A trio is any triple $(a,b,c)$ of nonzero real numbers satisfying $ab+bc+ca=0$. A triple is said to be reduced if $a+b+c=1$. Part 1We denote by $C$ the set of points $(x,y,z)$ in coordinate space for which $(x,y,z)$ is a trio, and by $\Gamma$ the set of those for which $(x,y,z)$ is a reduced trio. Let $O$ be the origin and $P$ be the plane given by $x+y+z=1$. (a) Does there exist a trio $(a,b,c)$ such that $a+b+c=0$? (b) Prove that $C$ is a union of lines passing through $O$, with $O$ excluded. (c) Prove that $\Gamma$ is the intersection of a plane and a sphere with center $O$. Describe $\Gamma$ geometrically. (d) Describe $C$ geometrically and sketch it. (e) Let $L$ be a fixed point in $\Gamma$. If $L’$ and $L''$ are arbitrary points on $\Gamma$, prove that the volume $V$ of the tetrahedron $OLL’L''$ is maximal when the lines $OL,OL’,OL''$ are orthogonal, and express the coordinates of $L’$ and $L''$ in terms of those of $L$. (f) Prove that the product $abc$ attains its maximum and minimum values on $\Gamma$, and find the points at which those are attained. Part 2A trio $(a,b,c)$ is called rational if $a,b,c$ are rational, and integer if $a,b,c$ are integers. We say that an integer trio $(a,b,c)$ is primitive if the greatest common divisor of $a,b,c$ is $1$. (a) Describe the set $H_1$ of points $(x,y,1)$ such that $(a,b,1)$ is a trio. Show that the point $\Omega_1(-1,-1,1)$ is the center of symmetry of $H_1$. Find all points of $H_1$ with integer coordinates. (b) For each nonzero integer $h$, denote by $Z_h$ the set of integer trios $(a,b,c)$ with $c=h$. Determine $Z_h$ for $h=1$ and $h=2$ (c) Prove that $Z_h$ is a finite set and find the number $N(h)$ of its elements in terms of the number of divisors of $h^2$ in $\mathbb Z$. Prove that $4$ divides $N(h)-2$. (d) For every positive integer $h$, denote by $N’(h)$ the number of integer trios $(a,b,c)$ such that at least one of $a,b,c$ is equal to $h$. Express $N’(h)$ in terms of $N(h)$ depending on the parity of $h$. (e) Prove that every integer trio $(a,b,c)$ can be assigned a triple of integers $(r,s,t)$ such that $r$ and $s$ are coprime, $s$ is nonnegative, and $$a=r(r+s)t,\qquad b=s(r+s)t,\qquad c=-rst.$$State and verify the converse. For which trios $(a,b,c)$ is the triple $(r,s,t)$ not unique? (f) Determine all triples $(r,s,t)$ that are assigned to some primitive trios. Deduce that if $(a,b,c)$ is a primitive trio, then $|abc|$, $|a+b|$, $|b+c|$ and $|c+a|$ are perfect squares. (g) For each positive integer $h$, denote by $P(h)$ the number of primitive trios $(a,b,c)$ with $c=h$. Prove that $P(h)$ is a power of $2$. For which $h$ is $P(h)=N(h)$? Give a sequence of integers $(h_n)$ for which the sequence $\frac{P(h_n)}{N(h_n)}$ converges to zero. (h) Let $(a,b,1)$ be a trio. Show that there exist sequence $(x_n),(y_n)$, and $(z_n)$ converging respectively to $a,b$, and $c$ such that $(x_n,y_n,1)$ is a rational trio for all $n$. (i) Let $(a,b,1)$ be a reduced trio. Show that there exist sequence $(x_n),(y_n)$, and $(z_n)$ converging respectively to $a,b$, and $c$ such that $(x_n,y_n,z_n)$ is a rational reduced trio for all $n$. Part 3Denote $j=e^{2i\pi/3}=-\frac12+i\frac{\sqrt3}2$. For each trio $T=(a,b,c)$ we define $\mathcal T=(a,c,b)$, $S(T)=a+b+c$ and $z(T)=a+bj+cj^2$. (a) Express the modulus of $z(T)$ as a function of $S(T)$. Can we have $z(T)=0$? Compute the sine and the cosine of the argument $\theta$ of $z(T)$ in terms of $a,b,c$. (b) Let $z_0$ be a given nonzero complex number. Find all trios $T=(a,b,c)$ such that $z(T)=z_0$. (c) Given trios $T_1$ and $T_2$, prove that there is a unique trio, to be denoted as $T_1*T_2$, satisfying $S(T_1*T_2)=S(T_1)S(T_2)$ and $z(T_1*T_2)=z(T_1)z(T_2)$. Compute $T_1*T_2$ in terms of $T_1$ and $T_2$. What can be said about the argument of $z(T_1*T_2)$? What can be said about $z(T_1*\mathcal T_1)$? (d) If $T_1$ and $T_2$ are reduced trios, is $T_1*T_2$? The same question if the word ”reduced” is replaced by ”integer” and by ”primitive”. (e) Compare the trios $T_1*T_2$ and $T_2*T_1$; $T_1*(T_2*T_3)$ and $(T_1*T_2)*T_3$, $T_1$ and $T_1*(1,0,0)$. (f) Given trios $T_1$ and $T_2$, solve the equation $T_1*T=T_2$ in $T$. (g) Given a trio $T$, define the sequence of trios $(T_n)$ by $T_0=(1,0,0)$ and $T_{n+1}=T*T_n$. Calculate $S(T_n)$. Given an integer $p$, find all $T$ for which $T_p=T_0$. Part 4Denote by $A$ the set of integers $m$ that are of the form $u^2+3v^2$, where $u,v$ are integers. Denote by $A’$ the set of nonzero complex numbers $z=u+iv\sqrt3$, where $u,v$ are integers (note that $|z|^2=u^2+3v^2$). Denote by $B$ the set of nonzero integers $n$ of the form $r^2+rs+s^2$, where $r,s$ are integers. (a) Prove that a product of two elements of $A’$ belongs to $A’$, and that a product of two elements of $A$ belongs to $A$. (b) Show that if $p\in A$ is a prime number, then $p=3$ or $3\mid p-1$. (c) Prove that $A=B$ (you may note that $r^2+rs+s^2=(r+s)^2-(r+s)s+s^2$). (d) Prove that every even element of $A$ is divisible by $4$ and that its quarter belongs to $A$; then prove that each element of $A$ is the product of a power of $4$ and an odd element of $A$. (e) i. Suppose that there is an odd integer $m=u^2+3v^2$, where $u,v$ are coprime integers, and there exists a prime divisor $p$ of $m$ not belonging to $A$. Prove that there exists the smallest positive integer $n_0$ such that $n_0p$ is in $A$, and that $n_0$ is odd. ii. Verify the existence of integers $u’,v’$ less than $\frac p2$ in absolute value such that $u-u’$ and $v-v’$ are divisible by $p$. Prove that $p$ divides the nonzero number $u’^2+3v’^2$ and hence that $n_0<p$. iii. Verify the existence of coprime nonzero integers $u_0,v_0$ such that $n_0p=u_0^2+3v_0^2$. iv. Verify the existence of integers $u_1,v_1$ less than $\frac{n_0}2$ in absolute value such that $u_1-u_0$ and $v_1-v_0$ are divisible by $n$. Prove that $n_0$ divides the nonzero integer $u_1^2+3v_1^2$ which we’ll denote by $n_0n_1$. v. Deduce that such an integer $m$ cannot exist (you may consider the number $n_0^2n_1p$). (f) Prove that every element of $A$ can be written in the form $m=C^2p_1\cdots p_k$, where $C$ is a positive integer and $p_i$ distinct prime factors of $A$. (g) i. Let $p$ be a prime number with $3\mid p-1$, and $K$ the set of triples $(x,y,z)$ of integers with $0<x,y,z<p$ such that $p\mid xyz-1$. Prove that $k$ has exactly $(p-1)^2$ elements and that the number of those with $x,y,z$ not equal is divisible by $3$. (h) Let $D$ be the set of integers $d$ for which there is an integer trio $(a,b,c)$ satisfying $a + b + c = d$ and $abc\ne 0$. Prove, using question (e) of part $2$, that every element of $D$ has a prime divisor in $A$. Conversely, what can be said about the nonzero integers having a prime divisor in $A$? (i) Find the elements of $D$ between $2001$ and $2010$ inclusive.