Search problems

14 problems match “combinatorics”. Clear

Participants in a chess tournament ●●●●○ 100 ptsEach participant in a chess tournament played one game against each of the other participants. A win earns one point, a draw earns half a point, and a loss earns no points. At the end of the tournament, it was determined that each participant had earned exactly half of their points in games against the three competitors who finished in the last three places. How many participants were there in the tournament?
Guaranteed score erasing blackboard numbers ●●●●● 200 ptsThere are 68 pairs of non-zero integers on a blackboard. Some pairs may be identical. For each positive integer $k$ , at most one of the pairs $(k,k)$ and $(-k,-k)$ appears on the blackboard. A student erases some of the 136 numbers so that the sum of any two erased numbers is not $0$ . For each of the 68 pairs, if at least one number from that pair is erased, the student scores one point. Find the largest score that the student can guarantee, whatever the 68 pairs on the blackboard are.
Maximum rounds guaranteeing a four-team condition ●●●●● 200 ptsThere are 18 teams in a single round-robin tournament. In each round the 18 teams are divided into 9 pairs, and the two teams in each pair play one match. The teams are regrouped from round to round, and over 17 rounds each team plays exactly one match against each of the other 17 teams. Suppose that after $n$ rounds, played according to any feasible schedule, there always exist 4 teams that have played exactly 1 match among themselves in total. Find the maximum possible value of $n$ .
Maximum points removing stones from piles ●●●●● 200 ptsThere are 100 piles, each containing 400 stones. On each move, Petya selects two piles, removes one stone from each, and earns as many points as the absolute difference between the numbers of stones in these two piles (before the removal). Petya must remove all the stones. What is the maximum total number of points he can earn?