Advanced book on Mathematics Olympiad

288 6 Combinatorics and Probability Example.Prove that anynpoints in the plane can be covered by finitely many disks with the su ...

6.1 Combinatorial Arguments in Set Theory and Geometry 289 845.Inside a square of side 38 lie 100 convex polygons, each with an ...

290 6 Combinatorics and Probability connected. The number of edges has decreased by 1; that of faces has also decreased by By t ...

6.1 Combinatorial Arguments in Set Theory and Geometry 291 4.m= 4 ,n=3, in which caseE=12,V=6,F=8; this is the regular octahedro ...

292 6 Combinatorics and Probability number, such that whenever the edges of a complete graph are colored red and blue, there is ...

6.1 Combinatorial Arguments in Set Theory and Geometry 293 order. These numbers, together with the differencesxi−x 1 ,i= 2 , 3 , ...

294 6 Combinatorics and Probability such that for any partition of{ 1 , 2 ,...,S(n)}intonsets one of the sets will contain a Sch ...

6.2 Binomial Coefficients and Counting Methods 295 ( n k ) = ( n− 1 k ) + ( n− 1 k− 1 ) allows the binomial coefficients to be a ...

296 6 Combinatorics and Probability Solution.Let us analyze the quotient pk,m(x)= (xk+m− 1 )(xk+m−^1 − 1 )···(xk+^1 − 1 ) (xm− 1 ...

6.2 Binomial Coefficients and Counting Methods 297 ( n m ) q = ( n− 1 m ) q +qn−m ( n− 1 m− 1 ) q , gives rise to what is called ...

298 6 Combinatorics and Probability 865.Prove the analogue of Newton’s binomial formula [x+y]n= ∑n k= 0 ( n k ) [x]k[y]n−k, wher ...

6.2 Binomial Coefficients and Counting Methods 299 The Laplace transform applied to the differential equation y′′+uy′+vy= 0 prod ...

300 6 Combinatorics and Probability = 1 − ∑∞ n= 1 ( 2 n− 3 )( 2 n− 5 )··· 1 n! ( 2 x)n= 1 − 2 ∑∞ n= 1 ( 2 n− 2 )! (n− 1 )!(n− 1 ...

6.2 Binomial Coefficients and Counting Methods 301 869.(a) Prove the identity ( m+n k ) = ∑k j= 0 ( m j )( n k−j ) . (b) Prove t ...

302 6 Combinatorics and Probability 877.The distinct positive integersa 1 ,a 2 ,...,an,b 1 ,b 2 ,...,bn, withn≥2, have the prope ...

6.2 Binomial Coefficients and Counting Methods 303 Next, a combinatorial identity. Example.Letmandnbe two integers,m≤n− 21. Prov ...

304 6 Combinatorics and Probability 880.Prove the combinatorial identity ∑n k= 1 k ( n k ) 2 =n ( 2 n− 1 n− 1 ) . 881.Prove the ...

6.2 Binomial Coefficients and Counting Methods 305 This formula can also be proved using induction onmfor arbitraryn. The casem= ...

306 6 Combinatorics and Probability i =jthe societies to whichCibelongs are all different from the societies to whichCj belong ...

6.2 Binomial Coefficients and Counting Methods 307 888.A numbernof tennis players take part in a tournament in which each of the ...