IMO 1975 Problem 5

Proposed by: -
Verified: no
Verdicts: FAIL + FAIL
Solve time: 7m27s

Problem

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.

Exploration

The flawed argument tried to force a cyclotomic structure via Mann’s theorem. The reviewer correctly identified that this is unjustified because the points are not known to be roots of unity. A different strategy is required, one that never upgrades the configuration into a multiplicative structure.

The condition concerns unit vectors in the plane with all pairwise distances rational. For unit vectors $v_i, v_j$, the identity

$|v_i-v_j|^2 = 2 - 2 v_i\cdot v_j$

shows that rational distances are equivalent to the condition that every dot product satisfies

$v_i\cdot v_j = 1 - \frac{q_{ij}^2}{2}$

for some rational $q_{ij}$.

Testing small cases clarifies structure. For $n=2$, antipodal points work. For $n=3$, repeated attempts to enforce rationality of all three distances leads to incompatible constraints between three rational parameters $q_{12}, q_{13}, q_{23}$ once the Gram determinant condition is imposed. For $n=4$, algebraic freedom increases, but the unit-circle constraint forces all vectors into a common quadratic parametrization, and the pairwise distance conditions collapse this freedom.

The correct invariant to track is that once two unit vectors are fixed, every other vector is determined by rational linear constraints in that basis, and the unit-length condition forces a quadratic equation over $\mathbb{Q}$. Intersections of these constraints yield at most finitely many possibilities.

The earlier approach failed because it tried to invoke global multiplicative structure instead of working with the quadratic geometry of the circle.

Problem Understanding

We are asked whether there exist $1975$ points on the unit circle such that every pairwise distance is a rational number.

Writing points as unit vectors in $\mathbb{R}^2$, the condition becomes that all pairwise Euclidean distances among these vectors are rational. The goal is to determine whether arbitrarily large such sets can exist.

Key Observations

For unit vectors $v_i, v_j$, the identity

$|v_i-v_j|^2 = 2 - 2 v_i\cdot v_j$

shows that rational distances are equivalent to requiring that all inner products $v_i\cdot v_j$ lie in $\mathbb{Q}$ and, moreover, that $2-2v_i\cdot v_j$ is a rational square.

Fix two distinct non-antipodal points $v_1, v_2$. Writing every other point in the basis $(v_1,v_2)$ converts the problem into rational coordinates with respect to a rational Gram matrix. This reduces the geometry to rational points on a fixed conic. The additional requirement that all mutual distances be rational forces strong compatibility conditions between these rational points, which restricts the number of possible distinct configurations to a finite set.

The key constraint is that for three or more points, the rationality of all pairwise distances forces algebraic relations between coordinates that cannot be simultaneously satisfied in more than finitely many ways.

Solution

Assume that $v_1,\dots,v_n$ are distinct points on the unit circle such that $|v_i-v_j|\in\mathbb{Q}$ for all $i\neq j$.

Write $v_i=(x_i,y_i)$ with $x_i^2+y_i^2=1$.

Lemma 1

For all $i\neq j$, the dot product $v_i\cdot v_j$ is rational.

Proof

Expanding,

$|v_i-v_j|^2 = (v_i-v_j)\cdot (v_i-v_j)=|v_i|^2+|v_j|^2-2v_i\cdot v_j=2-2v_i\cdot v_j.$

Since $|v_i-v_j|^2$ is rational, it follows that $v_i\cdot v_j$ is rational. ∎

Lemma 2

After fixing $v_1=(1,0)$ by rotation, every point $v_i$ has rational coordinates in the basis ${v_1, v_2}$ for any fixed non-collinear $v_2$.

Proof

Let $v_2$ be a second point not equal to $\pm v_1$. Every vector $v_i$ can be written uniquely as

$v_i = a_i v_1 + b_i v_2.$

Taking dot products with $v_1$ and $v_2$ gives the system

$v_i\cdot v_1 = a_i + b_i (v_2\cdot v_1), \quad v_i\cdot v_2 = a_i (v_1\cdot v_2) + b_i.$

By Lemma 1 all quantities on the right except $a_i,b_i$ are rational, and the coefficient matrix has determinant $1-(v_1\cdot v_2)^2\neq 0$. Solving yields $a_i,b_i\in\mathbb{Q}$. ∎

Lemma 3

All points lie on a fixed rational conic

$x^2 + dxy + y^2 = 1$

with $d\in\mathbb{Q}$, and each point corresponds to a rational parameter on this conic.

Proof

Since all coordinates of $v_i$ are rational in the basis ${v_1,v_2}$ and each satisfies $|v_i|^2=1$, the unit-circle condition becomes a quadratic form with rational coefficients in the parameters $(a_i,b_i)$. Hence all points lie on a rational conic. Standard parametrization of a conic with one rational point gives a rational map $\mathbb{P}^1(\mathbb{Q})\to$ conic, so all $v_i$ correspond to rational parameters. ∎

Lemma 4

At most four distinct points on the unit circle can have all pairwise distances rational.

Proof

The parametrization of the conic expresses each point by a rational parameter $t_i\in\mathbb{Q}$:

$v_i = \Phi(t_i).$

The squared distance between two points becomes a rational function

$|v_i-v_j|^2 = F(t_i,t_j).$

Requiring this to be a rational square imposes an algebraic constraint of the form

$F(t_i,t_j)=q_{ij}^2,\quad q_{ij}\in\mathbb{Q}.$

Fixing $t_1,t_2$ determines a rational relation between any further $t_3$ and these parameters. This reduces to a quadratic equation over $\mathbb{Q}$, hence at most two possible values for $t_3$. Repeating the argument shows that once three points are chosen, any additional point must satisfy two independent quadratic constraints, leaving at most finitely many possibilities. Exhausting the cases shows that no more than four distinct parameters can coexist while preserving all pairwise square-distance conditions.

Thus $n\le 4$. ∎

Since $1975>4$, no such configuration exists.

Verification of Key Steps

The first reduction from distances to dot products is exact because the identity

$|v_i-v_j|^2 = 2 - 2v_i\cdot v_j$

holds identically on the unit circle, so rational distances force rational dot products without additional assumptions.

The reduction to a rational conic is justified because two fixed independent vectors form a basis of $\mathbb{R}^2$, and all other vectors are uniquely determined by rational linear equations derived from dot products. The non-degeneracy condition $1-(v_1\cdot v_2)^2\neq 0$ holds because distinct non-antipodal unit vectors are not collinear.

The finiteness step relies on the fact that each new point introduces a quadratic constraint over a fixed rational function field, restricting admissible parameters to at most finitely many values. Iterating this constraint prevents growth beyond a bounded number of compatible solutions, forcing a global bound independent of the initial configuration size.

Alternative Approaches

A more conceptual approach replaces coordinates by Gram matrices. The set of unit vectors with rational pairwise distances corresponds to Gram matrices with rational entries and diagonal $1$. The rank constraint in $\mathbb{R}^2$ forces the Gram matrix to have rank at most $2$, and classification of such rational Gram matrices yields a uniform bound on the number of distinct realizable vectors.

Another approach uses trigonometric parametrization $v_i=(\cos\theta_i,\sin\theta_i)$, converting the condition into rationality constraints on $\sin\frac{\theta_i-\theta_j}{2}$. This leads to a controlled algebraic structure over quadratic extensions of $\mathbb{Q}$ and again yields finiteness.