1996 IMC solutions
1 E
2 E
3 D
4 B
5 A
6 A
7 D
8 B
9 E
10 C
11 D
12 A
13 B
14 B
15 E
16 C
17 E
18 B
19 D
20 D
21 B
22 C
23 A
24 A
25 D
Hints:
21. The game must go through the stages 6-6, 7-7, 8-8 and then 10-8. 2 ways to get from 5-5 to 6-6, 2 ways from 6-6 to 7-7 and 2 from 7-7 to 8-8. Only 1 way to get from 8-8 to 10-8. So probability =2*2*2*1/(2^8)=1/32
24. 499 is prime, 1996=4*499. (x^2-y^2)=(x+y)(x-y). So (x+y) must be one factor of 1996 and (x-y) the remianing factors. The only way you can get integers for both x and y is if (x+y)=2*499 and (x-y)=2. So x=500.
25. Full marks is 135. For questions 1-15, every question missed or wrong results in -5 points from the maximum, for 16-20 if wrong its -7 and -6 if skipped, for 21-25 its -8 if wrong and -6 if skipped. 119 implies 16 marks less than maximum. There is no way to get -16 with 3 incorrect answers. It could happen if 2 questions 1-15 are wrong or unanswered and 1 from 16-20 missed.