NTSE-Scholars [An Addrress for Young Talents]

RMO – 1995

1. In triangle ABC , K and L are points on the side BC ( K being closer to B
   than L ) such that BC KL = BK CL , and AL bisects KAC . Show that AL is
   perpendicular to AB .
   
   
   
2. Call a positive integer n good, if there are n integers, positive
   or negative, and not necessarily distinct, such that their sum and product are
   both equal to n (e.g. 8 is good, since 8=4 2 1 1 1 1 ( – 1) ( – 1)=4 + 2 + 1 +
   1 + 1 + 1 + ( – 1) + ( – 1)) . Show that integers of the form 4k + 1(k >=
   0) and 4l(l >= 2) are good.
   
   
   
3. Prove that among any 18 consecutive three – digit numbers there is at
   least one number which is divisible by the sum of its digits.
   
   
   
4. Show that the quadratic equation x ² + 7x – 14(q ² +
   1)=0 , where q is an integer, has no integer root.
   
   
   
5. Show that for any triangle ABC , the following inequality is true: a
   ² + b ² + c ² > square root 3 {|a ²
   - b ^2| ,|b ² - c ^2| ,|c ² -
   a ^2| }, where a,b,c are, as usual, the sides of the triangle.
   
   
   
6. Let A _{1A _ 2A _ 3} … A ₂₁ be a 21 – sided regular
   polygon inscribed in a circle with centre O . How many triangles A _{iA _
   jA _ k} , 1 <= i<j<k <= 21, contain the point O in their
   interior?
   
   
   
7. Show that for any real number x , x ² x + x cos x + x ²
   + (1) / (2) . >0.
   

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