7.1 A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square.
7.2 Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17.
7.3 In a 1000-digit number, all but one digit is a five. Prove that this number is not a perfect square.
7.4 / 6.5 Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams.
7.5 In a pentagon ABCDE, K is the midpoint of AB, L is the midpoint of BC, M is the midpoint of CD, N is the midpoint of DE, P is the midpoint of KM, Q is the midpoint of LN. Prove that the segment PQ is parallel to side AE and is equal to its quarter.
https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png7.6 / 8.4 Several circles are arbitrarily placed in a circle of radius 3, the sum of their radii is 25. Prove that there is a straight line that intersects at least 9 of these circles.
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