500 bài toán học tiếng anh 1995 five hundred mathematical challenges
[...]... tangent at T and the point S is the foot of the perpendicular from Qto this same tangent Prove that OR= OS .-1'3\ IPE 15 FIVE HUNDRED MATHEMATICAL CHALLENGES 10 Problem 95 1 1 Problem 98 Observe that: 1 1 = + 2 1 2 2; - 1 4 1 1 = + 3 1 1 1 = 1 (i + 2}(i + 3) + + 1 j(j + 1)' (American Mathematical Monthly S S ( 1948), 427 = BC 3AB side BCwith Let ABCD be a rectangle with Show that if P, Q are the points... distance from P to side BC CA and AB respectively Let h1o h2, and h.3 de note respectively the length of the altitude from 4, B, C to the opposite side of the triangle Prove that FIGURE 4 4 FIVE HUNDRED MATHEMATICAL CHALLENGES Problem 28 A hoy lives in each of n houses on J straight line At what point should the n boys meet so that the sum of the distances that they Problem 33 [il + [";2] + [";4] = [�]... tables, calculators etc \h,•ulll �· l!� d Let tenns of the sequence s > t Three noncollinear points are given Find the triangle f which or ·f': rnidpoi'lts of the edges P, Q R P, Q R are 6 FIVE HUNDRED MATHEMATICAL CHALLENGES Problem 48 59!' is Prove that 199 +299 + 3�19+499 + situated that the angles of elevation, at each point, Show that there are no integers Problem 56 divisible by 5 Problem 49... Locate any possible stations In other words, how many points are there: in :11l' F.'-' ·j=,lt:a•t Problem 77 "· : ·· · ':- " Prove that f all positive integers or 3''- 611 is divisible by 1 0 FIVE HUNDRED MATHEMATICAL CHALLENGES 8 FIGURE 12 FIGURE 1 1 Problem 83 Problem 78 n points are given on the circum ference of a circle, and the chords determined by them are drawn If no three chords have a common... his chauffeur until S PM he starts walking home On his way he meets the chauffeur who picks him up promptly and returns home arriv FIGURE 1 !n;; �0 minutes earlier than usual Some weeks 2 FIVE HUNDRED MATHEMATICAL CHALLENGES later on another tine day Mr Smith takes an ear l ier train and arrives at his home station at 4:30 FC' =CD= H A prove that ABCD is also a square PM Again instead of waiting for... Problem 114 Observe the folJowing equations: R A F FIGURE 1 7 sets 1.1 1= 1 1 1 2· 1 - 2=1 + 2 1 1 1 1 3· 1 - 3 · 2 + 3 =1 + 2 + 3 1 1 1 1 1 1 6· - + 4· - - -= 1 + - + - + 2 3 4 2 3 4 of FIVE HUNDRED MATHEMATICAL CHALLENGES 12 B 1 =1 � =2·1-(1+D � =3·1-3(1 +D +(1+� + D � =4·1-6(1+D +4(1+ �+ D J F FIGURE 1 8 Problem 1 19 State and prove a generalization for each set Generalize the relationship between... the smallest positive number k(n) such that k(n) + x x2 y2 z 2 y z x -+-+->-+-+- y2 z2 x2 - x y z • • • where r1, r2, , rn are real numbers Prove that (n - 1)a� � 2na2 If , ' IIJ'Jf,f ?Q 14 FIVE HUNDRED MATHEMATICAL CHALLENGES B Problem 143 Prove that if all plane cross sections of a bounded solid figure are circles, then the solid is a sphere Problem 144 In how many ways can we stack different coins... Problem 169 f a b, c, dare positive real num I bers,prove that al +b2 + 2 b2 + c2 +� c2 +d2 +a2 + - + b+c+d a+b+c c+d+a d2 +02 +b2 �a+b+c+d d+a+b ,;th -.··.:t·al_!t)only if a= b = c =d + 16 FIVE HUNDRED MATHEMATICAL CHALLENGES Problem 1 70 a ) Find all positive inte grs with ( e initi l di g 6 a it such th t th e nt egr orm d by del t a i e f e e in g s 6i s thi :fr.o fth eori Problem 1 79 Asequence... circumradius CD = C' D' , DA = D' A' Problem 190 Let ABCD be a concyclic con :h·.11 Hr.e A'C' is perpendicular to line B' D' vex quadrilateral for which AP ' '! ' D::nt·tr a - ; i)�i'\ t FIVE HUNDRED MATHEMATICAL CHALLENGES 18 A desk calendar consists of a reg ular dodecahedron with a different month on each of its twelve pentagonal faces How many essen tially ditTerent ways are there of arranging... b such that nn + b is triangular whenever n is Problem 223 ger n ,\it : •: • Prove that for any positive inte (.fii + Jn+I] = (J.tn + 2 ] J.:notes the greatest integer function 20 FIVE HUNDRED MATHEMATICAL CHALLENGES Problem 22-1 Prove or disprove the following statement Given a line I and two points A and B not on I the point P on I for which LAPB is largest must lie between the feet of the