p1. Given a set of n the first natural number. If one of the numbers is removed, then the average number remaining is 2141 . Specify the number which is deleted.
p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring 3×3. Ipin gets first turn by playing X. Upin plays O. They must fill in the X or O 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 4th 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 to 10 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 16. 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.pngp4. Define S(n)=k=1∑n(−1)k+1,k=(−1)1+11+(−1)2+12+...+(−1)n+1n Investigate whether there are positive integers m and n that satisfy S(m)+S(n)+S(m+n)=2011
p5. Consider the cube ABCD.EFGH with side length 2 units. Point A,B,C, and D lie in the lower side plane. Point I is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid I.ABCD. If the pyramid I.ABCD is cut by a diagonal plane connecting the points A,B,G, and H, determine the volume of the truncated pyramid low part. algebrageometrycombinatoricsnumber theoryindonesia juniors