Math Olympiad 5

Number Thinking (2)( Week 28 Evaluation)

2. If 28 ≤ X ≤ 33, 15 ≤ Y ≤ 19, and 54 ≤ Z ≤ 58 with X, Y, and Z all being integers, what is the smallest value of

X − Y

3. Make a six-digit number using the digits 3,3,6,6,8,8. In the six-digit number, the identical digits must be separated by at least another non-identical digit and the larger digits must be separated by more digits than smaller ones. There are two such numbers, can you find them? The smaller one is . The larger one is .

4. 86's factorial is the product of all natural numbers from 1 through 86. Let M = 1 × 2 × 3... × 86. How many continuous zeroes will M have at its end?

5. A symmetrical number is a number that reads the same from either direction, e.g. 11, 121, 1221 are symmetrical numbers. How many 6-digit symmetrical numbers are divisible by 11?

6. 5A782B is a six-digit number in which A and B are digits, and the number is divisible by 88 without remainder. Find the value of A = and the value of B = .

7. If the digits of a natural number are reversed to form a new number, the two numbers are palimage each other. 123 and 321, 1234 and 4321 are examples of palimages. If a number ends with 0, its palimage assumes a leading 0, which has no effect to its value. For example, the palimage of 1230 will be 0321, which is equal to 321. Now add 504 and its palimage to get the sum A. Add A and its palimage to get the sum B. Add B and its palimage to get the sum C. What will be the value for C?

9. If p is a positive integer and q is a positive integer, which of the following statements must be true?

I. pq > 0

II.(p + q)² < 0

III.(p + q)³ > 0