Advanced book on Mathematics Olympiad

732 Combinatorics and Probability This product is positive, so by exchangingyσ(i)andyσ(j)we decrease the sum. This means that th ...

Combinatorics and Probability 733 The first two properties are obvious, while the third requires a proof. Arguing by contradicti ...

734 Combinatorics and Probability { σ ( n+ 3 2 ) ,σ ( n+ 5 2 ) ,...,σ(n) } ⊂ { 1 , 2 ,..., n+ 1 2 } . Letσ(n+ 21 )=k.Ifk≤n+ 21 , ...

Combinatorics and Probability 735 then^2 −n+2 spherical regions divides some spatial region into two parts. This allows us to wr ...

736 Combinatorics and Probability 842.We examine separately the casesn= 3 , 4 ,5. A triangle can have at most one right angle, a ...

Combinatorics and Probability 737 ⌊ 200 π √ 2 2 π ⌋ = 100 √ 2 = 141. (Ukrainian Mathematical Olympiad) 844.The solution is bas ...

738 Combinatorics and Probability This is a well-known property, true ind-dimensional space, where “disks’’ becomes “balls’’ and ...

Combinatorics and Probability 739 b w b b b b b b b w w w w w w w Figure 97 848.For finding the upper bound we employ Euler’s fo ...

740 Combinatorics and Probability F= 2 +E−V= 5. We have reached a contradiction, which shows that the answer to the problem is n ...

Combinatorics and Probability 741 852.(a) We use an argument by contradiction. The idea is to start with Euler’s formula V−E+F= ...

742 Combinatorics and Probability v w C C A B D B w A A B D D 1 3 w w 5 w 2 4 C E Figure 100 Next, let us focus onw 2 andw 4 (Fi ...

Combinatorics and Probability 743 Call a vertex that belongs only to outgoing edges a source, a vertex that belongs only to inco ...

744 Combinatorics and Probability Remark.The polyhedron can be thought of as a discrete approximation of a surface. The orientat ...

Combinatorics and Probability 745 then−1 edges starting atx. Among them there are eitherR(p− 1 ,q)red edges, or R(p, q− 1 )blue ...

746 Combinatorics and Probability 857.Yet another Olympiad problem related to Schur numbers. We can reformulate the problem as f ...

Combinatorics and Probability 747 x=j−i,y=k−j, andz=k−iform a Schur triple. The fact that they have the same color means that th ...

748 Combinatorics and Probability f(s)=d.Iffwere not one-to-one, that is, iff(s 1 )=f(s 2 )=d, for somes 1 ,s 2 ∈A, thendwould c ...

Combinatorics and Probability 749 Integrating by parts, we obtain Ik= ∫ π 2 0 (2 sinθ)^2 k−^1 (2 sinθ)dθ =(2 sinθ)^2 k(−2 cosθ)| ...

750 Combinatorics and Probability This formula follows inductively from the combinatorial identity ( m m ) + ( m+ 1 m ) +···+ ( ...

Combinatorics and Probability 751 and this isF 2 n. The identity is proved. (E. Cesàro) 862.Note that fork= 0 , 1 ,...,n, (ak+ 1 ...