Download rules of mathematical tennis

**Number of participants and rounds**

* Mathematical tennis* is an individual competition in which any number of participants can participate, let them be N, which sa

*t*isfies the condition – it is greater than or equal to 2

*^k*, but less tha

*n 2^(k+*1) (hereinafter we will consider N=75 as an ex

*ampl*e). At the beginning of the competition, a draw is held and each participant receives a number from 1 to

*N. The competition will last exac*

*t*ly k rounds.

*Format of the round*

In each round, participants are divided into two groups – those that compete with each other in a p* air (f*ight), and those that simply work on their overall rating

*p). All participants of the round (both in the fight and in the group) receive 2 tasks for 15 minutes. At the end of this time, the jury should hand over a piece of paper with recorded (or not recorded) answers to each task. For the correct answer for each of the two tasks, the participant receives 3 points, if the answer to the problem is wrong – receives 0 points, for the absence of an answer – has 1 point. That is, the participant can score 6, 4, 3, 2, 1 or 0 points.*

**(grou**For participants competing in a group, this result is simply added to its group result. For participants of the fight, their results are compared with each other. If someone scores more (6 vs. 4, and because 1 vs. 0, etc.), then he is considered the winner in this round. If they score the same number of points, then they compete in a tie

*y are simultaneously given 1 task for 5 minutes. Whoever gives an answer to it earlier decides the fate of the round. If his answer is correct, then he is a winner, if not – he will lose this round. If the opponents gave the correct or incorrect answer at the same time, or did not provide any answers in the allotted 5 minutes, then the winner is determined by the draw.*

**break – th**e*Previous (first) round*

The first round is conducted according to a slightly different scheme than all the others. This applies not to scoring, but to the organization of its conduct. After the first round, exactly 2^k of participant*s w*ho will compete for the victory should remain. Therefore, if the participants were exac*tly *2^k, then this round is not held at all (for example, if the participants were exactly* 6*4). Thus, exactly N-2^k participants must "fly out of participat*ion f*or victory". Therefore, in the zero round, pairs of particip*ants wit*h numbers 1 to 2 (N-2^k) are formed, and *participa*nts with adjacent numbers play with each other – the 1st with the 2nd, the 3rd of the 4th,* …, 2(N-2^*k)-1 o*f 2(N-2^k*)-m. For this example, 11 participants must fly after the first (preliminary) round, so participants with numbers from 1 to 22 are divided into 11 pairs. All other participants solve the same problems in the group. At the end of the first round, exactl*y 2*^k participants begin to compete in the main rounds and fight for victory or for higher places, and those who have withed ou*t N-2*^k participants form a group. This group further solves the problems and will fight for places with (*2^k+1)-th* by N-so*m*ething. For this example, the 11 participants who withed out in the first round will compete in the group for 65-75 places.

*Main rounds*

Before the main round, we have 2^m o*f p*articipants fighting in fights, as well as all the others that are divided into as many groups as the rounds have already passed. After the first round, we have *2^k* participants for fights an*d N-2*^k participants form a group. This group further solves the problems and will fight for places from* (2^k+1)*-th by N* –* that. For fights, pairs are formed in accordance with the previously held draw, namely, the pairs are combined from top to bottom after excluding the lost numbers. For example, if a participant with the number 1 in the first bet and number 4 in the second round wins in the first round, then the participants with the numbers 1 and 4 and so on similarly play in the second round. After the second round, it turns out* that th*e 2^(k-1) winners of this round continue to compete in fights, and those who lose, that is, the other 2^*(k-1) of* the participants of the fights form a group that will compete for places fr*om (2^(k-1)+1*)-th to *2^k–*that. In the example, after the second round, 32 participants remain to compete in the fight, and the remaining 32 participants will win places from the 33rd to the 64th. At the same time, all previous groups continue to solve the same problems and score points in accordance with the answers given.

Therefore, after each subsequent round, the number of participants in the fights is halved, and a new group is formed that will fight for places fro*m (2^(m-1) +1*)-th to* 2^*m–te.

For this example, we have the following groups:

After the first round: fights – 64 participants, group 1, fighting for places 65-75.

After round 2: fights – 32 participants, group 1, fighting for places 65-75 and group 2, fighting for places 33-64. .

After round 3: Fights – 16 participants, group 1, fighting for places 65–75, group 2, fighting for places 33-64 and group 3, fighting for places 17-32 .

After round 4: fights – 8 participants, group 1, fighting for places 65-75, group 2, fighting for places 33-64, group 3, fighting for places 17-32, and group 4, fighting for places 9-16 .

After Round 5: Fights – 4 Participants, Group 1, Fighting for Places 65–75, Group 2, Fighting for Places 33–64, Group 3, Fighting for Places 17–32, Group 4, Fighting for Places 9–16, and Group 5 Struggling for Places 5-8

Then there are 2 more rounds, where the finalists (winners of the 6th round of fights) and those who win places 3-4 are determined, and in the last round there is a final allocation of places 1-4.

*Places in a group*

After the formation of a certain group that wins places within the specified rules, each participant scores a certain number of points in each round (points in the first round are also added here). At the end of the competition, they are simply added and the one who scored a larger number of them takes the higher place. In the case of the same number – they must share a certain place, or a higher place is occupied by the one who lost in the fight to the participant who took the higher final place.

*Schematic diagram*