Advanced book on Mathematics Olympiad

4.1 Geometry 227 cosφ 1 dx−cosφ 1 d 1 −(x− 1 )sinφ 1 dφ 1 +sinφ 1 dy−sinφ 1 dη 1 +(y−η 1 )cosφ 1 dφ 1 = 0. This expression can ...

228 4 Geometry and Trigonometry 639.Two convex polygons are placed one inside the other. Prove that the perimeter of the polygon ...

4.1 Geometry 229 We will show that the only possible configuration is that in Figure 32. Consider a triangle that maximizes the ...

230 4 Geometry and Trigonometry A E B C D Figure 33 If all three segmentsAB,AC, andADhad rational lengths, this relation would i ...

4.2 Trigonometry 231 prove that the quadrilateral is a rhombus. Does the property hold ifOis some other point in the interior of ...

232 4 Geometry and Trigonometry tanx= sin 20◦ 2 cos 40◦−cos 20◦ . The tangent function is one-to-one on the interval( 0 , 90 ◦), ...

4.2 Trigonometry 233 Remark.It is interesting to know that Leonardo da Vinci’s manuscripts contain drawings of such decompositio ...

234 4 Geometry and Trigonometry 657.Show that the trigonometric equation sin(cosx)=cos(sinx) has no solutions. 658.Show that if ...

4.2 Trigonometry 235 4.2.2 Euler’s Formula.......................................... For a complex numberz, ez= 1 + z 1! + z^2 2 ...

236 4 Geometry and Trigonometry Solution.Let 1 , 2 ,...,kbe thekth roots of unity, that is,j=cos^2 jπk +isin^2 jπk , j= 1 , ...

4.2 Trigonometry 237 f(x)+f ( x+ 2 π 3 ) +f ( x+ 4 π 3 ) = 3 a. Ifa<0, thens=twould work. Ifa=0, then for somexone of the ter ...

238 4 Geometry and Trigonometry 673.For positive integersndefineF (n)=xnsin(nA)+ynsin(nB)+znsin(nC), where x, y, z, A, B, Care r ...

4.2 Trigonometry 239 Thenf(cost)=csintfor some constantcandtin( 0 ,π), i.e.,f(x)=c √ 1 −x^2 for allx∈(− 1 , 1 ). It follows that ...

240 4 Geometry and Trigonometry 677.Find the maximum value of S=( 1 −x 1 )( 1 −y 1 )+( 1 −x 2 )( 1 −y 2 ) ifx 12 +x 22 =y 12 +y ...

4.2 Trigonometry 241 (c) Prove there is a positive constantCsuch that one has 0<bn−an< 8 Cn for alln. 684.Two real sequenc ...

242 4 Geometry and Trigonometry allows us to prove inductively that 2 coshkt 20 is an integer once we show that 2 cosht 20 is an ...

4.2 Trigonometry 243 This telescopes to 1 4 [( − 1 3 )n cos ( 3 n+^1 a ) − ( − 1 3 )− 1 cosa ] . Fora= 3 −nπ, we obtain the iden ...

244 4 Geometry and Trigonometry Example.Prove that ∏∞ n= 1 1 1 −tan^22 −n =tan 1. Solution.The solution is based on the identity ...

5 Number Theory................................................. This chapter on number theory is truly elementary, although its ...

246 5 Number Theory We pursue the track off( 0 )=0 first. We have 2 f( 12 + 02 )=(f ( 1 ))^2 +(f ( 0 ))^2 , so 2f( 1 ) =f( 1 )^2 ...