Olympiad Problem Pool Contents Introduction 2 Algebra 2.1 Talent Search Algebra Problems 2.1.1 Equations 2.1.2 Inequalities 2.1.3 Polynomials 2.1.4 Sequences 2.1.5 Functional Equations 2.2 Stellenbosch Level Algebra Problems 2.2.1 Equations 2.2.2 Inequalities 2.2.3 Sequences 2.2.4 Polynomials 2.2.5 Functional Equations 2.3 Olympiad Type Algebra Problems 2.3.1 Equations 2.3.2 Inequalities 2.3.3 Functional Equations 2.3.4 Polynomials 2.3.5 Sequences 2.4 IMO and IMO Proposed Algebra Problems 2.4.1 Equations 2.4.2 Inequalities 2.4.3 Polynomials 2.4.4 Functional Equations 2.4.5 Sequences 2 5 6 6 10 10 11 13 13 15 22 27 30 31 31 32 34 35 37 Combinatorics 3.1 Talent Search Combinatorics Problems 3.2 Stellenbosch Level Combinatorics Problems 3.3 Olympiad Type Combinatorics Problems 3.4 IMO and IMO Proposed Combinatorics Problems 38 38 39 45 57 Geometry 4.1 Talent Search Geometry Problems 4.2 Stellenbosch Level Geometry Problems 4.3 Olympiad Type Geometry Problems 4.4 IMO and IMO Proposed Geometry Problems 63 63 65 70 83 Number Theory 5.1 Talent Search Number Theory Problems 5.1.1 Divisibility 5.1.2 Congruence 5.1.3 Diophantine Equations 5.2 Stellenbosch Level Number Theory Problems 89 89 89 89 89 90 5.3 5.4 5.2.1 Divisibility 5.2.2 Congruence 5.2.3 Diophantine Equations 5.2.4 Combinatorial Number Theory Olympiad Type Number Theory Problems 5.3.1 Divisibility 5.3.2 Congruence 5.3.3 Diophantine Equations 5.3.4 Combinatorial Number Theory IMO and IMO Proposed Number Theory Problems 5.4.1 Divisibility 5.4.2 Congruence 5.4.3 Diophantine Equations 5.4.4 Combinatorial Number Theory 90 92 92 93 94 94 96 97 100 101 101 103 103 104 Introduction The problems in this collection are taken from the training of the South African IMO team for the years 1992 through 2008 (excluding 2007) In this collection we try to mention the original sources, especially if they come from national Olympiads We also try to mention when the problem was used in training, for anyone who is interested in obtaining the solutions Therefore there might be two sources, e.g [Ireland 1993] [TS 1994], as the source This would mean that it was originally a problem in the Irish Mathematical Olympiad in 1993, and was used in the South African Talent Search in training for the 1994 IMO So, also TS 1994 is the talent search in the 1994 booklet, which actually took place in 1993 For the national Olympiads we not write for example Irish Mathematical Olympiad 1993, but merely Ireland 1993 This saves a little space and effort while being less confusing than obscure abbreviations As for the South African reference we use the following abbreviations: • TS - Talent Search • SU - Stellenbosch Test • MP - Monthly Problem Sets • April - April Camp Test • July - July Camp Test Algebra 2.1 2.1.1 Talent Search Algebra Problems Equations √ √ 3x2 − 4x + 34 + √ 3x2 − 4x − 11 = [TS 1994] Solve the equation √ Hint: Set a = 3x − 4x + 34 and b = 3x2 − 4x − 11 = Then a + b = and a2 − b2 = 45 Find all prime numbers p and q for which the equation 5x2 − px + q = has distinct rational roots [TS 1994] At what time between 12h00 and 13h00 are the hands of a clock in a straight line? [TS 1994] Solve √ 2x − + y + 2xy + 6x − y [TS 1995] = = Between 5am and 6am the hands of a clock are at right angles exactly twice How many minutes elapse between these times? [TS 1995] Prove that if a, b, c are positive numbers such that abc = 1, then b c a + + = ab + a + bc + b + ac + c + [TS 1995] The pages of a book are numbered consecutively from to 119 with the odd numbers appearing on the right side when the book is opened Determine the location of a bookmark so that the sum of the page numbers before the bookmark is the same as the sum of the page numbers after the bookmark [Canadian Invitational Challenge 1995] [TS 1996] Real numbers a, b, c satisfy b+c−a c+a−b a+b−c = = , a a b where abc = Determine the value of (a + b)(b + c)(c + a) abc [Canadian Invitational Challenge 1995] [TS 1996] Two candles have equal lengths One burns down in hours, the other in hours They are lit at the same time After how many hours will one be three times the length of the other? [TS 1997] 10 A party of thee hikers who walk at 6km/h and a motorcyclist who travels at 30km/h leave together on a journey from village A to village B, which are 45km apart The motorcyclist can carry one passenger Find the shortest time needed for the whole party to get to B [TS 1996] 11 Prove that if a, b, c are integers such that a + b + c = then 2a4 + 2b4 + 2c4 is a perfect square [TS 1997] 12 If x + x = 3, calculate the value of x3 + x3 [Stel 1997] 13 A wooden cube with sides n units is painted blue all over It is then cut into n3 identical small cubes The number of small cubes with just one face painted blue is equal to the number of small cubes with no faces painted Determine n [TS 1998] 14 Ann walks to school every morning The trip takes 20 minutes On the way to school one morning she realized she had left her homework behind She knew that if she continued walking she would get to school minutes before the bell, but if she went home for her homework she would arrive at school ten minutes late What fraction of the way to school had she walked when she realized she had left her homework behind? [TS 1998] 15 Prove that if a b+c + b c+a + c a+b = 1, then a2 b2 c2 + + = b+c c+a a+b [TS 1998] 16 At what time between o’clock and o’clock is the minute hand of a clock directly above the hour hand? [Stel 1999] 17 A number of small white cubes of the same size are assembled to form a single solid large cube, which is then painted red on some of its sides The large cube is then broken up into small cubes again, and it is found that 168 of the small cubes are painted red on at least one of their sides How many small cubes are still completely white? [Stel 1999] 18 If x3 + y = and x2 + y = 3, determine x + y [TS 2000] 19 Prove that for all positive integers n, the numbers 102n − 10n 102n − 10n + 1, and when rounded to the nearest integer, are not equal [TS 2000] 20 Solve the system of equations x2 + xy + 2y 2 x +y = = 21 Solve: x + y + xy = 19 (1) y + z + yz = 11 (2) z + x + zx = 14 (3) [TS 2002] 22 Evaluate √ 2a √1+x2 , x+ 1+x2 where x = 12 ( a b + b a ), and a and b are positive real numbers [TS 2002] 23 Thirteen numbers x1 , x2 , , x13 are given We know that x4 = 17, x8 = 20, that xi + xi+1 + xi+2 is the same for every value of i and that the sum x1 + x2 + · · · + x13 = 217 Determine the numbers x1 , , x13 [TS 2002] √ √ √ 24 Simplify + + + + 40 − + 20 [TS 2002] 25 Solve for x, y, z in terms of a, b, c: x(y + z) = a2 y(z + x) = b2 z(x + y) = c2 [TS 2002] 26 It is between 10:30 am and 11 am The angle between the hands of a clock is 80◦ What time is it? [TS 2003] 27 If x3 + x3 = 18 and x is real, calculate x4 + x4 [TS 2003] 28 Determine all ordered triples of integers (x, y, z) which satisfy the system: (x + y)(x + y + z) = 90 (4) (y + z)(x + y + z) = 105 (5) (z + x)(x + y + z) = 255 (6) [TS 2003] 29 Four real numbers are in geometric progression Their sum is 13 and the sum of their squares is 1261 Find the numbers [TS 2003] 30 For which values of a does the system of equations x2 − y 2 (x − a) + y = (7) = (8) have 0, 1, 2, 3, or solutions? [TS 2004] 31 A soldier is last in a 100m long line of troops that is marching forward at a steady pace of 2km/h He begins to run toward the front of the line When he gets there, he turns around and runs back to the end of the line He runs at a steady km/h, losing no time in changing direction Compute the total distance he has run [TS 2005] 32 For which values of n is 33 Evaluate √ 26 + 15 + n+ n+ √ n + · · · a positive integer? [TS 2005] √ 26 − 15 [TS 2006] 34 If a + b = c, b + c = d, c + d = a and b is a positive integer, compute the greatest possible value for a + b + c + d [TS 2008] 2.1.2 Inequalities Solve the inequality |x − 1| + |x + 2| < [TS 1994] • Prove that for all positive real numbers a and b, a+b ≥ √ ab When does equality occur? • Prove that for all positive real numbers a, b, c b c a + + ≤ a+b b+c c+a [TS 1997] • Factorize a3 + b3 + c3 − 3abc • Prove that, if a, b, c are positive real numbers, then a3 + b3 + c3 ≥ 3abc When does equality occur? [TS 2001] Prove that, if a + b + c = and a, b, c are positive, then 1 + + ≥ a b c [TS 2004] Prove that if a, b, c are non-zero real numbers such that 2005] 2.1.3 a b + b c + c a = 3, then a = b = c = [TS Polynomials If a, b and c are the roots of the cubic equation x3 + 2x2 − 3x − = 0, find the cubic equation with roots a1 , 1b and 1c [TS 1994] Hint: If x3 + 2x2 − 3x − = (x − a)(x − b)(x − c), then a + b + c = −2, ab + bc + ca = −3 and abc = Then (x − 1/a)(x − 1/b)(x − 1/c) = x3 − (ab + bc + ca)/(abc)x2 + (a + b + c)/(abc)x + 1/(abc) Solve the system of equations x + 2y + 4z = 12 xy + 4yz + 2xz = 22 xyz = [TS 1994] Hint: Multiply out (T − x)(T − 2y)(T − 4z) and solve the resulting cubic equation Suppose a, b, c are the three roots of the equation x3 − 8x2 + 5x + = What is the value of • a + b + c; • a2 + b2 + c2 ; • a3 + b3 + c3 ? [TS 1996] Hint: If x3 + 2x2 − 3x − = (x − a)(x − b)(x − c), then a + b + c = 8, ab + bc + ca = and abc = −7 The other quantities can be expressed in terms of these If abc = and a + b + c = 1996] a + b + 1c , prove that at least one of the numbers a, b, c equals [TS 5 • Prove that the equation x3 + 2x2 − 5x + = has three real irrational roots • If the roots of the equation are a, b, c determine the value of a3 + b3 + c3 [TS 1997] Given that f (x) = 6x4 − (2p2 − 1)x3 − p4 x2 + 12x has the value when x = and that p is a real number, determine all other values of x for which f (x) = [TS 1997] Find the values of a and b so that the polynomial ax3 + bx + is divisible by (x − 1)2 [TS 2002] Solve: x4 − 8x3 + 8x2 + 32x − 44 = [TS 2003] Solve: x4 + x3 − 10x2 + x + = [TS 2003] 2.1.4 Sequences Find √ 1√ 2+ + √ 1√ 3+ + √ 1√ 4+ + ··· + √ √ 100+ 99 [TS 1997] Numbers are arranged in diamonds as shown below 2 , 1, 3 , 4 5 4 , etc What is the sum of the numbers in the n-th diamond? [TS 1997] Four numbers, taken two at a time, give the sums 84, 88, 100, 112, 116 What are the four numbers? [TS 1997] From the set of consecutive integers {1, 2, 3, , n} five integers that form an arithmetic sequence are deleted The sum of the remaining integers is 5000 Determine all value of n for which this is possible and for each acceptable n, determine the number of five-integer sequences possible [Canadian Invitational Challenge] [TS 1997] What is the sum of all the digits of all the numbers from to 100, and 100 included? [TS 2005] The numbers in the following sequence are known as hexagonal numbers: 1, 7, 19, 37, 61, 91, Find a formula for Hn in terms of n and prove that H1 + H2 + · · · + Hn = n3 [TS 2005] What is the 2005-th integer in the sequence: 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, ? [TS 2006] 2.1.5 Functional Equations None at this moment 2.2 2.2.1 Stellenbosch Level Algebra Problems Equations Solve the system of equations 8x2 = 18y + 7xy 2 = 18x − 7x2 y 8y [TS 1994] √ √ √ Solve: 2x2 + x + + x2 + x + = x2 − 3x + 13 [TS 1995] Find the value of x2 + y + z if x + y = and xy − z = [TS 1996] Solve the following system of equations for all possible integral values of a, b, c ab + = c bc + = a ca + = b [Canadian Invitational Challenge 1995] [TS 1996] Find all real solutions of 5x2 − 2xy + 2y − 2x − 2y + = [TS 1997] Solve the following system of equations 6x2 − xy + 4y 2 4x + xy + 4y = 24 (9) = 18 (10) [TS 1997] Solve for x: x+ √ + x2 + x− √ + x2 = [TS 1998] I looked at my watch yesterday and noticed that each hand was exactly on a minute mark and the angle between the hands was 36◦ What time was it? [TS 1998] √ √ √ Solve: x + + 2x − + x − − 2x − = 2 [TS 1998] √ √ 10 What is the smallest integer larger than ( + 2)6 ? [TS 1998] 11 Prove that if m and n are rational numbers, and both m + n and mn are integers, then m and n are also integers [Stel 2000] 12 Prove that, if a is a real number such that a + 1/a is a positive integer, then an + 1/an is a positive integer for all positive integers n [Stel 2001] 13 Four vehicles travelled on a road with constant velocities The car overtook the scooter at 12h00, then met the bike at 14h00 and the motorcycle at 16h00 The motorcycle met the scooter at 17h00 and overtook the bike at 18h00 At what time did the bike and the scooter meet [Stel 2001] 14 If a + b + c = 3, a2 + b2 + c2 = 5, a3 + b3 + c3 = 7, determine a4 + b4 + c4 and a5 + b5 + c5 [Stel 2001] 15 Prove by mathematical induction that × × + × × + × × + · · · + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3) [Stel 2002] 16 If a3 − 3ab2 = 44 and b3 − 3a2 b = 8, find the value of a2 + b2 [Stel 2002] 17 Tom (a cyclist) and Jerry (a runner) start off at the same point and the same time, each at a constant speed, and in the same direction around a circular track Tom cycles faster than Jerry runs When Tom first catches up with Jerry, he turns around and cycles in the opposite direction, meeting Jerry at the point where they first started What is the ratio of their speeds [Stel 2002] = 19 If abc = and a + b + c = 2003] a + 18 Prove that 20 Prove that − + b √ − [Stel 2002] + 1c , prove that at least one of the numbers a, b, c equals [Stel 2n n k! = k=1 k=1 [Stel 2004] (2k + 1)! 2k 21 Let a1 , , an be n real numbers satisfying • a1 ≥ −1 for each i ∈ {1, 2, , n}; • a31 + a32 + · · · + a3n = n Prove that a1 + a2 + · · · + an ≤ n [Arne Smeets] [Stel 2005] 22 Three people can walk at a speed of kilometres per hour They have a car which travels at 50 kilometres per hour that can accommodate any two of them Can they reach a point 62 kilometres away in less than hours? [Sweden 2002] [Stel 2005] 23 Find all real numbers x for which 3x + 4x = 5x [SAMO 1996] [Stel 2006] 24 Find all values of a for which the equation |x − a| + |x + 3a − 8| = has infinitely many solutions [Auckland 2001] [Stel 2006] 2.2.2 Inequalities What is the smallest value that a b c + + b + 2c c + 2a a + 2b can take; a, b and c being positive real numbers [TS 1994] Given two positive real numbers whose sum is less than their product Prove that the sum is greater than [TS 1994] Suppose that ≤ x, y, z ≤ Show that ≤ x2 + y + z − xy − yz − zx ≤ [Wohascum County Problem Book] [MP 1994] Prove that if a2 + b2 + c2 = 1, then (a − b)2 + (b − c)2 + (c − a)2 ≤ [TS 1995] Prove that if a, b, c are positive numbers such that a < b + c then b c a < + 1+a 1+b 1+c [Stel 1995] Prove that if a > b > c, then a2 (b − c) + b2 (c − a) + c2 (a − b) > [TS 1996] Prove that √ 2n − 1 < · · ··· ≤√ 2n 4n + 3n + for all positive integers n [TS 1996] Prove that if a + b + c = 0, then ab + bc + ca ≤ [TS 1996] Prove that if a + b + c = 1, then ab + bc + ca ≤ 13 [TS 1996] 10 Determine a necessary and sufficient condition on the real numbers r1 , r2 , , rn such that x21 + x22 + · · · + x2n ≥ (r1 x1 + r2 x2 + · · · + rn xn )2 for all real numbers x1 , x2 , , xn [Crux 1995] [TS 1996] 11 Prove that (n!)2 > nn for all integers n ≥ [TS 1996] 12 Prove that 13 + 23 + 33 + ··· + m3 < for all m [TS 1997] 13 Find the sum of the lengths of the intervals of values of x which satisfy the inequality 1 + + > x−1 x−2 x−3 [Stel 1997] 14 Let a1 , a2 , , an be positive real numbers and let Sk be the sum of all products of a1 , a2 , , an taken k at a time Show that Sk Sn−k ≥ nk a1 a2 · · · an [APMO 1990 Q2] [Stel 1998] 15 Suppose the sequence a1 , a2 , , a2n+1 satisfies ai−1 − 2ai + ai+1 ≥ for ≤ i ≤ 2n Show that a1 + a2 + · · · + a2n+1 a2 + a4 + · · · + a2n ≥ n+1 n When does equality occur? [Stel 1999] 16 Find the smallest positive integer n such that √ n− √ n−1< 100 [Stel 2000] 17 Let x, y ≥ be real numbers with x + y = Prove that x2 y (x2 + y ) ≤ [July 2001] 18 If a, b, c are the side lengths of a triangle, prove that a b c ≤ + + < 2 b+c c+a a+b [Stel 2002] 19 Prove that √ n+1− √ √ √ n< √ < n− n−1 n for all positive integers n [April 2003] 20 Prove that the integer part of the expression 1 1 + √ + √ + ··· + √ +√ 2 m −1 m2 where m is a positive integer, equals 2m − or 2m − [April 2003] 21 Prove that a b c + + ≤2 bc + ca + ab + if < a, b, c ≤ [Stel 2004] 22 For all real numbers x, y, z, prove that x4 + y + z ≥ √ 8xyz [USSR 1992] [Stel 2005] 23 Given that the equation x4 + px3 + qx2 + rx + s = has four real positive roots, prove that • pr − 16s ≥ 0; • q − 36s ≥ [Paul Vaderlind] [Stel 2006] 24 Let a, b, c be the lengths of the sides of a triangle with a+b+c = Prove that a2 +b2 +c2 +4abc < 12 [Stel 2008] 25 Prove that x2 x21 x2 x2 + + · · · + n−1 + n ≥ x1 + x2 + · · · + xn , x2 x3 xn x1 where x1 , x2 , , xn are positive real numbers [Stel 2008] 26 Prove for any a, b > that 2.2.3 a2 b−1 + b2 a−1 ≥ [Stel 2008] Sequences A sequence of numbers is defined by a1 = 21 and an = 2n−3 2n · an−1 for n ≥ Prove that the sum n k=1 ak < for each positive integer n [IMO Proposed 1988] [SU 1994] The sequence (an ) is formed by the rule a1 = an+1 = 2an + if n is odd; 2an if n is even Find a formula for an (as a function of n) [Stel 1995] Let x1 , x2 , be a sequence of non-zero real numbers such that x2n − xn+1 xn−1 = 1(n = 2, 3, ) Prove that there exists a real number k such that xn+1 = kxn − xn−1 for all n > [TS 1996] Show that there is a unique sequence of positive integers (an ) satisfying the following conditions: a1 = 1, a2 = 2, a4 = 12; an+1 an−1 = a2n ± for n = 2, 3, 4, [Stel 2000] Which natural numbers can be expressed as the sum of one or more consecutive odd numbers? [Stel 2002] n+1 (a1 + a2 + · · · + an−1 ) For The sequence a1 , a2 , is defined by a1 = and for n > 1, an = n−1 example, a2 = and a3 = Determine a2004 [Britain 1997] [Stel 2005] 2.2.4 Polynomials For which positive integers k is the polynomial x2k + (x + 1)2k + divisible by the polynomial x2 + x + 1? [TS 1994] Solve: x4 − 4x3 − 2x2 + 12x + = [TS 1995] Prove that 2x4 + 21x3 − 6x2 − 9x − cannot be factorized into two polynomials with integer coefficients [TS 1995] Find all real solutions of the equation x4 + 4x = [TS 1996] Solve the equation x3 + x2 + x = − 31 [TS 1996] Let f (x) = x2 − |x| + Find all subsets A ⊂ R with the property that if x ∈ A then f (x) ∈ A [Bulgaria 1993] [Stel 1996] Determine for which integers a and n the system of equations x+y+z = xy + yz + zx = a xyz has integer solutions [SAMO 1972] [TS 1997] 10 = 2n ... mention the original sources, especially if they come from national Olympiads We also try to mention when the problem was used in training, for anyone who is interested in obtaining the solutions... was originally a problem in the Irish Mathematical Olympiad in 1993, and was used in the South African Talent Search in training for the 1994 IMO So, also TS 1994 is the talent search in the 1994... 101 101 103 103 104 Introduction The problems in this collection are taken from the training of the South African IMO team for the years 1992 through 2008 (excluding 2007) In this collection we