116 problems
TRANSCRIPT
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116 Problems in Algebra
Author: Mohammad Jafari
Copyright Β©2011 by Mobtakeran Publications. All rights reserved.
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Contents
Function Equation Problemsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3
Inequality Problemsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.10
Polynomial Problemsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦16
Other Problemsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦19
Solution to the Problemsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..22
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Function Equation Problems
1) Find all functions π:π β π such that:
π(2010π(π) + 1389) = 1 + 1389 + β―+ 13892010 + π (βπ β β).
(Proposed by Mohammad Jafari)
2) Find all functions π:β β β such that for all real numbers π₯,π¦:
π(π₯ + π¦) = π(π₯). π(π¦) + π₯π¦
(Proposed by Mohammad Jafari)
3) Find all functions π:π β {1} β π such that:
π(π₯π¦) = π(π₯)π(π) + π₯π¦ (βπ₯,π¦ β β β {1})
(Proposed by Mohammad Jafari)
4) Find all functions π:π β π such that:
π(π₯) = 2ποΏ½π(π₯)οΏ½ (βπ₯ β β€)
(Proposed by Mohammad Jafari)
5) Find all functions π,π:β€ β β€ such that:
π(π₯) = 3ποΏ½π(π₯)οΏ½ (βπ₯ β β€)
(Proposed by Mohammad Jafari)
6) Find all functions π:β€ β β€ such that:
7π(π₯) = 3ποΏ½π(π₯)οΏ½ + 2π₯ (βπ₯ β β€)
(Proposed by Mohammad Jafari)
7) Find all functions π:β β β such that:
ποΏ½π₯ + π¦ + π(π₯ + π¦)οΏ½ = 2π(π₯) + 2π(π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
4
8) For all functions π:β β β such that:
π(π₯ + π(π₯) + π¦) = π₯ + π(π₯) + 2π(π¦) (βπ₯,π¦ β β) Prove that π(π₯) is a bijective function.
(Proposed by Mohammad Jafari)
9) Find all functions π:β β β such that:
π(π₯ + π(π₯) + 2π¦) = π₯ + ποΏ½π(π₯)οΏ½ + 2π(π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
10) Find all functions π:β β β such that:
π(π₯ + π(π₯) + 2π¦) = π₯ + π(π₯) + 2π(π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
11) For all π:β β β such that : π(π₯ + π(π₯) + 2π¦) = π₯ + π(π₯) + 2π(π¦) (βπ₯,π¦ β β)
Prove that π(0) = 0.
(Proposed by Mohammad Jafari)
12) Find all functions π:β+ β β+ such that, for all real numbers π₯ > π¦ > 0 :
π(π₯ β π¦) = π(π₯) β π(π₯). π οΏ½1π₯οΏ½ .π¦
(Proposed by Mohammad Jafari)
13) Find all functions π:β β β such that:
π οΏ½ποΏ½π₯ + π(π¦)οΏ½οΏ½ = π₯ + π(π¦) + π(π₯ + π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
14) Find all functions π:β+β{0} β β+β{0} such that:
π(ποΏ½π₯ + π(π¦)οΏ½) = 2π₯ + π(π₯ + π¦) (βπ₯,π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
5
15) Find all functions π:β β β such that:
π(π₯ + π(π₯) + 2π(π¦)) = π₯ + π(π₯) + π¦ + π(π¦) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
16) Find all functions π:β β β such that:
ποΏ½2π₯ + 2π(π¦)οΏ½ = π₯ + π(π₯) + π¦ + π(π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
17) Find all functions π:β β β such that:
ποΏ½π(π₯) + 2π(π¦)οΏ½ = π(π₯) + π¦ + π(π¦) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
18) Find all functions π:β β β such that: i) ποΏ½π₯2 + π(π¦)οΏ½ = π(π₯)2 + π(π¦) (βπ₯, π¦ β β) ii) π(π₯) + π(βπ₯) = 0 (βπ₯ β β+) iii) The number of the elements of the set { π₯ β£β£ π(π₯) = 0, π₯ β β } is finite.
(Proposed by Mohammad Jafari)
19) For all injective functions π:β β β such that: ποΏ½π₯ + π(π₯)οΏ½ = 2π₯ (βπ₯ β β) Prove that π(π₯) + π₯ is bijective.
(Proposed by Mohammad Jafari)
20) Find all functions π:β β β such that:
ποΏ½π₯ + π(π₯) + 2π(π¦)οΏ½ = 2π₯ + π¦ + π(π¦) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
21) For all functions π,π, β:β β β such that π is injective and β is bijective satisfying ποΏ½π(π₯)οΏ½ =
β(π₯) (βπ₯ β β) , prove that π(π₯) is bijective function.
(Proposed by Mohammad Jafari)
6
22) Find all functions π:β β β such that:
π(2π₯ + 2π(π¦)) = π₯ + π(π₯) + 2π¦ (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
23) Find all functions π:β+ βͺ {0} β β+ such that:
π οΏ½π₯+π(π₯)2
+ π¦οΏ½ = π(π₯) + π¦ (βπ₯, π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
24) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that:
π οΏ½π₯+π(π₯)2
+ π(π¦)οΏ½ = π(π₯) + π¦ (βπ₯, π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
25) For all functions π:β+ βͺ {0} β β such that : i) π(π₯ + π¦) = π(π₯) + π(π¦) (βπ₯, π¦ β β+ βͺ {0}) ii) The number of the elements of the set οΏ½ π₯ β£β£ π(π₯) = 0, π₯ β β+ βͺ {0} οΏ½ is finite.
Prove that π is injective function.
(Proposed by Mohammad Jafari)
26) Find all functions π:β+ βͺ {0} β β such that:
i) π(π₯ + π(π₯) + 2π¦) = π(2π₯) + 2π(π¦) (βπ₯, π¦ β β+ βͺ {0}) ii) The number of the elements of the set οΏ½ π₯ β£β£ π(π₯) = 0, π₯ β β+ βͺ {0} οΏ½ is finite.
(Proposed by Mohammad Jafari)
27) Find all functions π:β β βsuch that:
i) π(π(π₯) + π¦) = π₯ + π(π¦) (βπ₯, π¦ β β) ii) βπ₯ β β+; βπ¦ β β+π π’πβ π‘βππ‘ π(π¦) = π₯
(Proposed by Mohammad Jafari)
7
28) Find all functions π:β{0} β β such that: i) π(π(π₯) + π¦) = π₯ + π(π¦) (βπ₯, π¦ β β) ii) The set {π₯ β£ π(π₯) = βπ₯, π₯ β β} has a finite number of elements.
(Proposed by Mohammad Jafari)
29) Find all functions π:β β β such that:
ποΏ½ποΏ½π(π₯)οΏ½ + π(π¦) + π§οΏ½ = π₯ + π(π¦) + ποΏ½π(π§)οΏ½ (βπ₯, π¦, π§ β β)
(Proposed by Mohammad Jafari)
30) Find all functions π:β β β such that:
π οΏ½π₯+π(π₯)2
+ π¦ + π(2π§)οΏ½ = 2π₯ β π(π₯) + π(π¦) + 2π(π§) (βπ₯, π¦, π§ β β)
(Proposed by Mohammad Jafari)
31) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that:
π οΏ½π₯+π(π₯)2
+ π¦ + π(2π§)οΏ½ = 2π₯ β π(π₯) + π(π¦) + 2π(π§) (βπ₯, π¦, π§ β β+β{0})
(Proposed by Mohammad Jafari)
32) (IRAN TST 2010) Find all non-decreasing functions π:β+ βͺ {0} β β+ βͺ {0} such that:
π οΏ½π₯+π(π₯)2
+ π¦οΏ½ = 2π₯ β π(π₯) + π(π(π¦)) (βπ₯, π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
33) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that:
π(π₯ + π(π₯) + 2π¦) = 2π₯ + π(2π(π¦)) (βπ₯, π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
34) Find all functions π:β β β such that:
π(π₯ + π(π₯) + 2π¦) = 2π₯ + 2π(π(π¦)) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
8
35) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that:
π οΏ½π₯+π(π₯)2
+ π¦ + π(2π§)οΏ½ = 2π₯ β π(π₯) + πΉ(π(π¦)) + 2π(π§) (βπ₯, π¦, π§ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
36) Find all functions π:β β β such that:
ποΏ½π₯ β π(π¦)οΏ½ = π(π¦)2 β 2π₯π(π¦) + π(π₯) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
37) Find all functions π:β β β such that:
(π₯ β π¦)οΏ½π(π₯) + π(π¦)οΏ½ = (π₯ + π¦)οΏ½π(π₯) β π(π¦)οΏ½ (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
38) Find all functions π:β β β such that:
π(π₯ β π¦)(π₯ + π¦) = (π₯ β π¦)(π(π₯) + π(π¦)) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
39) Find all functions π:β β β such that:
π(π₯ β π¦)(π₯ + π¦) = π(π₯ + π¦)(π₯ β π¦) (βπ₯,π¦ β β)
(Proposed by Mohammad Jafari)
40) Find all non-decreasing functions π,π:β+ βͺ {0} β β+ βͺ {0} such that: π(π₯) = 2π₯ β π(π₯) prove that π and π are continues functions.
(Proposed by Mohammad Jafari)
41) Find all functions π: { π₯ β£ π₯ β β , π₯ > 1 } β β such that :
π(π₯)2. π(π₯2)2 + π(2π₯). π οΏ½π₯2
2οΏ½ = 1 βπ₯ β {π₯ β£ π₯ β β, π₯ > 1}
(Proposed by Mohammad Jafari)
9
42) (IRAN TST 2011) Find all bijective functions π:β β β such that:
ποΏ½π₯ + π(π₯) + 2π(π¦)οΏ½ = π(2π₯) + π(2π¦) (βπ₯, π¦ β β)
(Proposed by Mohammad Jafari)
43) Find all functions π:β+ β β+ such that:
π(π₯ + π(π₯) + π¦) = π(2π₯) + π(π¦) (βπ₯, π¦ β β+)
(Proposed by Mohammad Jafari)
44) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that:
ποΏ½π₯ + π(π₯) + 2π(π¦)οΏ½ = 2π(π) + π¦ + π(π¦) (βπ₯, π¦ β β+ βͺ {0})
(Proposed by Mohammad Jafari)
45) Find all functions π:β+ βͺ {0} β β+ βͺ {0} such that: i) ποΏ½π₯ + π(π₯) + π(2π¦)οΏ½ = 2π(π₯) + π¦ + π(π¦) (βπ₯, π¦ β β+ βͺ {0}) ii) π(0) = 0
(Proposed by Mohammad Jafari)
46) Find all functions π:β+ β β+ such that: π(π₯ + π¦π + π(π¦)) = π(π₯) (βπ₯, π¦ β β+ , π β β , π β₯ 2)
(Proposed by Mohammad Jafari)
47) Find all functions π:β β β such that:
π(π β 1) + π(π + 1) < 2π(π) (βπ β β, π β₯ 2)
(Proposed by Mohammad Jafari)
48) Find all functions π: {π΄ β£ π΄ β β,π΄ β₯ 1} β β such that:
π(π₯π¦2) = π(4π₯). π(π¦) +π(8π₯)π(2π¦)
(Proposed by Mohammad Jafari)
10
Inequality Problems:
49) For all positive real numbers π₯,π¦, π§ such that π₯ + π¦ + π§ = 2 prove that : π₯
π₯4 + π¦ + π§ + 1+
π¦π¦4 + π§ + π₯ + 1
+π§
π§4 + π₯ + π¦ + 1β€ 1
(Proposed by Mohammad Jafari)
50) For all positive real numbers π, π, π such that π + π + π = 6 prove that :
οΏ½ οΏ½π
(π + π)(π + π)π
β¦3
βππππ ,π β β, π β₯ 3
(Proposed by Mohammad Jafari)
51) For all real numbers π, π, π β (2,4) prove that: 2
π + π2 + π3+
2π + π2 + π3
+2
π + π2 + π3<
3π + π + π
(Proposed by Mohammad Jafari)
52) For all positive real numbers π, π, π prove that: π
π4 + π2 + 1+
ππ4 + π2 + 1
+π
π4 + π2 + 1<
43
(Proposed by Mohammad Jafari)
53) For all real positive numbers π, π, π such that1 + 1π+π+π
< 1βπ2+π2+π2
+ 1βππ+ππ+ππ
prove that:
ππ + ππ + ππ < π + π + π
(Proposed by Mohammad Jafari)
54) For all real numbers π, π, π such that π β₯ π β₯ 0 β₯ π and π + π + π < 0 prove that : π3 + π3 + π3 + ππ2 + ππ2 + ππ2 β€ 2π2π + 2π2π + 2π2π
(Proposed by Mohammad Jafari)
55) For all real numbers 0 < π₯1 < π₯2 < β― < π₯1390 < π2 prove that :
π ππ3π₯1 + πππ 3π₯2+β¦+π ππ3π₯1389 + πππ 3π₯1390 < 695
(Proposed by Mohammad Jafari)
11
56) For all πΌπ β οΏ½0, π2οΏ½ (π = 1,2, β¦ ,π) prove that :
π2β₯ min {οΏ½π ππ3πΌπ . πππ 3πΌπ+1,οΏ½π ππ3πΌπ+1. πππ 3πΌπ
π
π=1
}π
π=1
(πΌπ+1 = πΌ1)
(Proposed by Mohammad Jafari)
57) For all real numbers π, π, π β [2,3] prove that: 1
ππ(2π β π) +1
ππ(2π β π) +1
ππ(2π β π) β₯19
(Proposed by Mohammad Jafari)
58) For all real numbers π, π, π β [1,2] prove that: 2
ππ(3π β π)+
2ππ(3π β π)
+2
ππ(3π β π)β€ 3
(Proposed by Mohammad Jafari)
59) For all real positive numbers π, π, π prove that: π
3π + π + π+
π3π + π + π
+π
3π + π + πβ€
35
(Proposed by Mohammad Jafari)
60) For all positive real numbers such that πππ = 1 prove that: π
βπ2 + 3+
πβπ2 + 3
+π
βπ2 + 3β₯
32
(Proposed by Mohammad Jafari)
61) For all positive real numbers π, π, π such that π + π + π = 1 prove that:
(ππ
βπ + π+
ππβπ + π
+ππ
βπ + π)2 + (
ππβπ + π
+ππ
βπ + π+
ππβπ + π
)2 β€13
(Proposed by Mohammad Jafari)
62) For all positive real numbers π, π, π such that π + π + π = 1 prove that: 1
βπ + π+
1βπ + π
+1
βπ + πβ€
1β2πππ
(Proposed by Mohammad Jafari)
12
63) For all positive real numbers π, π, π such that ππ + ππ + ππ = 23 prove that:
1βπ + π
+1
βπ + π+
1βπ + π
β€1
βπππ
(Proposed by Mohammad Jafari)
64) For all positive real numbers π, π, π such that π + π + π = 3prove that: 1
π + π + π2+
1π + π + π2
+1
π + π + π2β€ 1
(Proposed by Mohammad Jafari)
65) For all positive real numbers π, π, π prove that: π4 + π2 + 1π6 + π3 + 1
+π4 + π2 + 1π6 + π3 + 1
+π4 + π2 + 1π6 + π3 + 1
β€3
π2 + π + 1+
3π2 + π + 1
+3
π2 + π + 1
(Proposed by Mohammad Jafari)
66) For all positive real numbers π, π, π prove that: π4 + π2 + 1π6 + π3 + 1
+π4 + π2 + 1π6 + π3 + 1
+π4 + π2 + 1π6 + π3 + 1
β€ 4
(Proposed by Mohammad Jafari)
67) For all positive real numbers π, π, π prove that: 1 + π2 + π4
π + π2 + π3+
1 + π2 + π4
π + π2 + π3+
1 + π2 + π4
π + π2 + π3β₯ 3
(Proposed by Mohammad Jafari)
68) For all real numbers π, π, π β (1,2) prove that: 4
π + π + πβ₯
11 + π + π2
+1
1 + π + π2+
11 + π + π2
(Proposed by Mohammad Jafari)
69) For all positive real numbers π₯,π¦, π§ such that π₯ + π¦ + π§ + 1π₯
+ 1π¦
+ 1π§
= 10 prove that:
(π₯2 + 1)2
4π₯+
(π¦2 + 1)2
4π¦+
(π§2 + 1)2
4π§β₯ π₯ + π¦ + π§
(Proposed by Mohammad Jafari)
13
70) For all positive real numbers π₯,π¦, π§ such that π₯ + π¦ + π§ + 1π₯
+ 1π¦
+ 1π§
= 10 prove that:
4οΏ½π₯ + π¦ + π§ β₯ π₯ + π¦ + π§ + 3
(Proposed by Mohammad Jafari)
71) For all positive real numbers π, π, π prove that:
(οΏ½π2
π + π)(οΏ½
π(π + π)2
) β₯98
(Proposed by Mohammad Jafari)
72) For all positive real numbers π, π, π prove that:
(οΏ½π
(π + π)2)(οΏ½
(π + π)2
π) β₯ (οΏ½
2ππ + π
)2
(Proposed by Mohammad Jafari)
73) For all positive real numbers π, π, π prove that:
(οΏ½1
π2 + π2)(
π3
π2 + π2) β₯ (οΏ½
ππ2 + π2
)(οΏ½π2
π2 + π2)
(Proposed by Mohammad Jafari)
74) For all positive numbers π, π, π prove that :
2(οΏ½π
π + π)(οΏ½ππ2) β₯ (οΏ½π) (οΏ½ππ)
(Proposed by Mohammad Jafari)
75) For all π₯π β β (π = 1,2, β¦ ,π) such that π₯π β π₯π ,prove that: 1π₯112
+2
π₯112 + π₯2
22+ β―+
ππ₯112 + β―+ π₯π
π2β€π(π + 3)
2
(Proposed by Mohammad Jafari)
76) For all π β β (π β₯ 3) prove that :
ποΏ½(πβ1)!οΏ½1
πβ1 + ποΏ½(πβ2)!οΏ½1
πβ2 + β―+ π(1)11 < π
π2 + π
πβ12 + β―+ π1
(Proposed by Mohammad Jafari)
14
77) For all positive real numbers π, π, π such that π + π + π = 1 prove that:
οΏ½1
(π + 2π)(π + 2π)β₯ 3
(Proposed by Mohammad Jafari)
78) For all positive real numbers π, π, πsuch that π + π + π = 2 prove that:
οΏ½1
(π3 + π + 1)(1 + π + π3)β€ 1
(Proposed by Mohammad Jafari)
79) For all positive real numbers π, π, π such that βπ + βπ + βπ = 1π
+ 1π
+ 1π prove that:
min {π
1 + π2,
π1 + π2
,π
1 + π2} β€
12
(Proposed by Mohammad Jafari)
80) For all positive real numbers π, π, π such that 1ππ
+ 1ππ
+ 1ππ
= 3ππππππ (π,π β β) prove
that:
min οΏ½ππ
1 + ππ,ππ
1 + ππ,ππ
1 + πποΏ½ β€
12
(π, π β β)
(Proposed by Mohammad Jafari)
81) For all positive real numbers π₯,π¦ prove that:
οΏ½π₯π
π₯πβ1 + π₯πβ2π¦ + β―+ π¦πβ1β₯π₯ + π¦ + π§
π (π β β)
(Proposed by Mohammad Jafari)
82) For all positive real numbers π₯,π¦, π§ prove that:
οΏ½π₯2π
οΏ½π₯20 + π¦20οΏ½οΏ½π₯21 + π¦21οΏ½β¦ (π₯2πβ1 + π¦2πβ1)β₯π₯ + π¦ + π§
2π
(Proposed by Mohammad Jafari)
83) For all positive real numbers π₯,π¦, π§ such that π₯π₯2+4
+ π¦π¦2+4
+ π§π§2+4
= 15 prove that:
π₯π₯ + 6
+π¦
π¦ + 6+
π§π§ + 6
< 2.4
(Proposed by Mohammad Jafari)
15
84) Find minimum real number π such that for all real numbers π, π, π :
οΏ½οΏ½2(π2 + 1)(π2 + 1) + π β₯ 2οΏ½π + οΏ½ππ
(Proposed by Mohammad Jafari)
85) (IRAN TST 2011) Find minimum real number π such that for all real numbers π, π, π,π :
οΏ½οΏ½(π2 + 1)(π2 + 1)(π2 + 1) + π β₯ 2(ππ + ππ + ππ + ππ + ππ + ππ)
(Proposed by Dr.Amir Jafari and Mohammad Jafari)
86) For all positive real numbers π, π, π prove that:
1 +βππβπ2
β€οΏ½(π + π)2 + π2
2π2 + 2π2 + π2β€ 2 +
βππβπ2
(Proposed by Mohammad Jafari)
87) For all real numbers 1 β€ π, π, π prove that:
οΏ½(π + π)2 + π2π + 2π + π
β€οΏ½π
(Proposed by Mohammad Jafari)
16
Polynomial Problems:
88) Find all polynomials π(π₯) and π(π₯) with real coefficients such that:
οΏ½ [π(π₯ β π).π(π₯ β π β 1). (π₯ β π β 3)]2010
π=0
β₯ π(π₯). π(π₯ β 2010) ( βπ₯ β β)
(Proposed by Mohammad Jafari)
89) Find all polynomials π(π₯) and π(π₯) such that: i) ποΏ½π(π₯)οΏ½ = ποΏ½π(π₯)οΏ½ (βπ₯ β β) ii) π(π₯) β₯ βπ₯ , π(π₯) β€ βπ₯ (βπ₯ β β)
(Proposed by Mohammad Jafari)
90) Find all polynomials π(π₯) and π(π₯) such that: i) βπ₯ β β βΆ π(π₯) > π(π₯) ii) βπ₯ β β βΆ π(π₯). π(π₯ β 1) = π(π₯ β 1). π(π₯)
(Proposed by Mohammad Jafari)
91) Find all polynomials π(π₯) and π(π₯) such that: π2(π₯) + π2(π₯) = (3π₯ β π₯3). π(π₯). π(π₯) βπ₯ β (0,β3)
(Proposed by Mohammad Jafari)
92) The polynomial π(π₯) is preserved with real and positive coefficients. If the sum of its coefficient's inverse equals 1, prove that :
π(1).π(π₯) β₯ (οΏ½οΏ½π₯π44
π=0
)4
(Proposed by Mohammad Jafari)
93) The polynomial π(π₯) is preserved with real and positive coefficients and with degrees of π. If the sum of its coefficient's inverse equals 1 prove that :
οΏ½π(4) + 1 β₯ 2π+1
(Proposed by Mohammad Jafari)
94) The polynomial π(π₯) is increasing and the polynomial π(π₯)is decreasing such that: 2ποΏ½π(π₯)οΏ½ = ποΏ½π(π₯)οΏ½ + π(π₯) βπ₯ β β Show that there is π₯0 β β such that:
π(π₯0) = π(π₯0) = π₯0
17
(Proposed by Mohammad Jafari)
95) Find all polynomials π(π₯) such that for the increasing function π:β+ βͺ {0} β β+ 2ποΏ½π(π₯)οΏ½ = ποΏ½π(π₯)οΏ½ + π(π₯) ,π(0) = 0
(Proposed by Mohammad Jafari)
96) Find all polynomials π(π₯) such that, for all non zero real numbers x,y,z that 1π₯
+ 1π¦
=1π§
π€π βππ£π: 1
π(π₯) +1
π(π¦)=
1π(π§)
(Proposed by Mohammad Jafari)
97) π(π₯) is an even polynomial (π(π₯) = π(βπ₯)) such that π(0) β 0 .If we can write π(π₯) as a multiplication of two polynomials with nonnegative coefficients, prove that those two polynomials would be even too.
(Proposed by Mohammad Jafari)
98) Find all polynomials π(π₯) such that: π(π₯ + 2)(π₯ β 2) + π(π₯ β 2)(π₯ + 2) = 2π₯π(π₯) ( βπ₯ β β)
(Proposed by Mohammad Jafari)
99) If for polynomials π(π₯) and π(π₯) that all their roots are real: π ππποΏ½π(π₯)οΏ½ = π πππ(π(π₯))
Prove that there is polynomial π»(π₯) such thatπ(π₯)π(π₯) = π»(π₯)2.
(Proposed by Mohammad Jafari)
100) For polynomials π(π₯) and π(π₯) : [π(π₯2 + 1)] = [π(π₯2 + 1)] Prove that: π(π₯) = π(π₯).
(Proposed by Mohammad Jafari)
101) For polynomials π(π₯) and π(π₯) with the degree of more than the degree of the polynomial π(π₯), we have :
οΏ½π(π₯)π(π₯)οΏ½
= οΏ½π(π₯)π(π₯)οΏ½
(βπ₯ β { π₯ β£β£ π(π₯) β 0, π₯ β β+ })
Prove that: π(π₯) = π(π₯).
(Proposed by Mohammad Jafari)
18
102) The plain and desert which compete for breathing play the following game : The desert choses 3 arbitrary numbers and the plain choses them as he wants as the coefficients of the polynomial ((β¦ π₯2 + β―π₯ + β― )).If the two roots of this polynomial are irrational, then the desert would be the winner, else the plain is the winner. Which one has the win strategy?
(Proposed by Mohammad Jafari)
103) The two polynomials π(π₯) and π(π₯) have an amount in the interval [π β 1,π] (π β β) forπ₯ β [0,1].If π is non-increasing such that:ποΏ½π(ππ₯)οΏ½ = ππ(π(π₯)) ,prove that there is π₯0 β [0,1] such that π(π(π₯0) = π₯0.
(Proposed by Mohammad Jafari)
19
Other Problems
104) Solve the following system in real numbers :
οΏ½π2 + π2 = 2π1 + π2 = 2πππ2 = ππ
(Proposed by Mohammad Jafari)
105) Solve the following system in real numbers :
β©βͺβͺβ¨
βͺβͺβ§ππ =
π2
1 + π2
ππ =π2
1 + π2
ππ =π2
1 + π2
(Proposed by Mohammad Jafari)
106) Solve the following system in positive real numbers : (π,π β β)
οΏ½(2 + ππ)(2 + ππ) = 9(2 + ππ)(2 β ππ) = 3
(Proposed by Mohammad Jafari)
107) Solve the following system in real numbers :
οΏ½π₯π¦2 = π¦4 β π¦ + 1π¦π§2 = π§4 β π§ + 1π§π₯2 = π₯4 β π₯ + 1
(Proposed by Mohammad Jafari)
108) Solve the following system in real numbers :
οΏ½π₯π¦ = π¦6 + π¦4 + π¦2 + 1π¦π§3 = π§6 + π§4 + π§2 + 1π§π₯5 = π₯ 6 + π₯4 + π₯2 + 1
(Proposed by Mohammad Jafari)
20
109) Solve the following system in real numbers :
οΏ½π₯2. π ππ2π¦ + π₯2 = π πππ¦. π πππ§π¦2. π ππ2π§ + π¦2 = π πππ§. π πππ₯π§2. π ππ2π₯ + π§2 = π πππ₯. π πππ¦
(Proposed by Mohammad Jafari)
110) Solve the following system in real numbers :
οΏ½(π2 + 1). (π2 + 1). (π2 + 1) =
818
(π4 + π2 + 1). (π4 + π2 + 1). (π4 + π2 + 1) =818
(πππ)32
(Proposed by Mohammad Jafari)
111) Solve the following system in real numbers :
οΏ½π2 + ππ = π2 + πππ2 + ππ = π2 + πππ2 + ππ = π2 + ππ
(Proposed by Mohammad Jafari)
112) Solve the following system in positive real numbers :
β©βͺβ¨
βͺβ§οΏ½π +
1ποΏ½ οΏ½
π +1ποΏ½
= 2(π +1π
)
οΏ½π +1ποΏ½ οΏ½
π +1ποΏ½
= 2(π +1π
)
οΏ½π +1ποΏ½ οΏ½
π +1ποΏ½
= 2(π +1π
)
(Proposed by Mohammad Jafari)
113) Solve the following equation for π₯ β (0, π2
): 2βπ₯π
+βπ πππ₯ + βπ‘πππ₯ = 12βπ₯
+ βπππ‘π₯ + βπππ π₯
(Proposed by Mohammad Jafari)
114) Find all functions π,π: β+ β β+ in the following system :
οΏ½(π(π₯) + π¦ β 1)(π(π¦) + π₯ β 1) = (π₯ + π¦)2 βπ₯, π¦, π§ β β+
(βπ(π₯) + π¦)(π(π¦) + π₯) = (π₯ + π¦ + 1)(π¦ β π₯ β 1) βπ₯, π¦, π§ β β+
(Proposed by Mohammad Jafari)
21
115) Solve the following system in real positive numbers :
οΏ½βπ4 + π3 + π2 = π + π + πβπ4 + π3 + π2 = π + π + πβπ4 + π3 + π2 = π + π + πβπ4 + π3 + π2 = π + π + π
(Proposed by Mohammad Jafari)
116) Solve the following equation in real numbers : π₯π¦
2π₯π¦ + π§2+
π¦π§2π¦π§ + π₯2
+π§π₯
2π§π₯ + π¦2= 1
(Proposed by Mohammad Jafari)