Proposed by Titu Andreescu, University of Texas at Dallas, USA J176.. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania J179... Proposed by Titu Andreescu, Univers
Trang 1Junior problems J175 Let a, b ∈ (0,π2) such that sin2a + cos 2b ≥ 12sec a and sin2b + cos 2a ≥ 12sec b Prove that
cos6a + cos6b ≥ 1
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
x 1 + x1
2 + · · · + x1
1 x 2 ···x n = n3+ 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
r x + y
r y + z
r z + x
4
√ xyz ≤1
4(abc + 5a + 5b + 5c).
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
(2 +√5)n= an+ bn
1 +√5 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)(y3− z3) = 3(z − x)(z3+ x3) (y + z)(z3− x3) = 3(x − y)(x3+ y3) (z + x)(x3− y3) = 3(y − z)(y3+ z3) Proposed by Titu Andreescu, University of Texas at Dallas, USA J180 Let a, b, c, d be distinct real numbers such that
1
3
√
a − b+
1
3
√
b − c +
1
3
√
c − d+
1
3
√
d − a 6= 0.
Prove that √3a − b +√3b − c +√3c − d +√3d − a 6= 0
Trang 2Senior 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+ a1xn−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= KAB1C1, Kb = KBC1A1, Kc= KCA1B1 Prove that KA1B1C1 is a root of the equation
x3+ (Ka+ Kb+ Kc)x2− 4KaKbKc= 0
Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA S177 Prove that in any acute triangle ABC,
sinA
2 + sin
B
2 + sin
C
2 ≥
5R + 2r 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,
(xk+ yk√5)n= Fkn−1+ Fkn1 +
√ 5
2 , where (Fm)m≥1 is the Fibonacci sequence
Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania S179 Find all positive integers a and b for which (aab−12+1)2 is a positive integer
Proposed by Valcho Milchev, Petko Rachov Slaveikov Secondary School, Bulgaria S180 Solve in nonzero real numbers the system of equations
(
x4− y4 = 121x−122y4xy
x4+ 14x2y2+ y4 = 122x+121yx2 +y 2 Proposed by Titu Andreescu, University of Texas at Dallas, USA
Trang 3Undergraduate 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 many primes p such that p divides b
p−1 ai
i − 1 for all i = 1, 2, , n
Proposed by Gabriel Dospinescu, Ecole Normale Superieure, France
U178 Let k be a fixed positive integer and let Sn(j)= nj + n
j+k + n
j+2k + · · · , j = 0, 1, , k − 1 Prove that
Sn(0)+ Sn(1)cos2π
k + · · · + S
(k−1)
n cos2(k − 1)π
k
2
+
Sn(1)sin2π
k + S
(2)
n sin4π
k + · · · + S
(k−1)
n sin2(k − 1)π
k
2
=2 cosπ
k
2n
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
Z 1 0
(a1+ b1x)n1· · · (ak+ bkx)nkdx ≤ (a + b)
n+1− an+1
(n + 1)b . Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania
Trang 4Olympiad problems O175 Find all pairs (x, y) of positive integers such that x3− y3 = 2010(x2+ y2)
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
√
x1+ 2x3
+√ x3
x2+ 2x4
+ · · · +√ x1
xn+ 2x2
≥ √n
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
O180 Let p be a prime Prove that each positive integer n ≥ p, p2 divides n+pp 2− n+2p2p − n+p
2p Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania