, b, c, d) \in \mathrm{R}^{4}\). Prove that $$ || a-c|-| b-c|| \leq|a-b| \leq|a-c|+|b-c| . $$

Let \(x, y\) be real numbers. Then $$ 0 \leq x

Is the set of real irrational numbers closed under addition? Under multiplication?

Prove that there as many numbers in \([0 ; 1]\) as in any interval \([\boldsymbol{a} ; \boldsymbol{b}]\) with \(\vec{a}<\boldsymbol{b} .\)

Find all functions with domain \(\\{\boldsymbol{a}, \boldsymbol{b}\\}\) and target set \(\\{c, d\\}\)

For a fixed \(\boldsymbol{n} \in \mathbb{N}\) put \(A_{n}=\\{\boldsymbol{n k :} \boldsymbol{k} \in \mathbb{N}\\}\). 1\. Find \(A_{2} \cap A_{3}\). 2\. Find \(\bigcap_{n=1}^{\infty} A_{n}\). 3\. Find \(\bigcup_{n=1}^{\infty} A_{n}\).

Assume that \(\boldsymbol{A}\) is a subset of the strictly positive real numbers. Prove that if \(A\) is bounded above, then the set \(A^{-1}=\left\\{\frac{1}{x}\right.\) : \(x \in A\\}\) is bounded below and that \(\sup A=\frac{1}{\inf A^{-1}} .\)

Let \(\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}\) be such that \(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}=x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}\) Prove that \(x_{k} \in\\{0,1\\} .\)

Let \(t \geq 0 .\) Prove that \(|x| \geq t \Longleftrightarrow(x \geq t)\) or \(\quad(x \leq-t) .\)

Given that 1002004008016032 has a prime factor \(\boldsymbol{p}>250000\), find it.

Let \(A, B\) be finite sets with \(\operatorname{card}(A)=n\) and \(\operatorname{card}(B)=m .\) Prove that \- The number of functions from \(A\) to \(B\) is \(m^{n}\). \- If \(n \leq m\), the number of injective functions from \(A\) to \(B\) is \(m(m-1)(m-2) \cdots(m-n+1) .\) If \(n>m\) there are no injective functions from \(A\) to \(B\).

Prove the following properties of the empty set: $$ A \cap \emptyset=\varnothing, \quad A \cup \varnothing=A $$

Let \(\boldsymbol{n} \geq 2\) be an integer. Prove that $$ \max _{0 \leq x_{1} \leq x_{2} \leq \cdots \leq x_{n} \leq 1}\left(\sum_{1 \leq i

Let \(n \geq 2\) an integer. Let \(\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}\) be such that $$ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{n-1} x_{n}+x_{n} x_{1} $$ Prove that \(x_{1}=x_{2}=\cdots=x_{n} .\)

Let \((x, y) \in \mathbb{R}^{2} .\) Prove that \(\max (x, y)=\) \(-\min (-x,-y)\)

Prove that \((a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+\) \(2 c a .\)

(Putnam, 1971) Let \(\mathbf{S}\) be a set and let o be a binary operation on \(\mathbf{S}\) satisfying the two laws $$ (\forall x \in S)(x \circ x=x) $$ and $$ \left(\forall(x, y, z) \in S^{3}\right)((x \circ y) \circ z=(y \circ z) \circ x) $$ Shew thato is commutative.

Let \(A\) and \(B\) be two finite sets with \(\operatorname{card}(A)=n\) and \(\operatorname{card}(B)=m .\) If \(n

Prove the following commutative laws: $$ A \cap \boldsymbol{B}=\boldsymbol{B} \cap A, \quad A \cup \boldsymbol{B}=\boldsymbol{B} \cup A . $$

If \(\boldsymbol{b}>\mathbf{0}\) and \(\boldsymbol{B}>\mathbf{0}\) prove that $$ \frac{a}{b}<\frac{A}{B} \Rightarrow \frac{a}{b}<\frac{a+A}{b+B}<\frac{A}{B} $$ Further, if \(\boldsymbol{p}\) and \(\boldsymbol{q}\) are positive integers such that $$ \frac{7}{10}<\frac{p}{q}<\frac{11}{15} $$ what is the least value of \(\boldsymbol{q}\) ?

Let \(x, y, z\) be real numbers. Prove that \(\max (x, y, z)=x+y+z-\min (x, y)-\min (y, z)-\min (z, x)+\min (x, y, z)\)

Let \(a, b, c\) be real numbers. Prove that $$ a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right) $$

(Putnam, 1972) Let \(\mathscr{S}\) be a set and let * be a binary operation of \(\mathscr{F}\) satisfying the laws \(\forall(x, y) \in \mathscr{S}^{2}\) $$ \begin{aligned} &x *(x * y)=y \\ &(y+x) * x=y . \end{aligned} $$ Shew that \(*\) is commutative, but not necessarily associative.

Let \(h: \mathbb{R} \rightarrow \mathbb{R}\) be given by \(h(1-x)=2 x .\) Find \(h(3 x)\).

Prove by means of set inclusion the following distributive law: $$ (A \cup B) \cap C=(A \cap C) \cup(B \cap C) $$

Prove that the integers $$ \left\lfloor(1+\sqrt{2})^{n}\right. $$ with \(n\) a positive integer, are alternately even or odd.

Prove that if \(r \geq s \geq t\) then $$ r^{2}-s^{2}+t^{2} \geq(r-s+t)^{2} $$

Let \(\boldsymbol{a}<\boldsymbol{b} .\) Demonstrate that $$ |x-a|<|x-b| \Longleftrightarrow x<\frac{a+b}{2} $$

Prove that $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n}{k}\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right) $$

On \(\mathbb{Q} \cap]-1 ; 1[\) define the binary operation \(\otimes\) by $$ a \otimes b=\frac{a+b}{1+a b}, $$ where juxtaposition means ordinary multiplication and \(+\) is the ordinary addition of real numbers. Prove that \((\mathrm{Q} \cap]-1 ; 1[, \otimes)\) is an abelian group by following these steps. 1\. Prove that \(\otimes\) is a closed binary operation on \(\mathrm{Qn}]-1 ; 1[.\) 2\. Prove that \(\otimes\) is both commutative and associative. 3\. Find an element \(e \in \mathbb{Q} \cap]-1 ; 1[\) such that \((\forall a \in Q \cap]-1 ; 10\) (e\otimes \(a=a)\) 4\. Given e as above and an arbitrary element \(a \in \mathrm{Q} \cap]-1 ; 1[\), solve the equation \(\boldsymbol{a} \otimes \boldsymbol{b}=e\) for \(\boldsymbol{b}\).

Consider the polynomial $$ \left(1-x^{2}+x^{4}\right)^{2003}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{8012} x^{8012} $$ Find (1) \(a_{0}\) (2) \(a_{0}+a_{1}+a_{2}+\cdots+a_{8012}\) 3 \(a_{0}-a_{1}+a_{2}-a_{3}+\cdots-a_{8011}+a_{8012}\) (4) \(a_{0}+a_{2}+a_{4}+\cdots+a_{8010}+a_{8012}\) 5 \(a_{1}+a_{3}+\cdots+a_{8009}+a_{8011}\)

Prove the following associative laws: $$ A \cap(B \cap C)=(A \cap B) \cap C, \quad A \cup(B \cup C)=(A \cup B) \cup C . $$

Assume that \(a_{k}, b_{k}, c_{k}, k=1, \ldots, n\), are positive real numbers. Shew that $$ \left(\sum_{k=1}^{n} a_{k} b_{k} c_{k}\right)^{4} \leq\left(\sum_{k=1}^{n} a_{k}^{4}\right)\left(\sum_{k=1}^{n} b_{k}^{4}\right)\left(\sum_{k=1}^{n} c_{k}^{2}\right)^{2} $$

Prove that $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n}{k} \cdot \frac{n-1}{k-1} \cdot\left(\begin{array}{l} n-2 \\ k-2 \end{array}\right) $$

Let \(\boldsymbol{G}\) be a group satisfying \((\forall a \in G)\) $$ a^{2}=e $$ Prove that \(\boldsymbol{G}\) is an abelian group.

Let \(f: \mathrm{R} \rightarrow \mathbb{R}\), be a function such that \(\forall x \in] 0 ;+\infty[,\), $$ \left[f\left(x^{3}+1\right)\right]^{\sqrt{x}}=5 $$ find the value of $$ \left[f\left(\frac{27+y^{3}}{y^{3}}\right)\right]^{\sqrt{\frac{27}{y}}} $$ for \(y \in] 0 ;+\infty[\)

Prove that $$ A \cap B=A \Longleftrightarrow A \subseteq B . $$

(Putnam 1948) If \(n\) is a positive integer, demonstrate that $$ \lfloor\sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4 n+2} \rrbracket $$

Prove that for integer \(\boldsymbol{n}>2\), $$ n^{n / 2}

Prove that $$ \sum_{k=1}^{n} k\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k}=n p $$

Let \(\boldsymbol{G}\) be a group where \(\left(\forall(a, b) \in G^{2}\right.\) ) $$ \left((a b)^{3}=a^{3} b^{3}\right) \text { and }\left((a b)^{5}=a^{5} b^{5}\right) $$ Shew that \(\mathbf{G}\) is abelian.

Let \(\boldsymbol{f}\) satisfy \(f(\boldsymbol{n}+1)=(-1)^{n+1} n-2 f(n), n \geq 1\) If \(f(1)=f(1001)\) find $$ f(1)+f(2)+f(3)+\cdots+f(1000) $$

Prove that $$ A \cup B=A \Longleftrightarrow B \subseteq A . $$

Find a formula for the \(\boldsymbol{n}\) -th non-square.

Prove that for integer \(n>2\), $$ n^{n / 2}

Prove that $$ \sum_{k=2}^{n} k(k-1)\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k}=n(n-1) p^{2} $$

Suppose that in a group \(\mathbf{G}\) there exists a pair \((a, b) \in\) \(G^{2}\) satisfying $$ (a b)^{k}=a^{k} b^{k} $$ for three consecutive integers \(k=i, i+1, i+2 .\) Prove that \(a b=\) ba.

If \(f(a) f(b)=f(a+b) \forall a, b \in \mathbb{R}\) and \(f(x)>0 \forall x \in\) \(\mathbb{R}\), find \(f(0) .\) Also, find \(f(-a)\) and \(f(2 a)\) in terms of \(f(a)\).

Prove that $$ A \subseteq B \Longrightarrow A \cap C \subseteq B \cap C . $$