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Junior problemsJ217. If a, b, c are integers such that a 2 + 2bc = 1 and b 2 + 2ca = 2012, find all possible values of c 2 + 2ab. Proposed by Titu Andreescu, University of Texas at Dallas, USA J218. Prove that in any triangle with sides of lenghts a, b, c, circumradius R, and inradius r, the following inequality holds √ ab a +b −c + √ bc b +c −a + √ ca c +a −b ≤ 1 + R r . Proposed by Cezar Lupu, University of Pittsburgh, USA, and Virgil Nicula, Bucharest, Romania J219. Trying to solve a problem, Jimmy used the following ”formula”: log ab x = log a xlog b x, where a, b, x are positive real numbers different from 1. Prove that this is correct only if x is a solution to the equation log a x + log b x = 1. Proposed by Titu Andreescu, University of Texas at Dallas, USA J220. Find the least prime p for which p = a 2 k +kb 2 k , k = 1, . . . , 5, for some (a k , b k ) in Z ×Z. Proposed by Cosmin Pohoata, Princeton University, USA J221. Solve in integers the system of equations xy − z 3 = xyz + 1 yz − x 3 = xyz −1 zx − y 3 = xyz −9. Proposed by Titu Andreescu, University of Texas at Dallas, USA J222. Give a ruler and straightedge construction of a triangle ABC given its orthocenter and the intersection points of the internal and external angle bisectors of one of its angles with the corresponding opposite side. Proposed by Cosmin Pohoata, Princeton University, USA Mathematical Reflections 1 (2012) 1 Senior problems S217. Find all integer solutions of the equation 2x 2 −y 14 = 1. Proposed by Nairi Sedrakyan, Yerevan, Armenia S218. Let ABC be a triangle with incircle C and incenter I. Let D, E, F be the tangency points of C with the sides BC, CA, and AB, respectively, and furthermore, let S be the intersection of BC and EF. Let P, Q be the intersection points of SI with C such that P, Q lie on the small arcs DE and FD respectively. Prove that the lines AD, BP, CQ are concurrent. Proposed by Marius Stanean, Zalau, Romania S219. Let ABCD be a quadrilateral and let {P} = AC ∩ BD, {E} = AD ∩ BC, and {F} = AB ∩ CD. Denote by isog XY Z (P) the isogonal conjugate of P with respect to triangle XY Z. Prove that isog ABE (P) = isog CDE (P) = isog ADF (P) = isog BCF (P) if and only if AC and BD are perpendicular. Proposed by Cosmin Pohoata, Princeton University, USA S220. Let a, b, c be nonnegative real numbers. Prove that 3 _ a 3 +b 3 +c 3 − 1 2 (ab(a +b) +bc(b +c) +ca(c +a)) ≥ _ a 2 +b 2 +c 2 −ab −bc −ca. Proposed by Mircea Lascu and Marius Stanean, Zalau, Romania S221. Let ABC be a triangle with centroid G and let F be a point that minimizes the quantity PA+PB +PC over all points P lying in the plane of ABC. Prove that FG ≤ min(AG, BG, CG). Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA S222. Solve the equation 3φ(n) = 4τ(n) where φ(n) is the Euler totient function and τ(n) is the number of divisors of n. Proposed by Roberto Bosch Cabrera, Florida, USA Mathematical Reflections 1 (2012) 2 Undergraduate problems U217. Define an increasing sequence (a k ) k∈Z + to be attractive if ∞ k=1 1 a k diverges and ∞ k=1 1 a 2 k converges. Prove that there is an attractive sequence a k such that a k √ k is also an attractive sequence. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA U218. Let ∗ be an associative and ”totally non-commutative” (x = y implies x ∗ y = y ∗ x) binary operation on a set S. Prove that x ∗ y ∗ z = x ∗ z for all x, y, z in S. Proposed by Titu Andreescu, University of Texas at Dallas, USA and Bogdan Enescu, ”B. P. Hasdeu” National College, Buzau, Romania U219. a) Let f : [0, ∞) →R be a convex differentiable function with f(0) = 0. Prove that _ x 0 f(t)dt ≤ x 2 2 f (x) for all x ∈ [0, ∞). b) Find all differentiable functions f : [0, ∞) → R for which we have equality in the above inequality. Proposed by Dorin Andrica, Babes-Bolyai University of Cluj-Napoca, and Mihai Piticari National College ”Dragos Voda” Campulung Moldovenesc, Romania U220. Evaluate lim n→∞ _ (n + 1) n+1 ¸ Γ _ 1 n + 1 _ −n n ¸ Γ _ 1 n _ _ , where Γ denotes the classical Gamma function. Proposed by Cezar Lupu, University of Pittsburgh, USA and Moubinool Omarjee, Lycee Jean Murcat, Paris, France U221. Let f : R → R be a continuous periodic function such that f(2012) = 0. Prove that there exists c > 0 such that for all n ≥ 2012 we have n k=1 |f(k)| k > c · ln n. Proposed by Gabriel Dospinescu, Ecole Polytechnique, Paris, France U222. Let p and q be distinct odd primes and let d be a divisor of q −1. Prove that a) Q[ζ q ] has a unique subfield K d that has degree d over Q, where ζ q denotes a primitive q-th root of unity. b) p splits completely in K d if and only if q splits completely in Q _ d √ p ¸ . Proposed by Cosmin Pohoata, Princeton University, USA Mathematical Reflections 1 (2012) 3 Olympiad problems O217. Equilateral triangles ACB and BDC are drawn on the diagonals of a convex quadri- lateral ABCD so that B and B are on the same side of AC, and C and C are on the same side of BD. Find ∠BAD +∠CAD if B C = AB +CD. Proposed by Nairi Sedrakyan, Yerevan, Armenia O218. Find all integers n such that 2 n + 3 n + 13 3 −14 n is the cube of an integer. Proposed by Titu Andreescu, University of Texas at Dallas, USA O219. Let a, b, c, d be positive real numbers that satisfy 1 −c a + 1 −d b + 1 −a c + 1 −b d ≥ 0. Prove that a(1 −b) +b(1 −c) +c(1 −d) +d(1 −a) ≥ 0. Proposed by Gabriel Dospinescu, Ecole Polytechnique, Paris, France O220. Let A 1 , . . . , A n be distinct points in the plane and let G be their center of gravity. Consider a point F in plane for which A 1 F +· · · +A n F is minimal. Prove that n i=1 A i G ≤ 2 _ 1 − 1 n _ n i=1 A i F. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA O221. Suppose that an ant starts at a vertex of complete graph K 4 and moves on edges with probability 1 3 . Determine the probability that the ant returns to the original vertex within n moves. Proposed by Antonio Blanca Pimentel, UC Berkeley and Roberto Bosch Cabrera, Florida, USA O222. Let a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . b n be nonnegative reals, where n is a positive integer. Let σ be a permutation of {1, 2, . . . , n}. For every k ∈ {1, 2, . . . , n}, let c k = max ({a 1 b k , a 2 b k , . . . , a k b k } ∪ {a k b 1 , a k b 2 , . . . , a k b k }) . Prove that a 1 b σ(1) +a 2 b σ(2) +· · · +a n b σ(n) ≤ c 1 +c 2 +· · · +c n . Proposed by Darij Grinberg, Massachusetts Institute of Technology, USA Mathematical Reflections 1 (2012) 4 Let D. {E} = AD ∩ BC. c be nonnegative real numbers. Proposed by Ivan Borsenco. and AB.Senior problems S217. Prove that 3 a3 + b3 + c3 − 1 (ab(a + b) + bc(b + c) + ca(c + a)) ≥ 2 a2 + b2 + c2 − ab − bc − ca. Romania S219. Armenia S218. CG). and {F } = AB ∩ CD. Prove that the lines AD. and furthermore. b. USA S220. Proposed by Cosmin Pohoata. Proposed by Marius Stanean. Zalau. Yerevan. CA. Prove that F G ≤ min(AG. Let P . Massachusetts Institute of Technology. Florida. Proposed by Nairi Sedrakyan. BG. CQ are concurrent. Let ABCD be a quadrilateral and let {P } = AC ∩ BD. Zalau. Q lie on the small arcs DE and F D respectively. Proposed by Mircea Lascu and Marius Stanean. USA S222. Let ABC be a triangle with incircle C and incenter I. BP . Find all integer solutions of the equation 2x2 − y 14 = 1. E. Let ABC be a triangle with centroid G and let F be a point that minimizes the quantity P A + P B + P C over all points P lying in the plane of ABC. Princeton University. Let a. let S be the intersection of BC and EF . respectively. F be the tangency points of C with the sides BC. Denote by isogXY Z (P ) the isogonal conjugate of P with respect to triangle XY Z. USA Mathematical Reflections 1 (2012) 2 . Solve the equation 3φ(n) = 4τ (n) where φ(n) is the Euler totient function and τ (n) is the number of divisors of n. Proposed by Roberto Bosch Cabrera. Q be the intersection points of SI with C such that P . Romania S221. Prove that isogABE (P ) = isogCDE (P ) = isogADF (P ) = isogBCF (P ) if and only if AC and BD are perpendicular. Proposed by Titu Andreescu. Prove that a) Q [ζq ] has a unique subfield Kd that has degree d over Q. Ecole Polytechnique. USA U218. ∞) → R be a convex differentiable function with f (0) = 0. USA and Bogdan Enescu. Define an increasing sequence (ak )k∈Z+ to be attractive if diverges and ∞ a12 k=1 k √ converges. France U221. √ b) p splits completely in Kd if and only if q splits completely in Q d p . Paris. France U222. z in S. y. Paris.Undergraduate problems U217. Romania U220. USA Mathematical Reflections 1 (2012) 3 . Proposed by Dorin Andrica. where Γ denotes the classical Gamma function. 2 b) Find all differentiable functions f : [0. and Mihai Piticari National College ”Dragos Voda” Campulung Moldovenesc. Romania U219. Evaluate n→∞ lim (n + 1) n+1 Γ 1 n+1 −nn Γ 1 n . Prove that there exists c > 0 such that for all n ≥ 2012 we have n k=1 |f (k)| > c · ln n. Prove that there is an attractive sequence ak such that ak k is also an attractive sequence. Prove that x f (t)dt ≤ 0 x2 f (x) for all x ∈ [0. USA and Moubinool Omarjee. Proposed by Cezar Lupu. Proposed by Cosmin Pohoata. ∞ 1 k=1 ak Proposed by Ivan Borsenco. k Proposed by Gabriel Dospinescu. ”B. Princeton University. ∞). Let p and q be distinct odd primes and let d be a divisor of q − 1. Buzau. Babes-Bolyai University of Cluj-Napoca. a) Let f : [0. ∞) → R for which we have equality in the above inequality. Let f : R → R be a continuous periodic function such that f (2012) = 0. University of Texas at Dallas. Hasdeu” National College. Lycee Jean Murcat. Prove that x ∗ y ∗ z = x ∗ z for all x. University of Pittsburgh. Massachusetts Institute of Technology. P. where ζq denotes a primitive q-th root of unity. Let ∗ be an associative and ”totally non-commutative” (x = y implies x ∗ y = y ∗ x) binary operation on a set S. USA O221. b. n}. France O220. . ak b2 . Let A1 . USA O219. . . 2. For every k ∈ {1. Florida. . . where n is a positive integer. Armenia O218. Find all integers n such that 2n + 3n + 133 − 14n is the cube of an integer. Let a1 . Yerevan. USA O222. USA Mathematical Reflections 1 (2012) 4 . b2 . Prove that a1 bσ(1) + a2 bσ(2) + · · · + an bσ(n) ≤ c1 + c2 + · · · + cn . Massachusetts Institute of Technology. . Consider a point F in plane for which A1 F + · · · + An F is minimal. Proposed by Titu Andreescu. Massachusetts Institute of Technology. . bn be nonnegative reals. ak bk }) . Suppose that an ant starts at a vertex of complete graph K4 and moves on edges with probability 1 . Let a. a2 bk . Determine the probability that the ant returns to the original vertex 3 within n moves. . UC Berkeley and Roberto Bosch Cabrera. an and b1 . a2 . Find ∠BAD + ∠CAD if B C = AB + CD. a b c d Prove that a(1 − b) + b(1 − c) + c(1 − d) + d(1 − a) ≥ 0. University of Texas at Dallas. Proposed by Nairi Sedrakyan. Equilateral triangles ACB and BDC are drawn on the diagonals of a convex quadrilateral ABCD so that B and B are on the same side of AC. . .Olympiad problems O217. . i=1 Proposed by Ivan Borsenco. Proposed by Darij Grinberg. . . n}. Proposed by Gabriel Dospinescu. . let ck = max ({a1 bk . and C and C are on the same side of BD. . . Ecole Polytechnique. . . . 2. d be positive real numbers that satisfy 1−c 1−d 1−a 1−b + + + ≥ 0. c. . ak bk } ∪ {ak b1 . An be distinct points in the plane and let G be their center of gravity. Prove that n Ai G ≤ 2 1 − i=1 1 n n Ai F. Let σ be a permutation of {1. . . Paris. . . . Proposed by Antonio Blanca Pimentel. . . 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