**RMO - 1995**

- 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 .
- 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. - Prove that among any 18 consecutive three - digit numbers there is at least one number which is divisible by the sum of its digits.
- Show that the quadratic equation x
^{ 2 }+ 7x - 14(q^{ 2 }+ 1)=0 , where q is an integer, has no integer root. - 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. - 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? - Show that for any real number x , x
^{ 2 }x + x cos x + x^{ 2 }+ (1) / (2) . >0.

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