Multiple Choice Questions Singapore Mathematical Olympiad (SMO) 2018 (Junior Section)

1.  How many zeroes does the product of 255, 1254 and 2018end with?

(A) 5
(B) 9
(C) 10
(D) 12
(E) 13

25= 510
125= 2x 3x 58
20183 = 2x 2513

That each product contains a factor

510 x 5= 513 x 55
2x 29 = 213

So which produce 1013.

2. Given that (√2𝐱 + 𝐲) + (√𝐱2 + 9) = 0, find the value(s) of 𝐲 - 𝐱.

For (√2𝐱 + 𝐲) + (√𝐱2 + 9) = 0,
(√2𝐱 + 𝐲)= 0 and (√𝐱2 + 9) = 0
So we have 𝐱 = 3 or -3
and 𝐲 = -2𝐱 = -6 or 6
So,  𝐲 - 𝐱 = -9 or 9

3. The number of integers between 208 and 2008 ending with 1 is

(A) 101
(B) 163
(C) 179
(D) 180
(E) 200

The first integers is 211 and the  is 2001.
So,
211 + (n-1)10 = 2001
(n-1)10 = 2001- 211
(n-1) = 1790/10
n = 179 + 1
n = 180

4. The remainder when is 72008 + 92008 divided by 64 is

(A) 2
(B) 4
(C) 8
(D) 16
(E) 32

72008  = (8 - 1)2008 = 64k1 + 1 for some integers k1
Similarly, we have
92008  = (8 +1)2008 = 64k2 + 1 for some integers k2

Hence, the remainder is (1 + 1) or 2.

5. John has two 20 cent coins and three 50 cent coins in his pocket. He take 2 coins out of his pocket, at random, one after the other without replacement. What is the propability that the total value of the two coins taken out is 70 cents?

(A) 6/25
(B) 3/10
(C) 12/25
(D) 3/5
(E) 13/20

(3/5 x 2/4) + (2/5 x 3/4) = 3/5

6. Let x, y, and z be non-negative numbers. Suppose x  + y = 10 and y + z = 8. Let S = x + z. What is the sum of the maximum and the minimum value of ?

(A) 16
(B) 18
(C) 20
(D) 24
(E) 26