Maths Olympiad Prep

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Combinatorics Difficulty 6.8 National olympiad Find the answer

a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?

b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges?

(Kyiv mathematical festival 2014)

A number or a short expression. Spacing and $ signs are ignored.

Solution

a) Let W1B1=a1 |W_1B_1| = a_1 , W2B1=a2 |W_2B_1| = a_2 , W1B2=b1 |W_1B_2| = b_1 , and W2B2=b2 |W_2B_2| = b_2 .

We are given the following conditions:
1. a1+a2=4 a_1 + a_2 = 4
2. b1+b2=8 b_1 + b_2 = 8
3. b1+a1=9 b_1 + a_1 = 9
4. b2+a2=3 b_2 + a_2 = 3

From these conditions, we can derive the following steps:

1. Subtract equation (4) from equation (3):
(b1+a1)(b2+a2)=93 (b_1 + a_1) - (b_2 + a_2) = 9 - 3
b1+a1b2a2=6 b_1 + a_1 - b_2 - a_2 = 6

2. Substitute a1+a2=4 a_1 + a_2 = 4 into the equation:
b1b2=6 b_1 - b_2 = 6

3. From equation (3), solve for b1 b_1 :
b1=9a1 b_1 = 9 - a_1

4. From equation (4), solve for b2 b_2 :
b2=3a2 b_2 = 3 - a_2

5. Substitute b1=9a1 b_1 = 9 - a_1 and b2=3a2 b_2 = 3 - a_2 into b1b2=6 b_1 - b_2 = 6 :
(9a1)(3a2)=6 (9 - a_1) - (3 - a_2) = 6
9a13+a2=6 9 - a_1 - 3 + a_2 = 6
6a1+a2=6 6 - a_1 + a_2 = 6
a1+a2=0 -a_1 + a_2 = 0
a1=a2 a_1 = a_2

6. Since a1+a2=4 a_1 + a_2 = 4 and a1=a2 a_1 = a_2 :
2a1=4 2a_1 = 4
a1=2 a_1 = 2
a2=2 a_2 = 2

7. Substitute a1=2 a_1 = 2 into b1=9a1 b_1 = 9 - a_1 :
b1=92=7 b_1 = 9 - 2 = 7

8. Substitute a2=2 a_2 = 2 into b2=3a2 b_2 = 3 - a_2 :
b2=32=1 b_2 = 3 - 2 = 1

Since b1=7 b_1 = 7 is greater than the other distances, the extremes are W1 W_1 (first white cat) and B2 B_2 (second black cat).

The final answer is W1 \boxed{ W_1 } and B2 B_2 .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.