The document contains 54 mathematics problems across four categories: Junior problems, Senior problems, Undergraduate problems, and Olympiad problems. The problems cover a range of mathematical topics and involve proving statements, evaluating expressions, finding solutions to equations, and more. Many of the problems were proposed by mathematicians from various countries and institutions.
The document contains 54 mathematics problems across four categories: Junior problems, Senior problems, Undergraduate problems, and Olympiad problems. The problems cover a range of mathematical topics and involve proving statements, evaluating expressions, finding solutions to equations, and more. Many of the problems were proposed by mathematicians from various countries and institutions.
The document contains 54 mathematics problems across four categories: Junior problems, Senior problems, Undergraduate problems, and Olympiad problems. The problems cover a range of mathematical topics and involve proving statements, evaluating expressions, finding solutions to equations, and more. Many of the problems were proposed by mathematicians from various countries and institutions.
The document contains 54 mathematics problems across four categories: Junior problems, Senior problems, Undergraduate problems, and Olympiad problems. The problems cover a range of mathematical topics and involve proving statements, evaluating expressions, finding solutions to equations, and more. Many of the problems were proposed by mathematicians from various countries and institutions.
J349. Prove that for each positive integer n, 6n+1 + 8n+1 + 27n 1 has at least 11 proper positive divisors. Proposed by Titu Andreescu, The University of Texas at Dallas, USA J350. Let a, b, c be positive real numbers such that ab + bc + ca = 1. Prove that p p p a4 + b2 + b4 + c2 + c4 + a2 2. Proposed by Titu Zvonaru, Comnes, ti, Romnia J351. Find the sum of all six-digit positive integers such that if a and b are adjacent digits of such an integer, then |a b| 2. Proposed by Neelabh Deka, India J352. Let ABC be a triangle and let D be a point on side AC such that 13 BCA = 41 ABD = DBC, and AC = BD. Find the angles of triangle ABC. Proposed by Marius Stnean, Zalu, Romnia J353. Let a, b, c be nonnegative real numbers and let 1 1 1 + + , 4a + 1 4b + 1 4c + 1 1 1 1 B= + + , 3a + b + 1 3b + c + 1 3c + a + 1 1 1 1 C= + + . 2a + b + c + 1 2b + c + a + 1 2c + a + b + c A=
Prove that A B C. Proposed by Nguyen Viet Hung, Hanoi, Vietnam J354. Evaluate
X 3n + 1 2n1 2n + 1
n1
Proposed by Cody Johnson, Carnegie Mellon University, USA
Mathematical Reflections 5 (2015)
Senior problems S349. Each face of eight unit cubes is colored in one of the k colors, where k {2, 3, 4, 5, 6, 8, 12, 24}, so that there are 48 k faces of each color. Prove that from these unit cubes, we can assemble a 2 2 cube that has on its surface equal amount of squares of each color. Proposed by Nairi Sedrakyan, Armenia S350. Let a1 , a2 , . . . , a15 be positive integers such that (a1 + 1)(a2 + 1) (a15 + 1) = 2015a1 a2 a15 . Prove that there are at least six and at most ten numbers among a1 , a2 , . . . , a15 that are equal to 1. Proposed by Titu Zvonaru, Comnes, ti and Neculai Stanciu, Buzu, Romnia S351. Let a, b, c be positive real numbers such that abc = 1. Prove that a+b+c
S352. In the triangle ABC, let denote its Brocard angle, and let satisfy the identity tan = tan A + tan B + tan C. Prove that
cos 2A + cos 2B + cos 2C
1 = (cot + 3 cot ) . sin 2A + sin 2B + sin 2C 4 Proposed by Oleg Faynshteyn, Leipzig, Germany
S353. Let a, b, c, x, y be positive real numbers such that xy 1. Prove that
ab bc ca a+b+c + +
xa + yb + 2c xb + yc + 2a xc + ya + 2b x+y+2 Proposed by Yong Xi Wang S354. Find all functions f : R R such that for all real numbers x, y, (f (x + y))2 = (f (x))2 + 2f (xy) + (f (y))2 . Proposed by Oleksiy Klurman, Universit de Montral, Canada
Mathematical Reflections 5 (2015)
Undergraduate problems U349. Let 0 < x, y, z < 1. Prove that 1 1 1 1 1 1 + + + + +
1 x4 1 y 4 1 z 4 1 x2 yz 1 y 2 zx 1 z 2 xy 1 1 1 1 1 1 + + . + + + 1 x3 y 1 xy 3 1 y 3 z 1 yz 3 1 x3 z 1 xz 3 Proposed by Mehtaab Sawhney, Commack High School, New York, USA U350. Let a and b be real numbers such that a 1 and b > a2 a + 1. Prove that the equation x5 ax3 + a2 x b = 0 has a unique real solution x0 , and 2b a3 < x60 < b2 + a a3 . Proposed by Corneliu Mnescu-Avram, Ploes, ti, Romnia U351. Let a 0. Evaluate 1 lim n 2n na
n X n
ka .
k=0
Proposed by Albert Stadler, Herrliberg, Switzerland
U352. Evaluate
X n1
. 2n n=1 n
Proposed by Cody Johnson, Carnegie Mellon, University, USA
U353. Let a R and let f : (1, 1) R be a function differentiable at 0. Evaluate
lim an
n X
f
k . n2
k=1
Proposed by Dorin Andrica, Babes, -Bolyai University, Romnia
U354. Let f , g : [1, 1] R be increasing functions. Prove that if f (x) = f (x), for all x [1, 1], then
Z1
f (x)g(x)dx 0. 1
Proposed by Marcel Chirit, , Bucharest, Romnia
Mathematical Reflections 5 (2015)
Olympiad problems O349. Find all positive integers n such that
n X n
k k=1
is an even integer. Proposed by Dorin Andrica, Babes, -Bolyai University, Romnia O350. Find all triples (x, y, z) of integers satisfying the equation x3 + 3xy + y 3 = 2z + 1. Proposed by Titu Andreescu, The University of Texas at Dallas, USA O351. Let ABC be a triangle with ABC = 60 and BCA = 70 , and let point D lie on side BC. Prove that BAD = 20 if and only if AB + BD = AD + DC. Proposed by Mircea Lascu and Titu Zvonaru, Romnia O352. Solve in positive integers the system of equations
1 2 3
+ + =1 x y z 50yz
x + 2y + 3z = . 8 + yz Proposed by Titu Andreescu, The University of Texas at Dallas, USA O353. Let a, b, c, d be nonnegative real nubers such that a b 1 c d and a + b + c + d = 4. Prove that 4(a2 + b2 + c2 + d2 ) 12 + a3 + b3 + c3 + d3 . Proposed by Marius Stnean, Zalu, Romnia O354. Find all primes p such that 1+
1 2
p2 + +
1 p1
is an integer. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA
