IT-CESENATICO 2010

In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The test results are the following: The average of all participants is $66$ , that of the promoted $71$ and that of the repeaters $56$ . However, due to an error in the wording of a question, all scores are increased by $5$ . At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$ . (a) Find all possible values of $N$ . (b) Find all possible values of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$ .

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$ . If $M$ be the midpoint of $AB$ , prove that $\angle BCM = \angle DBA$ .

In a trapezium $ABCD$ , the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$ .

In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is $01$ , one may replace this by substring $100$ . The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from $0101$ , one can win using move (iii) first in the last two digits, resulting in $01100$ , then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?