Junior problems
J289. Let a be a real number such that 0 ≤ a < 1. Prove that
1
1
a 1+
+1=
.
1−a
1−a
Proposed by Arkady Alt, San Jose, California, USA
J290. Let a, b, c be nonnegative real numbers such that a + b + c = 1. Prove that
p
p
p
3
3
3
13a3 + 14b3 + 13b3 + 14c3 + 13c3 + 14a3 ≥ 3.
Proposed by Titu Andreescu, University of Texas at Dallas, USA
J291. Let ABC be a triangle such that ∠BCA = 2∠ABC and let P be a point in its interior
such that P A = AC and P B = P C. Evaluate the ratio of areas of triangles P AB and
P AC.
Proposed by Panagiote Ligouras, Noci, Italy
J292. Find the least real number k such that for every positive real numbers x, y, z, the following
inequality holds:
Y
(2xy + yz + zx) ≤ k(x + y + z)6 .
cyc
Proposed by Dorin Andrica, Babes-Bolyai University, Romania
J293. Find all positive integers x, y, z such that
(x + y 2 + z 2 )2 − 8xyz = 1.
Proposed by Aaron Doman, University of California, Berkeley , USA
J294. Let a, b, c be nonnegative real numbers such that a + b + c = 3. Prove that
1 ≤ (a2 − a + 1)(b2 − b + 1)(c2 − c + 1) ≤ 7.
Proposed by An Zhen-ping, Xianyang Normal University, China
Mathematical Reflections 1 (2014)
1
Senior problems
S289. Let x, y, z be positive real numbers such that x ≤ 4, y ≤ 9 and x + y + z = 49. Prove
that
1
1
1
√ + √ + √ ≥ 1.
y
x
z
Proposed by Marius Stanean, Zalau, Romania
S290. Prove that there is no integer n for which
1
1
1
+
+ ··· + 2 =
22 32
n
2
4
.
5
Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA
S291. Let a, b, c be nonnegative real numbers such that ab + bc + ca = 3. Prove that
5
(2a2 − 3ab + 2b2 )(2b2 − 3bc + 2c2 )(2c2 − 3ca + 2a2 ) ≥ (a2 + b2 + c2 ) − 4.
3
Proposed by Titu Andreescu, USA and Marius Stanean, Romania
S292. Given triangle ABC, prove that there exists X on the side BC such that the inradii of
triangles AXB and AXC are equal and find a ruler and compass construction.
Proposed by Cosmin Pohoata, Princeton University, USA
S293. Let a, b, c be distinct real numbers and let n be a positive integer. Find all nonzero
complex numbers z such that
az n + bz +
c
a
b
= bz n + cz + = cz n + az + .
z
z
z
Proposed by Titu Andreescu, University of Texas at Dallas, USA
S294. Let s(n) be the sum of digits of n2 + 1. Define the sequence (an )n≥0 by an+1 = s(an ),
with a0 an arbitrary positive integer. Prove that there is n0 such that an+3 = an for all
n ≥ n0 .
Proposed by Roberto Bosch Cabrera, Havana, Cuba
Mathematical Reflections 1 (2014)
2
Undergraduate problems
1
U289. Let a ≥ 1 be such that (ban c) n ∈ Z for all sufficiently large integers n. Prove that a ∈ Z.
Proposed by Mihai Piticari, Campulung Moldovenesc, Romania
U290. Prove that there are infinitely many consecutive triples of primes (pn−1 , pn , pn+1 ) such
that 12 (pn+1 + pn−1 ) ≤ pn .
Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA
U291. Let f : R → R be a bounded function and let S be the set of all increasing maps
ϕ : R → R. Prove that there is a unique function g in S satisfying the conditions:
a) f (x) ≤ g(x), for all x ∈ R.
b) If h ∈ S and f (x) ≤ h(x) for all x ∈ R, then g(x) ≤ h(x) for all x ∈ R.
Proposed by Marius Cavachi, Constanta, Romania
U292. Let r be a positive real number. Evaluate
Z
0
π/2
1
dx.
1 + cotr x
Proposed by Ángel Plaza, Universidad de Las Palmas de Gran Canaria, Spain.
U293. Let f : (0, ∞) → R be a bounded continuous function
P and let α ∈ [0, 1). Suppose there
exist real numbers a0 , . . . , ak , with k ≥ 2, so that kp=0 ap = 0 and
k
X
α
lim x ap f (x + p) = α.
x→∞
p=0
Prove that α = 0.
Proposed by Marcel Chirita, Bucharest, Romania
U294. Let p1 , p2 , . . . , pn be pairwise distinct prime numbers. Prove that
√ √
√
√
√
√
Q( p1 , p2 , . . . , pn ) = Q( p1 + p2 + · · · + pn ).
Proposed by Marius Cavachi, Constanta, Romania
Mathematical Reflections 1 (2014)
3
Olympiad problems
O289. Let a, b, x, y be positive real numbers such that x2 − x + 1 = a2 , y 2 + y + 1 = b2 , and
(2x − 1)(2y + 1) = 2ab + 3. Prove that x + y = ab.
Proposed by Titu Andreescu, University of Texas at Dallas, USA
O290. Let Ω1 and Ω2 be the two circles in the plane of triangle ABC. Let α1 , α2 be the circles
through A that are tangent to both Ω1 and Ω2 . Similarly, define β1 , β2 for B and γ1 , γ2
for C. Let A1 be the second intersection of circles α1 and α2 . Similarly, define B1 and
C1 . Prove that the lines AA1 , BB1 , CC1 are concurrent.
Proposed by Cosmin Pohoata, Princeton University, USA
O291. Let a, b, c be positive real numbers. Prove that
√
b2
c2
a+b+c
a2
+√
+√
≥
.
2
2
2
2
2
2
3
4a + ab + 4b
4b + bc + 4c
4c + ca + 4a
Proposed by Titu Andreescu, University of Texas at Dallas, USA
O292. For each positive integer n let
Tn =
n
X
k=1
1
.
k · 2k
Find all prime numbers p for which
p−2
X
Tk
≡0
k+1
(mod p).
k=1
Proposed by Gabriel Dospinescu, Ecole Normale Superieure, Lyon
O293. Let x, y, z be positive real numbers and let t2 =
xyz
max(x,y,z) .
Prove that
4(x3 + y 3 + z 3 + xyz)2 ≥ (x2 + y 2 + z 2 + t2 )3 .
Proposed by Nairi Sedrakyan, Yerevan, Armenia
O294. Let ABC be a triangle with orthocenter H and let D, E, F be the feet of the altitudes
from A, B and C. Let X, Y , Z be the reflections of D, E, F across EF , F D, and
DE, respecitvely. Prove that the circumcircles of triangles HAX, HBY , HCZ share a
common point, other than H.
Proposed by Cosmin Pohoata, Princeton University, USA
Mathematical Reflections 1 (2014)
4