Junior problems - Phần 3

Chia sẻ: Nguyễn Tất Thu | Ngày: | Loại File: PDF | Số trang:4

Thêm vào BST

Báo xấu

104
lượt xem 13
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Tham khảo tài liệu 'junior problems - phần 3', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Junior problems - Phần 3

Junior problems J175. Let a, b ∈ (0, π ) such that sin2 a + cos 2b ≥ 1 1 sec a and sin2 b + cos 2a ≥ sec b. Prove that 2 2 2 1 cos6 a + cos6 b ≥ . 2 Proposed by Titu Andreescu, University of Texas at Dallas, USA J176. Solve in positive real numbers the system of equations x1 + x2 + · · · + xn = 1 1 1 1 1 3 x1 + x2 + · · · + xn + x1 x2 ···xn = n + 1. Proposed by Neculai Stanciu, George Emil Palade Secondary School, Buzau, Romania J177. Let x, y, z be nonnegative real numbers such that ax + by + cz ≤ 3abc for some positive real numbers a, b, c. Prove that z+x √ x+y y+z 1 + 4 xyz ≤ (abc + 5a + 5b + 5c). + + 2 2 2 4 Proposed by Titu Andreescu, University of Texas at Dallas, USA J178. Find the sequences of integers (an )n≥0 and (bn )n≥0 such that √ √n 1+ 5 (2 + 5) = an + bn 2 for each n ≥ 0. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania J179. Solve in real numbers the system of equations  (x + y )(y 3 − z 3 ) = 3(z − x)(z 3 + x3 )  (y + z )(z 3 − x3 ) = 3(x − y )(x3 + y 3 )  (z + x)(x3 − y 3 ) = 3(y − z )(y 3 + z 3 )  Proposed by Titu Andreescu, University of Texas at Dallas, USA J180. Let a, b, c, d be distinct real numbers such that 1 1 1 1 √ +√ +√ +√ = 0. 3 3 3 3 a−b b−c c−d d−a √ √ √ √ Prove that 3 a − b + 3 b − c + 3 c − d + 3 d − a = 0. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania 1 Mathematical Reflections 6 (2010)
Senior problems S175. Let p be a prime. Find all integers a1 , . . . , an such that a1 + · · · + an = p2 − p and all solutions to the equation pxn + a1 xn−1 + · · · + an = 0 are nonzero integers. Proposed by Titu Andreescu, University of Texas at Dallas, USA and Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania S176. Let ABC be a triangle and let AA1 , BB1 , CC1 be cevians intersecting at P . Denote by Ka = KAB1 C1 , Kb = KBC1 A1 , Kc = KCA1 B1 . Prove that KA1 B1 C1 is a root of the equation x3 + (Ka + Kb + Kc )x2 − 4Ka Kb Kc = 0. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA S177. Prove that in any acute triangle ABC, A B C 5R + 2r + sin + sin ≥ sin . 2 2 2 4R Proposed by Titu Andreescu, University of Texas at Dallas, USA S178. Prove that there are sequences (xk )k≥1 and (yk )k≥1 of positive rational numbers such that for all positive integers n and k , √ √n 1+ 5 (xk + yk 5) = Fkn−1 + Fkn , 2 where (Fm )m≥1 is the Fibonacci sequence. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania (a2 +1)2 S179. Find all positive integers a and b for which is a positive integer. ab−1 Proposed by Valcho Milchev, Petko Rachov Slaveikov Secondary School, Bulgaria S180. Solve in nonzero real numbers the system of equations 121x−122y x4 − y 4 = 4xy 14x2 y 2 + y 4 = 122x+121y . x4 + x2 +y 2 Proposed by Titu Andreescu, University of Texas at Dallas, USA 2 Mathematical Reflections 6 (2010)
Undergraduate problems U175. What is the maximum number of points of intersection that can appear after drawing in a plane l lines, c circles, and e ellipses? Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania U176. In the space, consider the set of points (a, b, c) where a, b, c ∈ {0, 1, 2}. Find the maximum number of non-collinear points contained in the set. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA U177. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be integers greater than 1. Prove that there are inﬁnitely p−1 ai − 1 for all i = 1, 2, . . . , n. many primes p such that p divides bi Proposed by Gabriel Dospinescu, Ecole Normale Superieure, France (j ) n n n + · · · , j = 0, 1, . . . , k − 1. U178. Let k be a ﬁxed positive integer and let Sn = + + j j +k j +2k Prove that 2 2(k − 1)π 2π + · · · + Snk−1) cos (0) (1) ( Sn + Sn cos k k 2 2(k − 1)π 2π 4π π 2n + · · · + Snk−1) sin (1) (2) ( + Sn sin + Sn sin = 2 cos . k k k k Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania U179. Let f : [0, ∞] → R be a continuous function such that f (0) = 0 and f (2x) ≤ f (x) + x for all x ≥ 0. Prove that f (x) < x for all x ∈ [0, ∞]. Proposed by Samin Riasat, University of Dhaka, Bangladesh U180. Let a1 , . . . , ak , b1 , . . . , bk , n1 , . . . , nk be positive real numbers and a = a1 + · · · + ak , b = b1 + · · · + bk , n = n1 + · · · + nk , k ≥ 2. Prove that 1 (a + b)n+1 − an+1 (a1 + b1 x)n1 · · · (ak + bk x)nk dx ≤ . (n + 1)b 0 Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania 3 Mathematical Reflections 6 (2010)
Olympiad problems O175. Find all pairs (x, y ) of positive integers such that x3 − y 3 = 2010(x2 + y 2 ). Proposed by Titu Andreescu, University of Texas at Dallas, USA O176. Let P (n) be the following statement: for all positive real numbers x1 , x2 , . . . , xn such that x1 + x2 + · · · + xn = n, x2 x3 x1 n √ +√ + ··· + √ ≥√ . x1 + 2 x3 x2 + 2 x4 xn + 2 x2 3 Prove that P (n) is true for n ≤ 4 and false for n ≥ 9. Proposed by Gabriel Dospinescu, Ecole Normale Superieure, France O177. Let P be point situated in the interior of a circle. Two variable perpendicular lines through P intersect the circle at A and B . Find the locus of the midpoint of the segment AB . Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania O178. Let m and n be positive integers. Prove that for each odd positive integer b there are inﬁnitely many primes p such that pn ≡ 1 (mod b)m implies bm−1 | n. Proposed by Vahagn Aslanyan, Yerevan, Armenia O179. Prove that any convex quadrilateral can be dissected into n ≥ 6 cyclic quadrilaterals. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania n+p 2 n+2p n+p O180. Let p be a prime. Prove that each positive integer n ≥ p, p2 divides − − . p 2p 2p Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania 4 Mathematical Reflections 6 (2010)