IMSO 2009 Mathematics - International Exploration

Find as many five-digit numbers as possible using each of 1,2,3,4,5 once. The first and fifth digits are neither 1 nor 5; the second digit is not 2; the third is not 3; the fourth is not 4.

Let C be outside a line l. Three distinct points X,Y,Z on l give triangles CXY,CXZ,CYZ. What is the maximum number of triangles with C and two points on l if l contains (a) 5, (b) 10, (c) 51 distinct points?

Muchlis does five different tasks A,B,C,D,E, one per day on five consecutive days. Each friend Ali, Benny, Charlie, Dennis, Ethan helps on exactly one task.

- Task A is one day after Ali helps.
- Task B is three days after Benny helps.
- On day 4 the task is C or D.
- Charlie helps with A, D or E.
- The helper for E is Dennis or Benny.

Complete the day–task–helper table for days 1 through 5.

A move chooses two boxes and redistributes their marbles equally, using whole marbles. Boxes with an odd combined total cannot be chosen. The game ends when no move changes the distribution. (The source’s examples require excluding a no-change move between equal boxes.)

Example: starting A–F at 10,11,12,14,19,24, merge B,E; A,C; D,F; A,D; C,F. The successive states are shown below.

Now start A–H at 4,5,6,8,9,31,32,35. Finish in the least number of moves. Record the pair and all eight box counts after each move.

In $1\star2\star3\star4\star5\star6\star7\star8\star9\star10$, replace each star by + or − so that the result is 25. Find as many different replacements as possible. Example: $1-2-3-4+5-6+7+8+9+10=25$.

A rectangular box has dimensions 2 × 1 × 1 units. One example net is shown. On one sheet of 12 × 15 unit-grid paper, construct as many nets as possible. Use colours to distinguish the nets.