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Indonesia Juniors 2011 day 2 OSN SMP - Tic Tac Toe game for p2

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November 3, 2021
algebrageometrycombinatoricsnumber theoryindonesia juniors

Problem Statement

p1. Given a set of nn the first natural number. If one of the numbers is removed, then the average number remaining is 211421\frac14 . Specify the number which is deleted.
p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring 3×33 \times 3. Ipin gets first turn by playing XX. Upin plays OO. They must fill in the XX or OO mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the 44th step. For example, one of the positions the end is like the picture on the side. https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png
p3. Numbers 1 1 to 1010 are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are 1616. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png

p4. Define S(n)=∑k=1n(−1)k+1  ,  k=(−1)1+11+(−1)2+12+...+(−1)n+1nS(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n Investigate whether there are positive integers mm and nn that satisfy S(m)+S(n)+S(m+n)=2011S(m) + S(n) + S(m + n) = 2011
p5. Consider the cube ABCD.EFGHABCD.EFGH with side length 22 units. Point A,B,CA, B, C, and DD lie in the lower side plane. Point II is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid I.ABCDI.ABCD. If the pyramid I.ABCDI.ABCD is cut by a diagonal plane connecting the points A,B,GA, B, G, and HH, determine the volume of the truncated pyramid low part.
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