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Junior problems - Phần 3 ppt
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Junior problems
J175. Let a, b ∈ (0,
π
2
) such that sin2 a + cos 2b ≥
1
2
sec a and sin2
b + cos 2a ≥
1
2
sec b. Prove that
cos6
a + cos6
b ≥
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
x1
+
1
x2
+ · · · +
1
xn
+
1
x1x2···xn
= n
3 + 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
2
+
r
y + z
2
+
r
z + x
2
+
√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)(y
3 − z
3
) = 3(z − x)(z
3 + x
3
)
(y + z)(z
3 − x
3
) = 3(x − y)(x
3 + y
3
)
(z + x)(x
3 − 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
√3
a − b
+
1
√3
b − c
+
1
√3
c − d
+
1
√3
d − a
6= 0.
Prove that √3
a − b +
√3
b − c +
√3
c − d +
√3
d − a 6= 0.
Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania
Mathematical Reflections 6 (2010) 1