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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