216 BOOK OF MATHEMATICAL PROBLEMS. 



1033. Trove that, c being < 1, 



-,/2ccoso; 



1034. On a straight line of length a + b + c are measured at 

 random two distances a + c, b + c ; prove that the mean value of 

 the part common to the two is 



I 2 



3~a' ^ )' 



1035. A point is taken at random on a given finite straight 



line of length a, prove that the mean value of the sum of the 



o 

 squares on the two parts of the line is a* ; and that the chance 



of the sum being less than this mean value is p> . 



*J3 



1036. A triangle is inscribed in a given circle whose radius 

 is a, prove that, if all positions of the angular points be equally 



probable, the mean value of the perimeter is , and the mean 



value of the radius of the inscribed circle is 



12 



1037. If 2a be the given perimeter of a triangle, and all 

 values of the sides for which the triangle is possible be equally 

 probable, the mean value of the radius of the circumscribed circle 

 is five times the mean value of the radius of the inscribed circle. 



1038. If 2a the perimeter, and c the side of a triangle, be 

 given, find the mean value of its area ; and prove that the mean 

 value of these mean values, c being equally likely to have any 



value from to a, is -_- - . 

 5U 



1039. The mean value of the area of all acute angled tri- 



02 



angles inscribed in a given circle of radius a is . 



