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Problems in Elementary Mathematics PROBLEMS IN ELEMENTARY MATHEMATICS MIR PUBLISHERS MOSCOW 3AAA 4l9 B. B..NACKIIA, IIO 3JIEMEHTAPHOf1 A.B OBCRHHHKOB, A.H. TYJIAAKOB. MATEMATHKE M. H WABYHHH H3.1 ATE.BbCTBO 4HAYKA. MOCKBA V. LIDSKY, PROBLEMS L. OVSYANNIKOV, A. TULAIKOV, M. SHABUNIN IN ELEMENTARY MATHEMATICS MIR PUBLISHERSI MOSCOW UDC 511+512+513+514(0.75.4)=20 Translated frori the Russian by V. VOLOSOV First published 1973 Revised from the 1970 Russian edition Ha axaau1cxoet R3btKe Q English translation, Mir Publishers, 1973 0223 - 374 041(01) - 73 CONTENTS ProblemsSolutions Algebra 1. Arithmetic and Geometric Progressions (1-23) 7 87 2. AlgebraicEquations and Systems ofEquations (24-95) 10 95 3. Algebraic Inequalities (96-123) . 20 134 4. Logarithmic and ExponentialEquations,Identities and Inequalities (124-169) . 24 142 5. Combinatorial Analysis and Newton's Binomial Theorem (170-188) . 29 157 6. Problems in Forming Equations (189-228) . 32 162 7. Miscellaneous Problems (229-291) ....... .. .. 38 180 Geometry A. Plane Geometry 1. Computation Problems (292-324).. .......... 47 202 2. Construction Problems (325.338) . ....... 51 217 3. Proof Problems (339-408) . .... 52 223 4. Loci of Points (409-420). ... ... 59 254 5. The Greatest and Least Values (421-430).. ...... 61 260 B. Solid Geometry 1. Computation Problems (431-500) . ........... 62 266 2. Proof Problems (501-523) . ... .. 70 309 3. Loci of Points (524-530) . .... ... 72 322 4. The Greatest and Least Values (531-532)..... ... 72 325 Trigonometry 1. Transforming Expressions Containing Trigonometric Func- tions (533-554) . 74 327 2. Trigonometric Equations and Systems of Equations (555-618) 77 333 3. Inverse Trigonometric Functions (619628) . .. 82 363 4. Trigonometric Inequalities (629-645) . 83 366 5. Miscellaneous Problems (646-658). ... 85 372 PROBLEMSALGEBRA 1. Arithmetic and Geometric Progressions Preliminaries Let a, d and Sn be, respectively, the nth term, the common difference and the sum of the first n terms of an arithmetic pro- gression. Then an=a1+d(n-1) (1) and Sn- 2 2 (2) Ifu,,,q and S are the nth term, the common ratio and the sum of the first n terms of a geometric progression, then an = ulgn-1 (3) and S= unq-ul = u1(qn- 1) n q-1 q-1 (4) Finally,if S isthe sum of an infinite geometric series with I qI < 1then S= . (5) uql 1. Prove that if positive numbers a, b and c form an arithmetic progression then the numbers 1 I 1 1 14V ' Y+a Ya+Yb also form an arithmetic progression. 2. Positive numbers a,, a2,..., anform an arithmetic progres- sion. Prove that I I I +I +Yan-I+Yan_n-l Ya1+Ya2 a2+Ya3 Ya1+Yan 3. Prove that if numbers a az, .,an are different from zero and form an arithmetic progression then alas a,a2 aaa` alan 8 PROBLEMS IN ELEMENTARY MATHEMATICS 4. Prove that any sequence of numbers a a2,... , ansatisfying the condition I I I I _n-1 -{- --j a-f -.. a1a2 a2a3 asaa an-1an alan for every n > 3 is an arithmetic progression. 5. Prove that for every arithmetic progression a a2, as, ars we have the equalities a, - 2a2 -1- a3 = 0, a,-3a2+3a3-a,=0, a,-4a2+6a3-4a4-{-a;=0; and, generally, a,-CRa2-{ Cna3-...(-l)'-' Cn-lars{ (-1)"Cna0 (where n > 2). Hint. Here and in the problem below it is advisable to apply the identity Cn =Cn_1+Cn-f which can be readily verified. 6. Given an arithmetic progression a... , ars, an+ ... prove that the equalities of -Cna2 '+.. + (-l )" Cqa;+1= 0 hold for n > 3. 7. Prove thatifthe numbers log* x,log. x and logn,x (x 1) form an arithmetic progression then 1'12 =_ (kn) 1091, m. 8. Find an arithmetic progression ifitis known that the ratio of the sum of the first n terms to the sum of the kn subsequent terms is independent of n. 9. The numbers x x2r... ,x, form an arithmetic progression. Find this progression if x, -l" x2 -l- ... +x,=a, xf -l- x2 -l-. .-}- xn = b2. Hint. Here and in the problem below use the equality 12+22 +32-{- ... + n2=n(n+1 )(2n+1) 10. The number sequence 1, 4, 10, 19, ... satisfies the condition that the differences of two subsequent terms form an arithmetic progression.Find the nth term and the sum of the first n terms of this sequence. PROBLEMS ALGEBRA 9 11. Consider the table 1 2, 3, 4 3, 4, 5, 6, 7 4, 5, 6, 7, 8, 9, 10 Prove that the sum of the terms in each row is equal to the square of an odd number. 12. Given the terms an,+n = A and a,n_n = B ofa geometric progression al, a2, a,,... ,find an and an (A 0 0). 13. Let Sn be the sum of the first n terms of a geometric pro- gression (S,0, q / 0). Prove that Sn _S2n-Sn Stn-SnS3n-S2n 14. Knowing the sum S. ofthefirst n terms of a geometric progression and the sum Sn of the reciprocals of these terms find the product nn of the first n terms of the progression. 15. Find the sum 1 +2x-i-3x2-j--4x3+... H-(n+ 16. Find the sum 1+11+111... { 111... 1 if the last summand is an n-digit number. 17. Find the sum nx+(n-1)x2+... +2x"-'+1x". 18. Find the sum 22L } 2 ... + 2"l 19. Prove that the numbers 49, 4489, 444889, ... obtained by inserting 48 into the middle of the preceding number are squares of integers. 20. Construct a geometric progression with Iql < 1 whose every term differs from the sum of all subsequent terms by a given constant factor k. For what values of k is the problem solvable? 10 PROBLEMS IN ELEMENTARY MATHEMATICS 21. An infinite number sequence x,, x2, x3,... , xn, ...(x, 0) satisfies the condition (x1 +X21+ ... +xrt_1) (x2 + x3 + ... -+- xn) _ = (x,x2 + x2x3 + ... -r X. _ lxrs)2 for any n 3. Prove that the numbers x1, x2,... ,x7l,...form an infinite geometric progression. Hint. Use the method of complete induction. 22. Given an arithmetic progression with general term a and a geometric progression with general term b,,. Prove that a < b for n > 2ifa, = b1,a2 = b2,a, r a2 and a > 0 for all natural numbers n. 23. Prove thatifthetermsofa geometric progression a,, a2,... ,a,,, ..and of an arithmetic progression b,, b2,... ,b,,,.. satisfy the inequalities a,>0,a2>0,b2-b1>0 then there exists a number a such that the difference is independent of n. 2. Algebraic Equations and Systems of Equations Preliminaries In the problems below the original systems of equations should be simplified and reduced to equivalent systems whose all solu- tions either are known or can readily be found. In some cases itis necessary to introduce redundant equations which are a priori satisfied by the solutions of the original systems but may have, inthe general case, some extraneous solutions. Then the values of the unknowns thus obtained must be tested by substituting them into the original systems. In some problems one should use Vieta's theorem for the equation of the third degree x3+px2--qx+r=0. (1) The theorem establishes the following relations between the coef- ficients p,q and rof the equation and its roots x1, x2 and x3: x,x2 x1x2x3 = - r. (2) X, + x2 + x3 = - p, 1 x2x3 ± x3x1 = q, Formulas (2)are derived by equating the coefficients in the equal powers of x on both sides of the identity x3 -l- px2 -1- qx -}- r (x-x1) (x-x2) (x-x3). PROBLEMS. ALGEBRA 11 24. Find all real solutions of the system of equations x3+y3=1, xzy+2xya+y3=2. J 25. Solve the system of equations x2+xy+y2=4, x+xy+y=2. 26. Find the real solutions of the system of equations x3+y3 =50, x3y + xy2 = a3 J provided a is real and different from zero. 27. Solve the system of equations xz x=12,y2 Y x+y=s1 1 1 28. Solve the system of equations x4 +x2y2 + y4 91 x2-xy+y2=7. 29. Solve the system of equations x3-y3= 19(x-y), x3+y3=7(x+y). 30. Find all real solutions of the system of equations 2(x+y)=5xy, 8(x3+y3)=65. 31. Find the real solutions of the system of equations (x+y)(x2-y2)=9, , (x-y) (x2+y2)=5. 32. Find all real solutions of the system of equations x+y 1, x4 + y4 = 7. 33. Solve the system of equations x+y 1, xb+y6=31. PROBLEMS IN ELEMENTARY MATHEMATICS 34. Find the real solutions of the system of equations x4+y4-x2y2= 13, x2-y2_-2xy=1, satisfying the condition xy>0. 35. Solve the system of equations (x2+1)(y2+1)=10, (x+y)(xy-1)=3. Hint. Put xy=v and x+y=u. 36. Solve the system of equations (x2+y2) x 6, y (x2 - y2) = 1. 37. Solve the system of equationsz x2 + y2 = axy, x4+ y4 = bx2 38. Solve the equation (x+b)2 (a + F ab )x2-b40 by factorizing its left member. 39. Solve the equation x2 48 x 4 3+x2 101 3 x 40. Solve the system of equations x yy+ x y I 7y-}x_yb b 41. Find all the solutions of the equation (x-4.5)4 + (x-5.5)2 = 1. 42. Solve the system of equations jx-1J-fly-51=1, y=5-i--lx=-l1*. * The absolute value ofa number x (denoted asl x I)is the non-negative number determined by the conditions _-x for x < 0, ixl xfor x:_::- 0.
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