Pre RMO

Pre-Regional Mathematical Olympiad 2016 West Bengal Region

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11 September 2016

ANSWERS TO THE PROBLEMS

1. Consider all possible integers n ≥ 0 such that (5 × 3m ) + 4 = n2 holds for some corresponding integer m ≥ 0. Find the sum of all such n.

2. Find the number of integer solutions of the equation x2016 + (2016! + 1!) · x2015 + (2015! + 2!) · x2014 + · · · + (1! + 2016!) = 0, where n! = n × (n − 1) × · · · × 3 × 2 × 1, for n ≥ 1.

3. Suppose N is any positive integer. Add the digits of N to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit number n. Find the number of positive integers N ≤ 1000, such that the final single-digit number n is equal to 5. Example: N = 563 → (5 + 6 + 3) = 14 → (1 + 4) = 5 will be counted as one such integer.

4. Consider a right-angled triangle ABC with 6 C = 90◦ . Suppose that the hypotenuse AB is divided into four equal parts by the points D, E, F , such that AD = DE = EF = F B. If CD2 + CE 2 + CF 2 = 350, find the length of AB.

5. Consider a triangle ABC with AB = 13, BC = 14, CA = 15. A line perpendicular to BC divides the interior of 4ABC into two regions of equal area. Suppose that the aforesaid perpendicular cuts BC at D, and cuts 4ABC again at E. If L is the length of the line segment DE, find L2 . √ 6. Suppose a circle C of radius 2 touches the Y -axis at the origin (0, 0). A ray of light L, parallel to the X-axis, reflects on a point P on the circumference of C, and after reflection, the reflected ray L0 becomes parallel to the Y -axis. Find the distance between the ray L and the X-axis.

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7. Find the coefficient of a5 b5 c5 d6 in the expansion of the following expression. 7

(bcd + acd + abd + abc)

8. Find the number of integer solutions of h x h x ii 100 100

= 5.

9. Let α and β be the roots of the equation x2 + x − 3 = 0. Find the value of the expression 4β 2 − α3 .

10. Let M be the maximum value of (6x−3y−8z), subject to 2x2 +3y 2 +4z 2 = 1. Find [M ].

11. For real numbers x and y, let M be the maximum value of the expression x4 y + x3 y + x2 y + xy + xy 2 + xy 3 + xy 4 ,

subject to x + y = 3.

Find [M ].

1 1 1 1 1 . Find [S]. 12. Let S = 1 + √ + √ + √ + · · · + √ + √ 2 3 4 99 100
√ √ √ 1 √ You may use the fact that

n<

2

n+

n+1 <

n + 1 for all integers n ≥ 1.

13. Find the total number of times the digit ‘2’ appears in the set of integers {1, 2, . . . , 1000}. For example, the digit ’2’ appears twice in the integer 229.

14. Find the minimum value of m such that any m-element subset of the set of integers {1, 2, . . . 2016} contains at least two distinct numbers a and b which satisfy |a − b| ≤ 3.

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15. Find the number of pairs of positive integers (m, n), with m ≤ n, such that the ‘least common multiple’ (LCM) of m and n equals 600.

16. For positive real numbers x and y, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers (x, y), with x ≤ y, from the set of numbers {1, 2, . . . , 2016}, such that the special mean of x and y is a perfect square.

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