Math Kangaroo 2006 Grade 7-8 #19,,#25 (1 Viewer)

[TuePhuong]

"19. Find a truly end of the sentence: If your reflection looks on me then
A) you look on mine reflection B) my reflection looks on you
C) you look on me D) I look on your reflection"

"25. A house has 10 rooms. Ten boys stay in different rooms and count the number of doors in them. After that they sum all results and receive 25. What a proposition can't be true about number N of doors which led outside the house?
A) N=7 B) N=5 C) N=3 D) N=2"

 

[InteGrand]

"19. Find a truly end of the sentence: If your reflection looks on me then
A) you look on mine reflection B) my reflection looks on you
C) you look on me D) I look on your reflection"

"25. A house has 10 rooms. Ten boys stay in different rooms and count the number of doors in them. After that they sum all results and receive 25. What a proposition can't be true about number N of doors which led outside the house?
A) N=7 B) N=5 C) N=3 D) N=2"

19. By symmetry, the answer is (A).

25. The answer is (D). To see this, note that each door either belongs to exactly two rooms (as it leads belongs to a room and leads to another room), or belongs to only one room (as it leads outside the house). We are told N is the number of doors that lead outside the house (belong to only one room). Let K be the no. of doors belonging to precisely two rooms. When the doors are counted and the counts added up, the K doors belonging to two rooms are each counted twice (once by the person in each of the rooms it belongs to), and the N doors belonging to only one room are each counted once. Therefore, the total no. of counts is N + 2K, which we are told is 25. So N + 2K = 25. Now, 2K is even and 25 is odd, so N must be odd. Therefore, N cannot be 2, so the answer is (D).