IMO 1975 (XVII International Mathematical Olympiad)

Permutation minimizes sum of squared differences

Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be real numbers such that $x_1\ge x_2\ge \cdots \ge x_n\text{and}{y}_1\ge y_2\ge \cdots \ge y_n.$ Prove that if $z_1,z_2,\dots ,z_n$ is any permutation of $y_1,y_2,\dots ,y_n$, then ${\sum }_{i=1}^n(x_i-y_i{)}^2\le {\sum }_{i=1}^n(x_i-z_i{)}^2.$

Topic: Disuguaglianze, Algebra Metodo: Simmetria, Induzione Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Permutation minimizes sum of squared differences

Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be real numbers such that $x_1\ge x_2\ge \cdots \ge x_n\text{and}{y}_1\ge y_2\ge \cdots \ge y_n.$ Prove that if $z_1,z_2,\dots ,z_n$ is any permutation of $y_1,y_2,\dots ,y_n$, then ${\sum }_{i=1}^n(x_i-y_i{)}^2\le {\sum }_{i=1}^n(x_i-z_i{)}^2.$

src_imho_1975__Q01

Infinite sequence expressible as xp + yq infinitely often

Let $a_1,a_2,a_3,\dots$ be an infinite increasing sequence of positive integers. Prove that for every $p\ge 1$ there are infinitely many $a_m$ which can be written in the form $a_m=x\cdot a_p+y\cdot a_q$ with $x,y$ positive integers and $q>p$.

Topic: Teoria dei Numeri, Combinatoria Metodo: Induzione, Ricorsione Abilita: Ragionamento geometrico, Lettura attenta, Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Infinite sequence expressible as xp + yq infinitely often

Let $a_1,a_2,a_3,\dots$ be an infinite increasing sequence of positive integers. Prove that for every $p\ge 1$ there are infinitely many $a_m$ which can be written in the form $a_m=x\cdot a_p+y\cdot a_q$ with $x,y$ positive integers and $q>p$.

src_imho_1975__Q02

Angle condition in triangle with externally constructed triangles

On the sides of an arbitrary triangle $ABC$, triangles $ABR$, $BCP$, $CAQ$ are constructed externally with $\angle CBP=\angle CAQ=45^{\circ }$, $\angle BCP=\angle ACQ=30^{\circ }$, $\angle ABR=\angle BAR=15^{\circ }$. Prove that $\angle QRP=90^{\circ }$ and $QR=RP$.

Topic: Geometria piana, Trigonometria Metodo: Trigonometria, Coordinate Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Angle condition in triangle with externally constructed triangles

On the sides of an arbitrary triangle $ABC$, triangles $ABR$, $BCP$, $CAQ$ are constructed externally with $\angle CBP=\angle CAQ=45^{\circ }$, $\angle BCP=\angle ACQ=30^{\circ }$, $\angle ABR=\angle BAR=15^{\circ }$. Prove that $\angle QRP=90^{\circ }$ and $QR=RP$.

src_imho_1975__Q03

Sum of digits of 4444^4444 iterated

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.)

Topic: Teoria dei Numeri Metodo: Congruenze, Casework Abilita: Manipolazione algebrica, Lettura attenta, Stima Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Sum of digits of 4444^4444 iterated

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.)

src_imho_1975__Q04

Points on circle with rational mutual distances

Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.

Topic: Teoria dei Numeri, Geometria piana Metodo: Congruenze, Induzione, Ricorsione Abilita: Modellizzazione, Ragionamento geometrico, Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri, Geometria Fonte: apri PDF p.1

Points on circle with rational mutual distances

Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.

src_imho_1975__Q05

Find all homogeneous polynomials satisfying cyclic sum condition

Find all polynomials $P$ in two variables, with the following properties: (i) for a positive integer $n$ and all real $t,x,y$: $P(tx,ty)=t^{n}P(x,y)$ (that is, $P$ is homogeneous of degree $n$), (ii) for all real $a,b,c$: $P(b+c,a)+P(c+a,b)+P(a+b,c)=0,$ (iii) $P(1,0)=1$.

Topic: Algebra, Insiemi e funzioni Metodo: Simmetria, Fattorizzazione Abilita: Manipolazione algebrica, Astrazione, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Find all homogeneous polynomials satisfying cyclic sum condition

Find all polynomials $P$ in two variables, with the following properties: (i) for a positive integer $n$ and all real $t,x,y$: $P(tx,ty)=t^{n}P(x,y)$ (that is, $P$ is homogeneous of degree $n$), (ii) for all real $a,b,c$: $P(b+c,a)+P(c+a,b)+P(a+b,c)=0,$ (iii) $P(1,0)=1$.

src_imho_1975__Q06