Cac bai toan Olympiad Philippin nam 2007-2008

J.M. Basilla E.P. Bautista I.J.L. Garces J.A. Marasigan A.R.L. Valdez The PMO 2007-2008 Problems and Solutions of the Tenth Philippine Mathematical Olympiad DOST-SEI • MSP • HSBC • VPHI Preface and Introduction This booklet contains the questions and answers in the Tenth Philippine Mathematical Olympiad, which was held during School Year 2007-2008. First held in 1984, the PMO was created as a venue for high school stu- dents with interest and talent in mathematics to come together in the spirit of friendly competition and sportsmanship. Its aims are: (1) to awaken greater interest in and promote the appreciation of mathematics among students and teachers; (2) to identify mathematically-gifted students and motivate them towards the development of their mathematical skills; (3) to provide a vehicle for the professional growth of teachers; and (4) to encourage the involvement of both public and private sectors in the promotion and devel- opment of mathematics education in the Philippines. The PMO is the first part of the selection process leading to participation in the International Mathematical Olympiad (IMO). It is followed by the Mathematical Olympiad Summer Camp (MOSC), a five-phase program for the twenty national finalists of PMO. The four selection tests given during the second phase of MOSC determine the tentative Philippine Team to the IMO. The final team is determined after the third phase of MOSC. The PMO is a continuing project of the Department of Science and Tech- nology - Science Education Institute (DOST-SEI), and is being implemented by the Mathematical Society of the Philippines (MSP). Though great effort was put in checking and editing the contents of this booklet, some errors may have slipped from the eyes of the reviewers. Should you find some errors, it would be greatly appreciated if these are reported to one of the authors at the following e-mail address: garces@math.admu.edu.ph Quezon City The Authors 20 June 2008 The Problems Area Stage 24 November 2007 1. Simplify:  2 −1 + 3 −1 2 −1 − 3 −1  −1 . 2. If 2A99561 is equal to the product when 3 ×(523 + A) is multiplied by itself, find the digit A. 3. The perimeter of a square inscribed in a circle is p. What is the area of the square that circumscribes the circle? 4. The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence? 5. It is given that CAB ∼ = EFD. If AC = x + y + z, AB = z + 6, BC = x+8z, EF = 3, DF = 2y −z, and DE = y + 2, find x 2 +y 2 +z 2 . 6. Container A contained a mixture that is 40% acid, while container B contained a mixture that is 60% acid. A chemist took some amount from each container, and mixed them. To produce 100 liters of mixture that is 17% acid, she needed to pour 70 liters of pure water to the mixture she got from containers A and B. How many liters did she take from container A? 7. If a and b are integers such that a log 250 2 + b log 250 5 = 3, what is the value of a + 2b? 8. Find all real values of x satisfying the inequality   1 2 −x + 1  2 ≥ 2. 2 9. Find the polynomial of least degree, having integral coefficients and leading coefficient equal to 1, with √ 3 − √ 2 as a zero. 10. Let x = cos θ. Express cos 3θ in terms of x. 11. Solve the system of equations:  x + y + √ xy = 28 x 2 + y 2 + xy = 336. 12. Let P be a point on the diagonal AC of the square ABCD. If AP is one-fourth of the length of one side of the square and the area of the quadrilateral ABP D is 1 square unit, find the area of ABCD. 13. A circle is inscribed in ABC with sides AB = 4, BC = 6, and AC = 8. If P and Q are the respective points of tangency of AB and AC with the circle, determine the length of chord P Q. 14. If 3 √ x + 5 − 3 √ x −5 = 1, find x 2 . 15. Let a, b, and c be real constants such that x 2 + x + 2 is a factor of ax 3 + bx 2 + cx + 5, and 2x −1 is a factor of ax 3 + bx 2 + cx − 25 16 . Find a + b + c. 16. Consider the function f defined by f(x) = 1 + 2 x . Find the roots of the equation (f ◦ f ◦ ···◦ f    10 times )(x) = x, where “◦” denotes composition of functions. 3 17. How many ordered pairs (x, y) of positive integers, where x < y, satisfy the equation 1 x + 1 y = 1 2007 . 18. Let ABC be an equilateral triangle. Let −→ AB be extended to a point D such that B is the midpoint of AD. A variable point E is taken on the same plane such that DE = AB. If the distance between C and E is as large as possible, what is ∠BED? 19. For what values of k does the equation |x −2007| + |x + 2007| = k have (−∞, −2007) ∪ (2007, +∞) as its solution set? 20. Find the sum of the maximum and minimum values of 1 1 + (2 cos x − 4 sin x) 2 . 21. Let k be a positive integer. A positive integer n is said to be a k-flip if the digits of n are reversed in order when it is multiplied by k. For example, 1089 is a 9-flip because 1089 ×9 = 9801, and 21978 is a 4-flip because 21978 ×4 = 87912. Explain why there is no 7-flip integer. 22. Let ABC be an acute-angled triangle. Let D and E be points on BC and AC, respectively, such that AD ⊥ BC and BE ⊥ AC. Let P be the point where −−→ AD meets the semicircle constructed outwardly on BC, and Q the point where −−→ BE meets the semicircle constructed outwardly on AC. Prove that P C = QC. 23. Two friends, Marco and Ian, are talking about their ages. Ian says, “My age is a zero of a polynomial with integer coefficients.” 4 Having seen the polynomial p(x) Ian was talking about, Marco ex- claims, “You mean, you are seven years old? Oops, sorry I miscalcu- lated! p(7) = 77 and not zero.” “Yes, I am older than that,” Ian’s agreeing reply. Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85. Ian sighs, “I am even older than that number.” Determine Ian’s age. National Stage Oral Competition 12 January 2008 15-Second Round 15.1. If        wxy = 10 wyz = 5 wxz = 45 xyz = 12 what is w + y? 15.2. Simplify: (x − 1) 4 + 4(x − 1) 3 + 6(x − 1) 2 + 4(x − 1) + 1. 15.3. By how much does the sum of the first 15 positive odd integers exceed the sum of the first 10 positive even integers? 15.4. Solve for x: 16 1/8 + x 1/4 = 23 5 − √ 2 . 5 15.5. The area of a trapezoid is three times that of an equilateral triangle. If the heights of the trapezoid and the triangle are both equal to 8 √ 3, what is the length of the median of the trapezoid? 15.6. If 1 2 sin 2 x + C = − 1 4 cos 2x is an identity, what is the value of C? 15.7. If ABCDEF is a regular hexagon with each side of length 6 units, what is the area of ACE? 15.8. Find the smallest positive integer x such that the sum of x, x+3, x+6, x + 9, and x + 12 is a perfect cube. 15.9. The length of one side of the square ABCD is 4 units. A circle is drawn tangent to BC and passing through the vertices A and D. Find the area of the circle. 15.10. If f(x + y) = f (x) ·f(y) for all positive integers x, y and f(1) = 2, find f(2007). 15.11. It is given that ABC ∼ DEF . If the area of ABC is 3 2 times that of DEF and AB = BC = AC = 2, what is the perimeter of DEF ? 15.12. For which real numbers x does the inequality 2 log x  a + b 2  ≤ log x a + log x b hold for all positive numbers a and b? 15.13. In Figure 1, what part of ABC is shaded? 15.14. In how many ways can the letters of the word SPECIAL be permuted if the vowels are to appear in alphabetical order? 15.15. Graph theory’s Four-Color Theorem says that four colors are enough to color the regions in a plane so that no two adjacent regions receive the 6 2 1 3 1 2 A B C Figure 1: Problem 15.13. same color. The theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken, 124 years after the Four-Color Problem was posed. Fermat’s Last Theorem in Number Theory was proved by Andrew Wiles in 1995, after 358 years of attempts by generations of mathe- maticians. In 2003, Grigori Perelman completed the proof of a conjecture in topol- ogy. Considered as one of the seven millennium prize problems, the conjecture says that the sphere is the only type of bounded three- dimensional surface that contains no holes. Mathematicians worked on this conjecture for almost a century. What is the name of this conjec- ture that earned Perelman the Fields Medal which he refused to accept in 2006? 30-Second Round 30.1. What is the least 6-digit natural number that is divisible by 198? 30.2. Given that x + 2 and x −3 are factors of p(x) = ax 3 + ax 2 + bx + 12, what is the remainder when p(x) is divided by x − 1? 30.3. The graphs of x 2 +y = 12 and x+y = 12 intersect at two points. What is the distance between these points? 7 30.4. In an arithmetic sequence, the third, fifth and eleventh terms are dis- tinct and form a geometric sequence. If the fourth term of the arith- metic sequence is 6, what is its 2007th term? 30.5. Let each of the characters A, B, C, D, E denote a single digit, and ABCDE4 and 4ABCDE represent six-digit numbers. If 4 ×ABCDE4 = 4ABCDE, what is C? 30.6. Let ABC be an isosceles triangle with AB = AC. Let D and E be the feet of the perpendiculars from B and C to AC and AB, respectively. Suppose that CE and BD intersect at point H. If EH = 1 and AD = 4, find DE. 30.7. Find the number of real roots of the equation 4 cos(2007a) = 2007a. 30.8. In ABC, ∠A = 15 ◦ and BC = 4. What is the radius of the circle circumscribing ABC? 30.9. Find the largest three-digit number such that the number minus the sum of its digits is a perfect square. 30.10. The integer x is the least among three positive integers whose product is 2160. Find the largest possible value of x. 60-Second Round 60.1. Three distinct diameters are drawn on a unit circle such that chords are drawn as shown in Figure 2. If the length of one chord is √ 2 units and the other two chords are of equal lengths, what is the common length of these chords? 8 [...]... A should be 1 This will imply two contradicting statements: (1) Z ≥ 7 and (2) the product 7Z should have a units digit of 1, making Z equal to 3 22 (This problem is taken from the British Mathematical Olympiad 2005.) Refer to Figure 3 By the Pythagorean Theorem, we have QC 2 = EQ2 + EC 2 On the other hand, since AQC is right-angled at Q and QE ⊥ AC, we have EQ2 = AE · EC It follows that QC 2 = EQ2 . Bautista I.J.L. Garces J.A. Marasigan A.R.L. Valdez The PMO 2007-2008 Problems and Solutions of the Tenth Philippine Mathematical Olympiad DOST-SEI • MSP • HSBC • VPHI Preface and Introduction This. Introduction This booklet contains the questions and answers in the Tenth Philippine Mathematical Olympiad, which was held during School Year 2007-2008. First held in 1984, the PMO was created as a venue. education in the Philippines. The PMO is the first part of the selection process leading to participation in the International Mathematical Olympiad (IMO). It is followed by the Mathematical Olympiad