[Tony]

\[ BRITISHMATHEMATICAL OLYMPIAD \]
\[ Round 1 : Wednesday 13th January 1993 \]

1. Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits
of n. (Thus n might look like 123124, although this is not a square.)

2.A square piece of toast ABCD of side length 1 and centre O is cut in half to form two equal pieces ABC and CDA. If the triangle ABC has to be cut into two parts of equal area, one would usually cut along the line of symmetry BO. However,
there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle ABC into two parts of equal area.

3. For each positive integer c, the sequence un of integers is defned by
\[ u₁=1,u₂=c, u_n=(2n+1)u_n-1-(n²-1)u_n-2, (nณ3) \]

4. Two circles touch internally at M. A straight line touchesthe inner circle at P and cuts the outer circle at Q and R.
Prove that \( ะQMP = ะRMP \)

5. Let x; y; z be positive real numbers satisfying
\[ \frac{1}{3}ฃ xy+yz+zx ฃ 3 \]
Determine the range of values for (i) xyz, and (ii) x+y +z.