Challenging Mathematical Problems with Elementary Solutions, 第 1 卷Holden-Day, 1964 - 440 頁 |
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第 46 頁
... shown in figs . 24a and 24b ) . Remark . Let A1 , A2 , A3 , А be the areas of the four faces of the tetrahedron T. It can be shown that if one of the three equations ( 1 ) A1 + A2 = A3 + A4 ( 2 ) A1 + A3 ( 3 ) A1 + А = - A2 + A A2 + A3 ...
... shown in figs . 24a and 24b ) . Remark . Let A1 , A2 , A3 , А be the areas of the four faces of the tetrahedron T. It can be shown that if one of the three equations ( 1 ) A1 + A2 = A3 + A4 ( 2 ) A1 + A3 ( 3 ) A1 + А = - A2 + A A2 + A3 ...
第 72 頁
... shown in fig . 26. Any solution of the equation x1 + ··· + xm = n in positive integers corresponds to a decomposition of this segment into m pieces whose lengths are positive integers . The m - 1 end points of these pieces ( other than ...
... shown in fig . 26. Any solution of the equation x1 + ··· + xm = n in positive integers corresponds to a decomposition of this segment into m pieces whose lengths are positive integers . The m - 1 end points of these pieces ( other than ...
第 73 頁
... shown in fig . 27a ( where 4 ) . There are 1 + 2 + 3 + ··· + m m ( m + 1 ) / 2 dots in this triangle . m = = Suppose now that this triangle is removed , and that the remaining dots are then shifted to the left as shown in fig . 27b ...
... shown in fig . 27a ( where 4 ) . There are 1 + 2 + 3 + ··· + m m ( m + 1 ) / 2 dots in this triangle . m = = Suppose now that this triangle is removed , and that the remaining dots are then shifted to the left as shown in fig . 27b ...
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A₁ A₂ An+m arrangements b₁ B₂ binomial coefficients binomial theorem bishops black squares C₁ chessboard chord circle coefficient color column compute the number Consequently consider corresponding customers denote determine the number diagonals digits dihedral angle divided divisible draw equally likely possible equation equidistant equivalence classes exactly example experiment favorable outcomes follows formula given Hence inclusion and exclusion intersection k-gons knights L₁ length mathematical induction maximum number n-gon number of different number of favorable number of paths number of shortest obtain pairs partition passengers plane points A1 polygons positive integers possible outcomes Pr{E probability theory problem 54 prove queens rectangle relatively prime remaining required probability rooks segment selected at random sequence shortest paths side solution to problem solved sphere square controlled Suppose T₂ total number triangle unfavorable values vertex vertices