SMO 2006 Senior Round 2

Let $a,d$ be integers such that $a,a+d,a+2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

Let $ABCD$ be a cyclic quadrilateral, let the angle bisectors at $A$ and $B$ meet at $E$, and let the line through $E$ parallel to side $CD$ intersect $AD$ at $L$ and $BC$ at $M$. Prove that $LA+MB=LM$.

Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Prove that the line $TP$ bisects $\angle ATB$.

You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to make a convex equiangular hexagon (i.e., one whose interior angles are $120^\circ$). Obviously, $n$ cannot be any positive integer. The first three feasible $n$ are $6$, $10$ and $13$. Determine if $19$ and $20$ are feasible.

It is claimed that the number

$$N=526315789473684210$$

is a persistent number, that is, if multiplied by any positive integer the resulting number always contains the ten digits $0,1,\ldots,9$ in some order with possible repetitions.

(a) Prove or disprove the claim.

(b) Are there any persistent numbers smaller than the above number?