Advanced book on Mathematics Olympiad

308 6 Combinatorics and Probability 896.A sheet of paper in the shape of a square is cut by a line into two pieces. One of the p ...

6.2 Binomial Coefficients and Counting Methods 309 The second example comes from I. Tomescu’s bookProblems in Combinatorics (Wil ...

310 6 Combinatorics and Probability 899.LetEbe a set withnelements andFa set withpelements,p≤n. How many surjective (i.e., onto) ...

6.3 Probability 311 Example.Show that the probability of getting a six when a die is rolled four times is greater than the proba ...

312 6 Combinatorics and Probability world situation, namely that about particles and states. The above considerations apply to b ...

6.3 Probability 313 906.Letvandwbe distinct, randomly chosen roots of the equationz^1997 − 1 =0. Find the probability that √ 2 + ...

314 6 Combinatorics and Probability 6.3.2 Establishing Relations Among Probabilities We adopt the usual notation:P (A)is the pro ...

6.3 Probability 315 Example.In a selection test, each of three candidates receives a problem sheet withn problems from algebra a ...

316 6 Combinatorics and Probability Let us find the probability that the first player wins in exactlya+kgames,k= 0 , 1 ,...,b−1. ...

6.3 Probability 317 915.An exam consists of 3 problems selected randomly from a list of 2nproblems, where nis an integer greater ...

318 6 Combinatorics and Probability already five propeller planes and one jet plane. Later, a farmer sees a jet plane flying out ...

6.3 Probability 319 Solution.The center of the coin falls on some tile. For the coin to lie entirely on that tile, its center mu ...

320 6 Combinatorics and Probability 930.Two airplanes are supposed to park at the same gate of a concourse. The arrival times of ...

SOLUTIONS ...

Methods of Proof 1.Assume the contrary, namely that √ 2 + √ 3 + √ 5 =r, whereris a rational number. Square the equality √ 2 + √ ...

324 Methods of Proof 3.The example 2^2 , 32 , 52 ,..., 432 , where we considered the squares of the first 14 prime numbers, show ...

Methods of Proof 325 33 =f( 2 )^3 =f( 23 )<f( 32 )=f( 3 )^2 =k^2 , hencek>5. Similarly, using 3^3 < 25 , we obtain k^3 ...

326 Methods of Proof 10.Assume the contrary. Our chosen numbersa 1 ,a 2 ,...,ak+ 1 must have a total of at mostkdistinct prime f ...

Methods of Proof 327 The induction is complete. 13.As in the solution to the previous problem we argue by induction onnusing tri ...

328 Methods of Proof ( 1 + 1 n )n < 3 , while the inequality on the right can be reduced to ( 1 + 1 n )n > 2. These are bo ...