1. If 2, 4 and 7 are three digits of a four-digit number

*N*, where*N*is divisible by 36, find the greatest possible value of*N*. [*Reminder: no calculators!! Use divisibility rules and your brain!*]2. In the triangle on the right, the areas are as shown.

Find the area

*x*of the quadrilateral at the top.3. For which positive integers

*n*is 2*–*^{n}*n*^{2}divisible by 7 ?4. According to legend there is a monster that wakes up every now and then to swallow everyone who is solving this problem, and then falls back asleep for as many years as the sum of the digits of that year. The monster first hit the UKMT Mentoring Schemes in the year 234 A.D. For how many years will it be safe to participate?

5. An equilateral triangle has sides of length 4√3. A point

*Q*is situated inside the triangle so that the perpendicular distances from two of the sides of the triangle are 1 and 2. What is the perpendicular distance to the third side?6. Solve:

x + y + xy = 19

y + z + yz = 11

z + x + zx = 14

7. The number 916238457 is an example of a nine-digit number which contains each of the digits 1 to 9 exactly once. It also has the property that the digits 1 to 5 occur in their natural order, while the digits 1 to 6 do not. How many such numbers are there?

8. If

*x*and*y*are positive integers such that 3*x*^{2}+*x*= 4*y*^{2}+*y*, prove that*x*–*y*is a perfect square.
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