# SMO JUNIOR SECTION 2012 SOLUTIONS PART 5

Problem 14 :

Find the four digit number

Solution :

From the picture above, it is very clear

2d = a, then 2d must be even and a is also an even.

2a+1 = d,

2a+1  ≤ 9

By applying any one of the following values of a,

a = 2 or a = 4 or a = 6, the above condition can be satisfied.

• If a = 2, then 2a + 1 ≤ d will become 5 ≤ d,
• This will work only if d = 6

2c + 1 = b

2b = 10 + c

will not give integer solution.

2c + 1 = 10 + b

If c = 9 and b = 9

19 = 19

a = 2, b = c = 9 and d = 6

So, the required number is 2996.

Problem 15 :

Suppose x and y are real number satisfying

x2 + y2 - 22x - 20y + 221 = 0. Find xy.

Solution :

x2 + y2 - 22x - 20y + 221 = 0

x2 - 22x + y2- 20y + 221 = 0

x2 - 2   11 + 112 - 11 + y2- 2  y  10 + 102 - 102 + 221 = 0

(x - 11)2 - 112 + (y - 10)2 - 102 + 221 = 0

(x - 11)2 + (y - 10)2 - 121 - 100 + 221 = 0

(x - 11)2 + (y - 10)2 = 0

Will become 0, only when x = 11 and y = 10

So, the value of xy = 110.

Problem 16 :

Let m and n be positive integers satisfying

mn2 + 876 = 4mn + 217 n

Find the sum of all possible values of m.

Solution :

mn2 + 876 = 4mn + 217 n

It is enough to find the value of m, so recreate the equation as n =

mn2 - 217n = 4mn - 876

n(mn - 217) = 4mn - 876

n = (4mn - 876) / (mn - 217)

n = 4 - [8 / (mn - 217)]

Here mn - 217 = ±1, ±2, ±4, ±8

 If mn -217 = 1mn = 218, n = -4If mn -217 = -1mn = 216, n = 12 If mn -217 = 2mn = 219, n = 0If mn -217 = -2mn = 215, n = 8 If mn -217 = 4mn = 221, n = 2If mn -217 = -4mn = 213, n = 6 If mn -217 = 8mn = 225, n = 3If mn -217 = -8mn = 209, n = 5

Since m is integer,

m = 216/12 ==> 18

m = 215/3 ==>  75

Sum of values of m = 75 + 18 ==> 93

Problem 17 :

For any real number x, let ⌊x⌋ denote the largest integer less than or equal to x. Find the value of ⌊x⌋ of the smallest x satisfying ⌊x2⌋ - ⌊x⌋2 = 100.

Solution :

x = ⌊x⌋ + {x}

100   (⌊x⌋ + {x})2 ⌊x⌋2

100   ⌊x⌋2 + 2⌊x⌋{x} + {x}2 -  ⌊x⌋2

100   2⌊x⌋{x} + {x}2

100   2⌊x⌋ + 1

S o  x  50 and x ⌊x2 = 100 + 5022600. On the other and, x = 2600 is a solution.

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