•
• L
A
T
E
X
ε
• 
•
a, b, c
a
b + c − a
+
b
c + a − b
+
c
a + b − c
≥ 3
d n
a
1
, a
2
, , a
n
s n(n − 1)/2

i<j
|a

i
− a
j
| (n − 1)d ≤ s ≤ n
2
d/4
n > 1
n n

8n/3
√
4n
n > 1000
2
n
1, 2, , n 2n
x
1
< x
2
< < x
n
√
x
2
− x
1
x
2
+

√
x
3
− x
2
x
3
+ +
√
x
n
− x
n−1
x
n
< 1 +
1
2
+
1
3
+ +
1
n
2
S(n) n
S(8n)
S(n)
≥
1

8
k c
k
S(kn)
S(n)
≥ c
k
n c
k
a
1
, a
2
, , a
n
k ≤ n
a
k
a
k
+ a
k−1
2
a
k
+ a
k−1
+ a
k−2
3

a
k
+ + a
2
+ a
1
k
a
1
, a
2
, , a
n
n x
1
, x
2
, , x
n
S
x
1
1 + x
1
x
2
1 + x
1
+ x
2

x
n
1 + x
1
+ + x
n
S
a
1
< a
2
< < a
n
m m ∈ (a
n
) m = a
k
+ a
l
k, l ∈ N a
n
≤ n
2
n ∈ N
a
1
, a
2
, , a
n

a
1
a
2
+ a
3
+ + a
n
+
a
2
a
1
+ a
3
+ + a
n
+ +
a
n
a
1
+ a
2
+ + a
n−1
≥
n
n − 1
x

1
, x
2
, , x
n
x
1
+ x
2
+ + x
k
k
k = 1, 2, , n
x
1
, x
2
, x
3
, x
4
, x
5
(x
1
+ x
2
+ x
3
+ x

4
+ x
5
)
2
≥ 4(x
1
x
2
+ x
2
x
3
+ x
3
x
4
+ x
4
x
5
+ x
5
x
1
)
a
1
, a
2

, , a
n
b
1
, b
2
, , b
n
a
i
< a
j
b
i
≤ b
j
a
i
≤
a
1
+ a
2
+ + a
n
n
< a
j
b
i

≤ b
j
i j
(a
1
+ a
2
+ + a
n
)(b
1
+ b
2
+ + b
n
) ≤ n(a
1
b
1
+ a
2
b
2
+ + a
n
b
n
)
n ≥ 2
x

2
1
+ x
2
2
+ + x
2
n
≥ p(x
1
x
2
+ x
2
x
3
+ + x
n−1
x
n
)
p = 1
p =
4
3
p =
6
5
a, b, c, d, x, y, u, v abcd > 0
(ax+bu)(av+by)(cx+dv)(cu+dy) ≥ ( uvx+bcuxy+advxy+bduvy)(acx+bcu+adv+bdy)

a
1
cosx + a
2
cos2x + + a
n
cosnx ≥ −1 x
a
1
+ a
2
+ + a
n
≤ n
a
1
, a
2
, , a
n
|a
1
| = 1 |a
k+1
| = |a
k
+ 1|
k = 1, 2, , n − 1 |a
1
+ a

2
+ + a
n
|
n = 1975.
n = 1976.
a, b, c a > c, b > c

c(a − c) +

c(b − c) ≤
√
ab
n
n(
n
√
n + 1 − 1) ≤ 1 +
1
2
+
1
3
+ +
1
n
≤ 1 + n(1 −
1
n
√

n
).
x, y, z
(x
2
+ y
2
− z
2
)(x
2
+ z
2
− y
2
)(y
2
+ z
2
− x
2
) ≤ (x + y −z)
2
(x + z −y )
2
(y + z − x)
2
x
n
+ a

1
x
n−1
+ + a
k−1
x
n−k+1
+ a
k+1
x
n−k−1
+ +
a
n−1
x + a
n
= 0 n R a
k+1
a
k−1
< 0
a
0
, a
1
, a
2
, , a
2n
a

k
≥
a
k−1
+ a
k+1
2
k = 1, 2, , 2n−
1
a
1
+ a
3
+ + a
2n−1
n
≥
a
0
+ a
2
+ + a
2n
n + 1
a, b, c, d
a
4
+ b
4
+ c

4
+ d
4
+ 2abcd ≥ a
2
b
2
+ a
2
c
2
+ a
2
d
2
+ b
2
c
2
+ b
2
d
2
+ c
2
d
2
.
a
1

< a
2
< < a
n
< 2n n > 5
min
1≤i<j≤n
lcm[a
i
, a
j
] ≤ 6([
n
2
] + 1)
max
1≤i<j≤n
gcd(a
i
, a
j
) >
38n
147
− c c n
6
38
147
0 < α < π n
sin α +

1
2
sin 2α +
1
3
sin 3α + +
1
n
sin nα > 0
x
1
, x
2
, , x
n
[a, b] 0 < a < b
(x
1
+ x
2
+ + x
n
)(
1
x
1
+
1
x
2

+ +
1
x
n
) ≤
(a + b)
2
n
2
4ab
x
1
, x
2
, , x
n
∈ [0, 1] n ≥ 3
x
1
+ x
2
+ + x
n
− x
1
x
2
− − x
n−1
x

n
− x
n
x
1
≤

n
2

a
n
=
√
n − 1 +
√
n b
n
=
√
4n + 2
[a
n
] = [b
n
]
0 < b
n
− a
n

<
1
16n
√
n
n = 1, 2, 3
ρ(x, y) =
|x − y|

(1 + x
2
)(1 + y
2
)
a, b, c
ρ(a, c) ≤ ρ(a, b) + ρ(b, c)
a, b, c




a
b
+
b
c
+
c
a
−

b
a
−
c
b
−
a
c




< 1
a
1
, a
2
, a
n
b
k
k a
1
, a
2
, , a
n
b
1
=

a
1
+ a
2
+ + a
n
n
,
b
2
=
a
1
a
2
+ a
1
a
3
+ + a
n−1
a
n
n(n − 1)/2
, ,
b
n
= a
1
a

2
a
n
b
1
≥

b
2
b
2
k
≥ b
k−1
b
k+1
k = 2, 3, , n − 1
k

b
k
≥
k+1

b
k+1
, k = 2, 3, , n − 1.
x
1
, x

2
, , x
n
,
x
1
1
+
x
4
2
+ +
x
n
2
n
≤ 1
n
x
1
1
+
x
2
2
+ +
x
n
n
< 2

2
x
1
, x
2
, , x
n
[0; 1]
(x
1
+ x
2
+ + x
n
+ 1)
2
≥ 4(x
2
1
+ x
2
2
+ + x
2
n
)
x n
|cos x| + |cos 2x| + |cos 4x| + + |cos 2
n
x| ≥

n
2
x
n
= 1 +
1
2
+
1
3
+ +
1
n
. γ = lim
m→∞
(x
n
− lnn)
γ < x
m
+ x
n
− x
mn
≤ 1 m, n
x
1
, x
2
, , x

n
x
2
1
+ x
2
2
+ + x
2
n
= 1
2
n
±x
1
± x
2
± ± x
n
+ − 2
n
1
2
≤ a ≤ b

b
2
− a
2
2


2
≥

a
2
+ b
2
2
−
a + b
2
.
n a
1
< a
2
< < a
n
na
1
m
2
+ n
2
m, n
m ≤ 1981, n ≤ 1981 n
2
− mn − m
2

= ±1
a
2
+ b
2
+ c
2
+ 2abc < 2 a, b, c
2
x
1
, x
2
, x
3
x
1
x
2
+ x
3
+
x
2
x
3
+ x
1
+
x

3
x
1
+ x
2
≥
3
2
x
1
, x
2
, , x
n
n ≤ 4
x
1
x
n
+ x
2
+
x
2
x
1
+ x
3
+ +
x

n−1
x
n−2
+ x
n
+
x
n
x
n−1
+ x
1
+
x
3
x
1
+ x
2
≥ 2
n = 4
n > 4
a, b, c
a + b + c ≤
a
2
+ b
2
2c
+

b
2
+ c
2
2a
+
c
2
+ a
2
2b
≤
a
3
bc
+
b
3
ca
+
c
3
ab
σ(n) n
n
σ(n) > 2n
σ(n) > 3n
n
σ(n) < log(2n + 1)
σ(n) < ln(n + 1)

n
1
2
+
1
3
√
2
+
1
4
√
3
+ +
1
(n + 1)
√
n
< 2
a, b, c
a − b
a + b
+
b − c
b + c
+
c − a
c + a
<
1

8
a, b, c
a
2
b(a − b) + b
2
c(b − c) + c
2
a(c − a) ≥ 0
a, b, c
a
3
c + b
3
c + c
3
a ≥ a
2
bc + b
2
ca + c
2
ab
a
0
< a
1
< < a
n−1
< a

n
1
[a
0
, a
1
]
+
1
[a
1
, a
2
]
+ +
1
[a
n−1
, a
n
]
≤ 1 −
1
2
n
[a, b] a, b
a, b, c, d
a
b + c
+

b
c + d
+
c
d + a
+
d
a + b
≥ 2
9 <

3
0
4

x
4
+ 1dx +

3
1
4

x
4
− 1dx < 9, 0001.
a, b, c
a
3
a

2
+ ab + b
2
+
b
3
b
2
+ bc + c
2
+
c
3
c
2
+ ca + a
2
≥
a + b + c
3
.
a, b 2
√
a + 3
3
√
b ≥ 5
5
√
ab

a
4
+ b
4
+ c
4
+ abc(a + b + c) ≥ k(ab + bc + ca)
2
a, b, c
a
1
, a
2
, , a
n
1
a
1
+
2
a
1
+ a
2
+ +
n
a
1
+ a
2

+ + a
n
< 4

1
a
1
+
1
a
2
+ +
1
a
n

n a
1
≥ a
2
≥ ≥ a
n
a
2
1
− a
2
2
+ a
2

3
≥ (a
1
− a
2
+ a
3
)
2
a
2
1
− a
2
2
+ a
2
3
− a
2
4
≥ (a
1
− a
2
+ a
3
− a
4
)

2
a
2
1
−a
2
2
+ + (−1)
n−2
a
2
n−1
+ (−1)
n−1
a
2
n
≥ (a
1
−a
2
+ + (−1)
n−2
a
n−1
+ (−1)
n−1
a
n
)

2
a
1
, a
2
, , a
n
(a
1
+ a
2
+ + a
n
)
2
2(a
2
1
+ a
2
2
+ + a
2
n
)
≤
a
1
a
2

+ a
3
+
a
2
a
3
+ a
4
+ +
a
n−1
a
n
+ a
1
+
a
n
a
1
+ a
2
h
n
= 1 +
1
2
+ +
1

n
1
h
2
1
+
1
2h
2
2
+
1
3h
2
3
+ +
1
nh
2
n
< 2
x
1
, x
2
, , x
n
(1 + x
1
)

1/x
2
(1 + x
2
)
1/x
3
(1 + x
3
)
1/x
4
(1 + x
n
)
1/x
1
≥ 2
n
ax + by + cz +

(a
2
+ b
2
+ c
2
)(x
2
+ y

2
+ z
2
) ≥
2
3
(a + b + c)(x + y + z)
x, y m
(x
2
+ y
2
)
m
≥ 2
m
x
m
y
m
+ (x
m
− y
m
)
2
x
1
, x
2

, , x
n
n > 1
(s − x
1
)
x
1
+ (s − x
2
)
x
2
+ + (s − x
n
)
x
n
> n − 1
s = x
1
+ x
2
+ + x
n
a, b, c
a
bc + 1
+
b

ca + 1
+
c
ab + 1
≤ 2
a
1
, a
2
, , a
n

1
a
2
1
− 1

1
a
2
2
− 1



1
a
2
n

− 1

≥ (n
2
− 1)
n
n a
1
, a
2
, , a
n

a
1
+ a
2
a
3
+

a
2
+ a
3
a
4
+ +

a

n−1
+ a
n
a
1
+

a
n
+ a
1
a
2
≥ n
√
2
x, y x.2
y
+ y.2
x
≥ x + y
x
1
, x
2
, , x
n
∈ [−1, 1] x
3
1

+ x
3
2
+ + x
3
n
= 0
x
1
+ x
2
+ + x
n
≤
n
3
m, n
1
n
√
m + 1
+
1
m
√
n + 1
> 1
a, b, c a + b + c = 1
4a
3

+ 4b
3
+ 4d
3
+ 15abc ≥ 1
a
3
+ b
3
+ c
3
+ abc ≥ min{
1
4
,
1
9
+
d
27
}
a
1
, a
2
, , a
n
, b
1
, , b

n
a
k
b
k
a
k
+ b
k
1 ≤ k ≤ n
AB
A + B
A = a
1
+ a
2
+ +a
n
B = b
1
+ b
2
+ + b
n
0 < x
1
≤ x
2
≤ ≤ x
n

n > 2
x
1
x
2
+
x
2
x
3
+ +
x
n
x
1
≥
x
2
x
1
+
x
3
x
2
+ +
x
1
x
n

x, y, z x + y + z = 0 xyz = 2
x
2
y
+
y
2
z
+
z
2
x
a, b
a + 1
b
+
b + 1
a
(lcm[a, b])
2
≤ a + b
0 < a
1
< a
2
< < a
m
≤ n
a
r

+ a
s
1 ≤ r ≤ s ≤ m n
{a
1
, a
2
, , a
m
}
a
1
+ a
2
+ + a
m
m
≥
n + 1
2
0 ≤ x
1
≤ x
2
≤ ≤ x
n
x
k
2
(x

1
− x
3
) + x
k
3
(x
2
− x
4
) + + x
k
1
(x
n
− x
2
)
k > 1 0 < k < 1
s(n) = 1
1
+ 2
2
+ + n
n
n > 3
3s(n) > (n + 1)
n
2s(n) < (n + 1)
n

1
n
n
>
1
s(n)
+
1
s(n + 1)
+
1
s(n + 3)

a, b, c
1
a
3
(b + c)
+
1
b
3
(c + a)
+
1
c
3
(a + b)
≥
3

2
a, b > 0 abc = 1
1
1 + 2a
+
1
1 + 2b
+
1
1 + 2c
≥ 1
1
1 + a + b
+
1
1 + b + c
+
1
1 + c + a
≤ 1
x, y, z x
2
+ xy + y
2
= 3 y
2
+ yz + z
2
= 16
xy + yz + zx

x
1
, x
2
, , x
n
|x
1
+ x
2
+ x
n
| = 1 |x
k
| ≤
n + 1
2
k = 1, 2, , n y
1
, y
2
, , y
n
x
1
, x
2
, , x
n
|y

1
+ 2y
2
+ + ny
n
| ≤
n + 1
2
a, b, c
a
2
+ 2bc
b
2
+ c
2
+
b
2
+ 2ca
c
2
+ a
2
+
c
2
+ 2ab
a
2

+ b
2
> 3
n
{
√
1} + {
√
2} + + {
√
n
2
} ≤
n
2
− 1
2
.
x, y > 0 x
2
+ y
3
≥ x
3
+ y
4
x
3
+ y
3

≤ 2
p, q, r, x, y, z p + q + r = 1 x
p
y
q
z
r
= 1
p
2
x
2
qy + rz
+
q
2
y
2
px + rz
+
r
2
z
2
px + qy
≥
1
2
a, b, c
a

√
a
2
+ 8bc
+
b
√
b
2
+ 8ac
+
c
√
c
2
+ 8ab
≥ 1
a, b, c

a
b + c
+

b
a + c
+

c
a + b
> 2

n



{
n
1
} − {
n
2
} + {
n
3
} − + (−1)
n
{
n
n
}



<
√
2n
a, b, c
1
a
+
1

b
+
1
c
≥ a + b + c
a + b + c ≥ 3abc.
x
6
y
6
+ y
6
z
6
+ z
6
x
6
+ 3x
4
y
4
z
4
≥ 2(x
3
+ y
3
+ z
3

)x
3
y
3
z
3
x
6
+ y
6
+ z
6
+ 3x
2
y
2
z
2
≥ 2(x
3
y
3
+ y
3
z
3
+ z
3
x
3

).
0 < x <
π
4
(cosx)
cox
2
x
> (sinx)
sin
2
x
(cosx)
cox
4
x
< (sinx)
sin
4
x
n, k ≤ n
(1 +
1
n
)
k
≤
n
2
+ nk + k

2
n
2
p p
2
= 2
n
.3
n
+ 1 m, n
p < 18
x, y x = y x
n
+
1
x
m
= y
n
+
1
y
m
m, n
x
2
+ y
2
>
n+m


16
9
a, b, c 1
1
1 − a
+
1
1 − b
+
1
1 − c
≥
2
1 + a
+
2
1 + b
+
2
1 + c
a, b, c a(b
2
+c
2
) = 2b
2
c 2b ≤ c+a
√
a

0 ≤ x ≤ y ≤ 1
2

(1 − x
2
)(1 − y
2
) ≤ 2(1 − x)(1 − y) + 1
a, p, q ap + 1 q aq + 1
p 2a(p + q) > pq.
x
1
, x
2
, , x
n
2
x
1
+ x
2
+ + x
n
≤
√
1 + x
1
x
2
+

√
1 + x
2
x
3
+ +
√
1 + x
n
x
1
≤ x
1
+ + x
n
+ n.
x
1
, x
2
, , x
n
max{x
1
, x
2
, , x
n
, −x
1

− x
2
− − x
n
} ≥
|x
1
| + |x
2
| + + |x
n
|
2n − 1
x
1
≤ x
2
≤ ≤ x
n
≤ y
1
≤ y
2
≤ ≤ y
n
(x
1
+ x
2
+ + x

n
+ y
1
+ y
2
+ + y
n
)
2
≥ 4n(x
1
y
1
+ x
2
y
2
+ + x
n
y
n
)
a, b, c a + b + c = 1
1
a
+
1
b
+
1

c
≥
25
1 + 48abc
tan
sin x
+ cot
cos x
≥ 2, 0 < x <
π
2
a, b, c
a
b + c − a
+
b
c + a − b
+
c
a + b − c
≥ 3
a = y + z, b = z + x, c = x + y x, y, z > 0
y + z
x
+
z + x
y
+
x + y
z

≥ 6
y + z
x
+
z + x
y
+
x + y
z
≥ 6
6

y
x
×
z
x
×
z
y
×
x
y
×
x
z
×
y
z
= 6

x = y = z a = b = c
∇
d n
a
1
, a
2
, , a
n
s n(n −1)/2

i<j
|a
i
− a
j
|
(n − 1)d ≤ s ≤ n
2
d/4
a
1
< a
2
< < a
n
d
k
= a
k+1

− a
k
d = a
n
− a
1
= d
1
+ d
2
+ + d
n−1
|a
j
− a
i
| = d
i
+ d
i+1
+ + d
j−1
s =

i<j
|a
i
− a
j
| =

n−1

k=1
k(n − k)d
k
k(n−k) ≥ n−1 k(n−k) ≤
n
2
4
k = 1, 2, , n−1
∇
n > 1 n
n

8n/3
√
4n
p
n
=
1 3.5 (2n−1)
2.4.6 (2n)
p
2
n
=
1
2
3
2

2.4
5
2
4.6

(2n − 1)
2
(2n − 2)2n
.
1
2n
k > 1
(2k−1)
2
(2k−2)2k
=
(2k−1)
2
(2k−1)
2
−1
> 1 p
2
n
≥
1
4n
p
2
n

=
3
2
2
.
3.5
4
2
.
5.7
6
2

(2n − 3)(2n − 1)
(2n − 2)
2
.
2n − 1
(2n)
2
p
2
n
≤
3
8n
(2k−1)(2k+1)
(2k)
2
=

(2k)
2
−1
(2k)
2
< 1
∇
a, b, c
a
2
+ 2bc
b
2
+ c
2
+
b
2
+ 2ca
c
2
+ a
2
+
c
2
+ 2ab
a
2
+ b

2
> 3
a, b, c a+b−c, b+c−a, c+a−b > 0
LHS −RHS =

cyc
a
2
− (b − c)
2
b
2
+ c
2
=

cyc
(a − b + c)(a + b − c)
b
2
+ c
2
> 0
∇
p, q, r, x, y, z p + q + r = 1 x
p
y
q
z
r

= 1
p
2
x
2
qy + rz
+
q
2
y
2
px + rz
+
r
2
z
2
px + qy
≥
1
2
p
2
x
2
qy + rz
+
q
2
y

2
px + rz
+
r
2
z
2
px + qy
≥
(px + qy + rz)
2
2(px + qy + zx)
=
px + qy + rz
2
px + qy + rz ≥ x
p
y
q
z
r
= 1
p = q = r =
1
3
x = y = z = 1
∇
a, b, c
a
√

a
2
+ 8bc
+
b
√
b
2
+ 8ac
+
c
√
c
2
+ 8ab
≥ 1
∇
a, b, c

a
b + c
+

b
a + c
+

c
a + b
> 2

∇
n



{
n
1
} − {
n
2
} + {
n
3
} − + (−1)
n
{
n
n
}



<
√
2n
a, b, c
1
a
+

1
b
+
1
c
≥ a + b + c
a + b + c ≥ 3abc.
(a + b + c)
2
≥ 3abc

1
a
+
1
b
+
1
c

≥ 3abc(a + b + c)
3abc

1
a
+
1
b
+
1

c

= 3(ab + bc + ca) ≤ (a + b + c)
2
a = b = c = 1
∇
x
6
y
6
+ y
6
z
6
+ z
6
x
6
+ 3x
4
y
4
z
4
≥ 2(x
3
+ y
3
+ z
3

)x
3
y
3
z
3
x
6
+ y
6
+ z
6
+ 3x
2
y
2
z
2
≥ 2(x
3
y
3
+ y
3
z
3
+ z
3
x
3

).
xy = a, yz = b, zx = c
x
6
+ y
6
+ z
6
+ 3x
2
y
2
z
2
≥ x
2
y
2
(x
2
+ y
2
) + y
2
z
2
(y
2
+ z
2

) + z
2
x
2
(z
2
+ x
2
)
x
2
y
2
(x
2
+ y
2
) +y
2
z
2
(y
2
+ z
2
) +z
2
x
2
(z

2
+ x
2
) −2(x
3
y
3
+ y
3
z
3
+ z
3
x
3
) =

cyc
x
2
y
2
(x − y)
2
≥ 0
x = y = z
∇
0 < x <
π
4

(cos x)
cos x
2
x
> (sin x)
sin
2
x
(cos x)
cos x
4
x
< (sin x)
sin
4
x
f(y) = cos
y
x − sin
y
x y ≥ 0 0 < x <
π
4
f(0) = 0 f(y) > 0 y > 0 lim
y→∞
f(y) = 0
f

(y) = cos
y

x ln(cos x) − sin
y
x ln(sin x) = cos
y
x(ln(cos x) − tan
y
x ln(sin x))
f

(y) y > 0 g(y) = tan
y
x
f(2) = f(2)(cos
2
x + sin
2
x) = f(4) f

(2) > 0 f

(4) < 0
f

(2) > 0 f

(4) < 0
cos
2
x ln(cos x) > sin
2

x ln(sin x)
cos
4
x ln(cos x) < sin
4
x ln(sin x)
∇
n, k ≤ n
(1 +
1
n
)
k
≤
n
2
+ nk + k
2
n
2
k = 1
k

1 +
1
n

k+1
≤


1 +
k
n
+
k
2
n
2

1 +
1
n

= 1 +
k + 1
n
+
k
2
+ k
n
2
+
k
2
n
3
< 1 +
k + 1
n

+
(k + 1)
2
n
2
∇
x, y x = y x
n
+
1
x
m
= y
n
+
1
y
m
m, n
x
2
+ y
2
>
n+m

16
9
x = a sin t y = a cos t a > 0 t ∈ (0,
π

4
) ∪(
π
2
)
a
n+m
sin
m
t cos
m
t.
sin
n
t − cos
n
t
sin
m
t − cos
m
t
= 1 (∗)
t ∈ (
π
4
,
π
2
) sin

k+2
t − cos
k+2
t ≤
sin
k
−cos
k
t k ≥ 2
sin
n
t − cos
n
t <
3
2
(sin t − cos t) (∗∗)
sin
k
t cos
k
t
sin
k
t − cos
k
t
≤
sin
k−2

t cos
k−2
t
sin
k−2
t − cos
k−2
t
k > 2
sin
2
t cos
2
t
sin
2
t − cos
2
t
<
sin t cos t
sin t − cos t
sin
m
t cos
m
t
sin
m
t − cos

m
t
≤
sin t cos t
sin t − cos t
(∗ ∗ ∗)
(∗∗) (∗ ∗ ∗)
sin
m
t cos
m
t.
sin
n
t − cos
n
t
sin
m
t − cos
m
t
<
3
2
sin t cos t <
3
4
(∗) x
2

+ y
2
= a
2
∇
a, b, c 1
1
1 − a
+
1
1 − b
+
1
1 − c
≥
2
1 + a
+
2
1 + b
+
2
1 + c
1
x
+
1
y
≥
4

x + y
∀x, y > 0
1
1 − a
+
1
1 − b
≥
4
2 − a − b
=
4
1 + c
1
1 − b
+
1
1 − c
≥
4
1 + a
,
1
1 − c
+
1
1 − a
≥
4
1 + b

a = b = c =
1
3
∇
a, b, c a(b
2
+ c
2
) = 2b
2
c 2b ≤ c + a
√
a
t =
c
b
c = bt a =
2bt
1+t
2
c = bt =
a(1+t
2
)
2
t =
m
n
gcd(m, n) = 1 a(m
2

+n
2
) = 2bmn a(m
2
+n
2
) = 2cn
2
gcd(m
2
+ n
2
, m) = 1 gcd(m
2
+ n
2
, n) = 1 a mn
2
a = kmn
2
2b = kn(m
2
+ n
2
) 2c = km(m
2
+ n
2
)
2b = km.mn + kn

3
≤ km.
m
2
+ n
2
2
+ k
3/2
m
3/2
n
3
= c + a
3/2
∇
0 ≤ x ≤ y ≤ 1
2

(1 − x
2
)(1 − y
2
) ≤ 2(1 − x)(1 − y) + 1
0 ≤ x ≤ y ≤ 1 x = sin a, y = sin b a, b ∈

0,
π
2


2 cos a cos b ≤ 2(1 − sin a)(1 − sin b) + 1 ⇐⇒ (cos a − cos b)
2
+ (sin a + sin b − 1)
2
≥ 0
x = y =
1
2
2

(1 − x
2
)(1 − y
2
) ≤ (1 − x
2
) + (1 − y
2
) = 2 − x
2
− y
2
2 − x
2
− y
2
≤ 2(1 − x)(1 − y) + 1 ⇐⇒ (x + y −1)
2
≥ 0
∇

a, p, q ap + 1 q aq + 1
p 2a(p + q) > pq.
d = gcd(p, q) d | ap + 1 d | ap d | ap + 1 −ap = 1
p q ap + aq + 1 p q
pq a(p + q) ≥ pq − 1 a(p + q) > 1 2a(p + q) > pq
∇
x
1
, x
2
, , x
n
2
x
1
+ x
2
+ + x
n
≤
√
1 + x
1
x
2
+
√
1 + x
2
x

3
+ +
√
1 + x
n
x
1
≤ x
1
+ + x
n
+ n.
|x
i
− x
j
| ≤ 2 ∀i, j ∈ {1 , ··· , n}
x
i
+ x
j
≤ 2

1 + x
i
x
j
≤ x
i
+ x

j
+ 2
(x
i
+ x
j
)
2
≤ 4(1 + x
i
x
j
) ≤ (x
i
+ x
j
+ 2)
2
⇐⇒ (x
i
− x
j
)
2
≤ 4 (x
i
− x
j
)
2

+ 4(x
i
+ x
j
) ≥ 0
∇
x
1
, x
2
, , x
n
max{x
1
, x
2
, , x
n
, −x
1
− x
2
− − x
n
} ≥
|x
1
| + |x
2
| + + |x

n
|
2n − 1
x
1
, x
2
, , x
n
x
1
, x
2
, , x
k
x
k+1
, , x
n
1 ≤ k ≤ n
S = |x
1
| + |x
2
| + + |x
n
| = −x
1
− x
2

− − x
k
+ x
k+1
+ + x
n
S
2n−1
> max{x
k+1
, , x
n
}
S
2n − 1
> −x
1
− x
2
− − x
n
= S − 2(x
k+1
+ + x
n
)
S
2n − 1
> S − 2(n − k)
S

2n − 1
S > (2k − 1)S
∇
x
1
≤ x
2
≤ ≤ x
n
≤ y
1
≤ y
2
≤ ≤ y
n
(x
1
+ x
2
+ + x
n
+ y
1
+ y
2
+ + y
n
)
2
≥ 4n(x

1
y
1
+ x
2
y
2
+ + x
n
y
n
)
P
i
(t) = t
2
− (x
i
+ y
i
)t + x
i
y
i
= (t − x
i
)(t − y
i
), i = 1, 2, , n
t ∈ [x

i
, y
i
] P
i
(t) ≤ 0
F (t) =
n

i=1
P
i
(t) = nt
2
− (x
1
+ + x
n
+ y
1
+ + y
n
)t + (x
1
y
1
+ + x
n
y
n

) = 0
F (t
0
) ≤ 0 t
0
=
x
n
+y
1
2
F (t) = 0 ∆ = (x
1
+ + x
n
+ y
1
+ + y
n
)
2
−4n(x
1
y
1
+ + x
n
y
n
)

∇
a, b, c a + b + c = 1
1
a
+
1
b
+
1
c
≥
25
1 + 48abc
1
a
+
1
b
+
1
c
+ 48 (ab + bc + ca) ≥ 25
a + b ≤
1
3
√
3
a + b >
1
3

√
3
, b + c >
1
3
√
3
, c + a >
1
3
√
3
2 >
3
3
√
3
f (a, b, c) =
1
a
+
1
b
+
1
c
+ 48 (ab + bc + ca)
f (a, b, c) ≥ f

a + b

2
,
a + b
2
, c

≥ 25
1
12
≥ ab(a + b)
a + b ≤
1
3
√
3
a + b
2
= l (z = 1 − 2l)
144l
2
− 168l
3
+ 73l
2
− 14l + 1 ≥ 0 ⇔ (3l −1)
2
(4l − 1)
2
∇
tan

sin x
+ cot
cos x
≥ 2, 0 < x <
π
2
sin x = a, cos x = b ⇒ 0 < a, b < 1

a
b

a
+

b
a

b
≥ 2 ⇐⇒
a
a+b
+ b
a+b
a
b
b
a
≥ 2
a
a+b

+ b
a+b
≥ 2

(ab)
a+b
a
a−b
2
˙
b
b−a
2
≥ 1 ⇐⇒

a
b

a−b
≥ 1
• a = b

a
b

a−b
= 1
• a > b
a
b

> 1, a − b > 0

a
b

a−b
> 1
• a < b
a
b
< 1, a − b < 0

a
b

a−b
> 1
∇