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Tham khảo tài liệu 'junior problems - phần 4', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Junior problems J181. Let a b c d be positive real numbers. Prove that a b 3 c d 3 a2 d2 3 b2 c2 3 2 2 a d b c Proposed by Pedro H. O. Pantoja Natal-RN Brazil J182. Circles C1 Opr and C2 O2 R are externally tangent. Tangent lines from O1 to C2 intersect C2 at A and B while tangent lines from O2 to C1 intersect C1 at C and D. Let O1A O2C E and O1B O2D F . Prove that EF O1O2 AD BC. Proposed by Roberto Bosch Cabrera Florida USA J183. Let x y z be real numbers. Prove that x2 y2 z2 2 xyz x y z 3 xy yz zx 2 x2y2 y2z2 z2x2 . Proposed by Neculai Stanciu George Emil Palade Buzau Romania J184. Find all quadruples x y z w of integers satisfying the system of equations x y z w xy yz zx w2 w xyz w3 1. Proposed by Titu Andreescu University of Texas at Dallas USA J185. Let H x y x y be the harmonic mean of the positive real numbers x and y. For n 2 find the greatest constant C such that for any positive real numbers a1 . an b1 . bn the following inequality holds C 1 1 H a1 ---- an b1 ---- bn - H a1 b1 H an bn Proposed by Dorin Andrica Babes-Bolyai University Cluj-Napoca Romania J186. Let ABC be a right triangle with AC 3 and BC 4 and let the median AA1 and the angle bisector BB1 intersect at O. A line through O intersects hypotenuse AB at M and AC at N. Prove that MB NC 4 MA NA - 9. Proposed by Valcho Milchev Kardzhali Bulgaria Mathematical Reflections 1 2011 1 Senior problems 5181. Let a and b be positive real numbers such that a 2b and 2a b p. ab Prove that a b 2. Proposed by Titu Andreescu University of Texas at Dallas USA 5182. Let a b c be real numbers such that a b c. Prove that for each real number x the following inequality holds X x a 4 b c a b b c a c a b 2 b c 2 c a 2 . 6 cyc Proposed by Dorin Andrica Babes-Bolyai University Cluj-Napoca Romania 5183. Let ao 2 0 1 and an an-i ff1 n 1. Prove that for all n n 1 1 n 1 n 2 an ao 2 Proposed by Arkady Alt San Jose California USA 5184. Let Hn 1 1 n n 2. Prove that e -n 2H. Proposed by Tigran Hakobyan Vanadzor Armenia 5185. Let
