116 problems
TRANSCRIPT
![Page 1: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/1.jpg)
1
116 Problems in Algebra
Author: Mohammad Jafari
Copyright ยฉ2011 by Mobtakeran Publications. All rights reserved.
![Page 2: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/2.jpg)
2
Contents
Function Equation Problemsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ3
Inequality Problemsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.10
Polynomial Problemsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ16
Other Problemsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ19
Solution to the Problemsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ..22
![Page 3: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/3.jpg)
3
Function Equation Problems
1) Find all functions ๐:๐ โ ๐ such that:
๐(2010๐(๐) + 1389) = 1 + 1389 + โฏ+ 13892010 + ๐ (โ๐ โ โ).
(Proposed by Mohammad Jafari)
2) Find all functions ๐:โ โ โ such that for all real numbers ๐ฅ,๐ฆ:
๐(๐ฅ + ๐ฆ) = ๐(๐ฅ). ๐(๐ฆ) + ๐ฅ๐ฆ
(Proposed by Mohammad Jafari)
3) Find all functions ๐:๐ โ {1} โ ๐ such that:
๐(๐ฅ๐ฆ) = ๐(๐ฅ)๐(๐) + ๐ฅ๐ฆ (โ๐ฅ,๐ฆ โ โ โ {1})
(Proposed by Mohammad Jafari)
4) Find all functions ๐:๐ โ ๐ such that:
๐(๐ฅ) = 2๐๏ฟฝ๐(๐ฅ)๏ฟฝ (โ๐ฅ โ โค)
(Proposed by Mohammad Jafari)
5) Find all functions ๐,๐:โค โ โค such that:
๐(๐ฅ) = 3๐๏ฟฝ๐(๐ฅ)๏ฟฝ (โ๐ฅ โ โค)
(Proposed by Mohammad Jafari)
6) Find all functions ๐:โค โ โค such that:
7๐(๐ฅ) = 3๐๏ฟฝ๐(๐ฅ)๏ฟฝ + 2๐ฅ (โ๐ฅ โ โค)
(Proposed by Mohammad Jafari)
7) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ๐ฅ + ๐ฆ + ๐(๐ฅ + ๐ฆ)๏ฟฝ = 2๐(๐ฅ) + 2๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
![Page 4: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/4.jpg)
4
8) For all functions ๐:โ โ โ such that:
๐(๐ฅ + ๐(๐ฅ) + ๐ฆ) = ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ) Prove that ๐(๐ฅ) is a bijective function.
(Proposed by Mohammad Jafari)
9) Find all functions ๐:โ โ โ such that:
๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = ๐ฅ + ๐๏ฟฝ๐(๐ฅ)๏ฟฝ + 2๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
10) Find all functions ๐:โ โ โ such that:
๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
11) For all ๐:โ โ โ such that : ๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ)
Prove that ๐(0) = 0.
(Proposed by Mohammad Jafari)
12) Find all functions ๐:โ+ โ โ+ such that, for all real numbers ๐ฅ > ๐ฆ > 0 :
๐(๐ฅ โ ๐ฆ) = ๐(๐ฅ) โ ๐(๐ฅ). ๐ ๏ฟฝ1๐ฅ๏ฟฝ .๐ฆ
(Proposed by Mohammad Jafari)
13) Find all functions ๐:โ โ โ such that:
๐ ๏ฟฝ๐๏ฟฝ๐ฅ + ๐(๐ฆ)๏ฟฝ๏ฟฝ = ๐ฅ + ๐(๐ฆ) + ๐(๐ฅ + ๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
14) Find all functions ๐:โ+โ{0} โ โ+โ{0} such that:
๐(๐๏ฟฝ๐ฅ + ๐(๐ฆ)๏ฟฝ) = 2๐ฅ + ๐(๐ฅ + ๐ฆ) (โ๐ฅ,๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
![Page 5: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/5.jpg)
5
15) Find all functions ๐:โ โ โ such that:
๐(๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ)) = ๐ฅ + ๐(๐ฅ) + ๐ฆ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
16) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ2๐ฅ + 2๐(๐ฆ)๏ฟฝ = ๐ฅ + ๐(๐ฅ) + ๐ฆ + ๐(๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
17) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ๐(๐ฅ) + 2๐(๐ฆ)๏ฟฝ = ๐(๐ฅ) + ๐ฆ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
18) Find all functions ๐:โ โ โ such that: i) ๐๏ฟฝ๐ฅ2 + ๐(๐ฆ)๏ฟฝ = ๐(๐ฅ)2 + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ) ii) ๐(๐ฅ) + ๐(โ๐ฅ) = 0 (โ๐ฅ โ โ+) iii) The number of the elements of the set { ๐ฅ โฃโฃ ๐(๐ฅ) = 0, ๐ฅ โ โ } is finite.
(Proposed by Mohammad Jafari)
19) For all injective functions ๐:โ โ โ such that: ๐๏ฟฝ๐ฅ + ๐(๐ฅ)๏ฟฝ = 2๐ฅ (โ๐ฅ โ โ) Prove that ๐(๐ฅ) + ๐ฅ is bijective.
(Proposed by Mohammad Jafari)
20) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ)๏ฟฝ = 2๐ฅ + ๐ฆ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
21) For all functions ๐,๐, โ:โ โ โ such that ๐ is injective and โ is bijective satisfying ๐๏ฟฝ๐(๐ฅ)๏ฟฝ =
โ(๐ฅ) (โ๐ฅ โ โ) , prove that ๐(๐ฅ) is bijective function.
(Proposed by Mohammad Jafari)
![Page 6: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/6.jpg)
6
22) Find all functions ๐:โ โ โ such that:
๐(2๐ฅ + 2๐(๐ฆ)) = ๐ฅ + ๐(๐ฅ) + 2๐ฆ (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
23) Find all functions ๐:โ+ โช {0} โ โ+ such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐ฆ๏ฟฝ = ๐(๐ฅ) + ๐ฆ (โ๐ฅ, ๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
24) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐(๐ฆ)๏ฟฝ = ๐(๐ฅ) + ๐ฆ (โ๐ฅ, ๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
25) For all functions ๐:โ+ โช {0} โ โ such that : i) ๐(๐ฅ + ๐ฆ) = ๐(๐ฅ) + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ+ โช {0}) ii) The number of the elements of the set ๏ฟฝ ๐ฅ โฃโฃ ๐(๐ฅ) = 0, ๐ฅ โ โ+ โช {0} ๏ฟฝ is finite.
Prove that ๐ is injective function.
(Proposed by Mohammad Jafari)
26) Find all functions ๐:โ+ โช {0} โ โ such that:
i) ๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = ๐(2๐ฅ) + 2๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ+ โช {0}) ii) The number of the elements of the set ๏ฟฝ ๐ฅ โฃโฃ ๐(๐ฅ) = 0, ๐ฅ โ โ+ โช {0} ๏ฟฝ is finite.
(Proposed by Mohammad Jafari)
27) Find all functions ๐:โ โ โsuch that:
i) ๐(๐(๐ฅ) + ๐ฆ) = ๐ฅ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ) ii) โ๐ฅ โ โ+; โ๐ฆ โ โ+๐ ๐ข๐โ ๐กโ๐๐ก ๐(๐ฆ) = ๐ฅ
(Proposed by Mohammad Jafari)
![Page 7: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/7.jpg)
7
28) Find all functions ๐:โ{0} โ โ such that: i) ๐(๐(๐ฅ) + ๐ฆ) = ๐ฅ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ) ii) The set {๐ฅ โฃ ๐(๐ฅ) = โ๐ฅ, ๐ฅ โ โ} has a finite number of elements.
(Proposed by Mohammad Jafari)
29) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ๐๏ฟฝ๐(๐ฅ)๏ฟฝ + ๐(๐ฆ) + ๐ง๏ฟฝ = ๐ฅ + ๐(๐ฆ) + ๐๏ฟฝ๐(๐ง)๏ฟฝ (โ๐ฅ, ๐ฆ, ๐ง โ โ)
(Proposed by Mohammad Jafari)
30) Find all functions ๐:โ โ โ such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐ฆ + ๐(2๐ง)๏ฟฝ = 2๐ฅ โ ๐(๐ฅ) + ๐(๐ฆ) + 2๐(๐ง) (โ๐ฅ, ๐ฆ, ๐ง โ โ)
(Proposed by Mohammad Jafari)
31) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐ฆ + ๐(2๐ง)๏ฟฝ = 2๐ฅ โ ๐(๐ฅ) + ๐(๐ฆ) + 2๐(๐ง) (โ๐ฅ, ๐ฆ, ๐ง โ โ+โ{0})
(Proposed by Mohammad Jafari)
32) (IRAN TST 2010) Find all non-decreasing functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐ฆ๏ฟฝ = 2๐ฅ โ ๐(๐ฅ) + ๐(๐(๐ฆ)) (โ๐ฅ, ๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
33) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = 2๐ฅ + ๐(2๐(๐ฆ)) (โ๐ฅ, ๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
34) Find all functions ๐:โ โ โ such that:
๐(๐ฅ + ๐(๐ฅ) + 2๐ฆ) = 2๐ฅ + 2๐(๐(๐ฆ)) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
![Page 8: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/8.jpg)
8
35) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐ ๏ฟฝ๐ฅ+๐(๐ฅ)2
+ ๐ฆ + ๐(2๐ง)๏ฟฝ = 2๐ฅ โ ๐(๐ฅ) + ๐น(๐(๐ฆ)) + 2๐(๐ง) (โ๐ฅ, ๐ฆ, ๐ง โ โ+ โช {0})
(Proposed by Mohammad Jafari)
36) Find all functions ๐:โ โ โ such that:
๐๏ฟฝ๐ฅ โ ๐(๐ฆ)๏ฟฝ = ๐(๐ฆ)2 โ 2๐ฅ๐(๐ฆ) + ๐(๐ฅ) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
37) Find all functions ๐:โ โ โ such that:
(๐ฅ โ ๐ฆ)๏ฟฝ๐(๐ฅ) + ๐(๐ฆ)๏ฟฝ = (๐ฅ + ๐ฆ)๏ฟฝ๐(๐ฅ) โ ๐(๐ฆ)๏ฟฝ (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
38) Find all functions ๐:โ โ โ such that:
๐(๐ฅ โ ๐ฆ)(๐ฅ + ๐ฆ) = (๐ฅ โ ๐ฆ)(๐(๐ฅ) + ๐(๐ฆ)) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
39) Find all functions ๐:โ โ โ such that:
๐(๐ฅ โ ๐ฆ)(๐ฅ + ๐ฆ) = ๐(๐ฅ + ๐ฆ)(๐ฅ โ ๐ฆ) (โ๐ฅ,๐ฆ โ โ)
(Proposed by Mohammad Jafari)
40) Find all non-decreasing functions ๐,๐:โ+ โช {0} โ โ+ โช {0} such that: ๐(๐ฅ) = 2๐ฅ โ ๐(๐ฅ) prove that ๐ and ๐ are continues functions.
(Proposed by Mohammad Jafari)
41) Find all functions ๐: { ๐ฅ โฃ ๐ฅ โ โ , ๐ฅ > 1 } โ โ such that :
๐(๐ฅ)2. ๐(๐ฅ2)2 + ๐(2๐ฅ). ๐ ๏ฟฝ๐ฅ2
2๏ฟฝ = 1 โ๐ฅ โ {๐ฅ โฃ ๐ฅ โ โ, ๐ฅ > 1}
(Proposed by Mohammad Jafari)
![Page 9: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/9.jpg)
9
42) (IRAN TST 2011) Find all bijective functions ๐:โ โ โ such that:
๐๏ฟฝ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ)๏ฟฝ = ๐(2๐ฅ) + ๐(2๐ฆ) (โ๐ฅ, ๐ฆ โ โ)
(Proposed by Mohammad Jafari)
43) Find all functions ๐:โ+ โ โ+ such that:
๐(๐ฅ + ๐(๐ฅ) + ๐ฆ) = ๐(2๐ฅ) + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ+)
(Proposed by Mohammad Jafari)
44) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that:
๐๏ฟฝ๐ฅ + ๐(๐ฅ) + 2๐(๐ฆ)๏ฟฝ = 2๐(๐) + ๐ฆ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ+ โช {0})
(Proposed by Mohammad Jafari)
45) Find all functions ๐:โ+ โช {0} โ โ+ โช {0} such that: i) ๐๏ฟฝ๐ฅ + ๐(๐ฅ) + ๐(2๐ฆ)๏ฟฝ = 2๐(๐ฅ) + ๐ฆ + ๐(๐ฆ) (โ๐ฅ, ๐ฆ โ โ+ โช {0}) ii) ๐(0) = 0
(Proposed by Mohammad Jafari)
46) Find all functions ๐:โ+ โ โ+ such that: ๐(๐ฅ + ๐ฆ๐ + ๐(๐ฆ)) = ๐(๐ฅ) (โ๐ฅ, ๐ฆ โ โ+ , ๐ โ โ , ๐ โฅ 2)
(Proposed by Mohammad Jafari)
47) Find all functions ๐:โ โ โ such that:
๐(๐ โ 1) + ๐(๐ + 1) < 2๐(๐) (โ๐ โ โ, ๐ โฅ 2)
(Proposed by Mohammad Jafari)
48) Find all functions ๐: {๐ด โฃ ๐ด โ โ,๐ด โฅ 1} โ โ such that:
๐(๐ฅ๐ฆ2) = ๐(4๐ฅ). ๐(๐ฆ) +๐(8๐ฅ)๐(2๐ฆ)
(Proposed by Mohammad Jafari)
![Page 10: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/10.jpg)
10
Inequality Problems:
49) For all positive real numbers ๐ฅ,๐ฆ, ๐ง such that ๐ฅ + ๐ฆ + ๐ง = 2 prove that : ๐ฅ
๐ฅ4 + ๐ฆ + ๐ง + 1+
๐ฆ๐ฆ4 + ๐ง + ๐ฅ + 1
+๐ง
๐ง4 + ๐ฅ + ๐ฆ + 1โค 1
(Proposed by Mohammad Jafari)
50) For all positive real numbers ๐, ๐, ๐ such that ๐ + ๐ + ๐ = 6 prove that :
๏ฟฝ ๏ฟฝ๐
(๐ + ๐)(๐ + ๐)๐
โฆ3
โ๐๐๐๐ ,๐ โ โ, ๐ โฅ 3
(Proposed by Mohammad Jafari)
51) For all real numbers ๐, ๐, ๐ โ (2,4) prove that: 2
๐ + ๐2 + ๐3+
2๐ + ๐2 + ๐3
+2
๐ + ๐2 + ๐3<
3๐ + ๐ + ๐
(Proposed by Mohammad Jafari)
52) For all positive real numbers ๐, ๐, ๐ prove that: ๐
๐4 + ๐2 + 1+
๐๐4 + ๐2 + 1
+๐
๐4 + ๐2 + 1<
43
(Proposed by Mohammad Jafari)
53) For all real positive numbers ๐, ๐, ๐ such that1 + 1๐+๐+๐
< 1โ๐2+๐2+๐2
+ 1โ๐๐+๐๐+๐๐
prove that:
๐๐ + ๐๐ + ๐๐ < ๐ + ๐ + ๐
(Proposed by Mohammad Jafari)
54) For all real numbers ๐, ๐, ๐ such that ๐ โฅ ๐ โฅ 0 โฅ ๐ and ๐ + ๐ + ๐ < 0 prove that : ๐3 + ๐3 + ๐3 + ๐๐2 + ๐๐2 + ๐๐2 โค 2๐2๐ + 2๐2๐ + 2๐2๐
(Proposed by Mohammad Jafari)
55) For all real numbers 0 < ๐ฅ1 < ๐ฅ2 < โฏ < ๐ฅ1390 < ๐2 prove that :
๐ ๐๐3๐ฅ1 + ๐๐๐ 3๐ฅ2+โฆ+๐ ๐๐3๐ฅ1389 + ๐๐๐ 3๐ฅ1390 < 695
(Proposed by Mohammad Jafari)
![Page 11: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/11.jpg)
11
56) For all ๐ผ๐ โ ๏ฟฝ0, ๐2๏ฟฝ (๐ = 1,2, โฆ ,๐) prove that :
๐2โฅ min {๏ฟฝ๐ ๐๐3๐ผ๐ . ๐๐๐ 3๐ผ๐+1,๏ฟฝ๐ ๐๐3๐ผ๐+1. ๐๐๐ 3๐ผ๐
๐
๐=1
}๐
๐=1
(๐ผ๐+1 = ๐ผ1)
(Proposed by Mohammad Jafari)
57) For all real numbers ๐, ๐, ๐ โ [2,3] prove that: 1
๐๐(2๐ โ ๐) +1
๐๐(2๐ โ ๐) +1
๐๐(2๐ โ ๐) โฅ19
(Proposed by Mohammad Jafari)
58) For all real numbers ๐, ๐, ๐ โ [1,2] prove that: 2
๐๐(3๐ โ ๐)+
2๐๐(3๐ โ ๐)
+2
๐๐(3๐ โ ๐)โค 3
(Proposed by Mohammad Jafari)
59) For all real positive numbers ๐, ๐, ๐ prove that: ๐
3๐ + ๐ + ๐+
๐3๐ + ๐ + ๐
+๐
3๐ + ๐ + ๐โค
35
(Proposed by Mohammad Jafari)
60) For all positive real numbers such that ๐๐๐ = 1 prove that: ๐
โ๐2 + 3+
๐โ๐2 + 3
+๐
โ๐2 + 3โฅ
32
(Proposed by Mohammad Jafari)
61) For all positive real numbers ๐, ๐, ๐ such that ๐ + ๐ + ๐ = 1 prove that:
(๐๐
โ๐ + ๐+
๐๐โ๐ + ๐
+๐๐
โ๐ + ๐)2 + (
๐๐โ๐ + ๐
+๐๐
โ๐ + ๐+
๐๐โ๐ + ๐
)2 โค13
(Proposed by Mohammad Jafari)
62) For all positive real numbers ๐, ๐, ๐ such that ๐ + ๐ + ๐ = 1 prove that: 1
โ๐ + ๐+
1โ๐ + ๐
+1
โ๐ + ๐โค
1โ2๐๐๐
(Proposed by Mohammad Jafari)
![Page 12: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/12.jpg)
12
63) For all positive real numbers ๐, ๐, ๐ such that ๐๐ + ๐๐ + ๐๐ = 23 prove that:
1โ๐ + ๐
+1
โ๐ + ๐+
1โ๐ + ๐
โค1
โ๐๐๐
(Proposed by Mohammad Jafari)
64) For all positive real numbers ๐, ๐, ๐ such that ๐ + ๐ + ๐ = 3prove that: 1
๐ + ๐ + ๐2+
1๐ + ๐ + ๐2
+1
๐ + ๐ + ๐2โค 1
(Proposed by Mohammad Jafari)
65) For all positive real numbers ๐, ๐, ๐ prove that: ๐4 + ๐2 + 1๐6 + ๐3 + 1
+๐4 + ๐2 + 1๐6 + ๐3 + 1
+๐4 + ๐2 + 1๐6 + ๐3 + 1
โค3
๐2 + ๐ + 1+
3๐2 + ๐ + 1
+3
๐2 + ๐ + 1
(Proposed by Mohammad Jafari)
66) For all positive real numbers ๐, ๐, ๐ prove that: ๐4 + ๐2 + 1๐6 + ๐3 + 1
+๐4 + ๐2 + 1๐6 + ๐3 + 1
+๐4 + ๐2 + 1๐6 + ๐3 + 1
โค 4
(Proposed by Mohammad Jafari)
67) For all positive real numbers ๐, ๐, ๐ prove that: 1 + ๐2 + ๐4
๐ + ๐2 + ๐3+
1 + ๐2 + ๐4
๐ + ๐2 + ๐3+
1 + ๐2 + ๐4
๐ + ๐2 + ๐3โฅ 3
(Proposed by Mohammad Jafari)
68) For all real numbers ๐, ๐, ๐ โ (1,2) prove that: 4
๐ + ๐ + ๐โฅ
11 + ๐ + ๐2
+1
1 + ๐ + ๐2+
11 + ๐ + ๐2
(Proposed by Mohammad Jafari)
69) For all positive real numbers ๐ฅ,๐ฆ, ๐ง such that ๐ฅ + ๐ฆ + ๐ง + 1๐ฅ
+ 1๐ฆ
+ 1๐ง
= 10 prove that:
(๐ฅ2 + 1)2
4๐ฅ+
(๐ฆ2 + 1)2
4๐ฆ+
(๐ง2 + 1)2
4๐งโฅ ๐ฅ + ๐ฆ + ๐ง
(Proposed by Mohammad Jafari)
![Page 13: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/13.jpg)
13
70) For all positive real numbers ๐ฅ,๐ฆ, ๐ง such that ๐ฅ + ๐ฆ + ๐ง + 1๐ฅ
+ 1๐ฆ
+ 1๐ง
= 10 prove that:
4๏ฟฝ๐ฅ + ๐ฆ + ๐ง โฅ ๐ฅ + ๐ฆ + ๐ง + 3
(Proposed by Mohammad Jafari)
71) For all positive real numbers ๐, ๐, ๐ prove that:
(๏ฟฝ๐2
๐ + ๐)(๏ฟฝ
๐(๐ + ๐)2
) โฅ98
(Proposed by Mohammad Jafari)
72) For all positive real numbers ๐, ๐, ๐ prove that:
(๏ฟฝ๐
(๐ + ๐)2)(๏ฟฝ
(๐ + ๐)2
๐) โฅ (๏ฟฝ
2๐๐ + ๐
)2
(Proposed by Mohammad Jafari)
73) For all positive real numbers ๐, ๐, ๐ prove that:
(๏ฟฝ1
๐2 + ๐2)(
๐3
๐2 + ๐2) โฅ (๏ฟฝ
๐๐2 + ๐2
)(๏ฟฝ๐2
๐2 + ๐2)
(Proposed by Mohammad Jafari)
74) For all positive numbers ๐, ๐, ๐ prove that :
2(๏ฟฝ๐
๐ + ๐)(๏ฟฝ๐๐2) โฅ (๏ฟฝ๐) (๏ฟฝ๐๐)
(Proposed by Mohammad Jafari)
75) For all ๐ฅ๐ โ โ (๐ = 1,2, โฆ ,๐) such that ๐ฅ๐ โ ๐ฅ๐ ,prove that: 1๐ฅ112
+2
๐ฅ112 + ๐ฅ2
22+ โฏ+
๐๐ฅ112 + โฏ+ ๐ฅ๐
๐2โค๐(๐ + 3)
2
(Proposed by Mohammad Jafari)
76) For all ๐ โ โ (๐ โฅ 3) prove that :
๐๏ฟฝ(๐โ1)!๏ฟฝ1
๐โ1 + ๐๏ฟฝ(๐โ2)!๏ฟฝ1
๐โ2 + โฏ+ ๐(1)11 < ๐
๐2 + ๐
๐โ12 + โฏ+ ๐1
(Proposed by Mohammad Jafari)
![Page 14: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/14.jpg)
14
77) For all positive real numbers ๐, ๐, ๐ such that ๐ + ๐ + ๐ = 1 prove that:
๏ฟฝ1
(๐ + 2๐)(๐ + 2๐)โฅ 3
(Proposed by Mohammad Jafari)
78) For all positive real numbers ๐, ๐, ๐such that ๐ + ๐ + ๐ = 2 prove that:
๏ฟฝ1
(๐3 + ๐ + 1)(1 + ๐ + ๐3)โค 1
(Proposed by Mohammad Jafari)
79) For all positive real numbers ๐, ๐, ๐ such that โ๐ + โ๐ + โ๐ = 1๐
+ 1๐
+ 1๐ prove that:
min {๐
1 + ๐2,
๐1 + ๐2
,๐
1 + ๐2} โค
12
(Proposed by Mohammad Jafari)
80) For all positive real numbers ๐, ๐, ๐ such that 1๐๐
+ 1๐๐
+ 1๐๐
= 3๐๐๐๐๐๐ (๐,๐ โ โ) prove
that:
min ๏ฟฝ๐๐
1 + ๐๐,๐๐
1 + ๐๐,๐๐
1 + ๐๐๏ฟฝ โค
12
(๐, ๐ โ โ)
(Proposed by Mohammad Jafari)
81) For all positive real numbers ๐ฅ,๐ฆ prove that:
๏ฟฝ๐ฅ๐
๐ฅ๐โ1 + ๐ฅ๐โ2๐ฆ + โฏ+ ๐ฆ๐โ1โฅ๐ฅ + ๐ฆ + ๐ง
๐ (๐ โ โ)
(Proposed by Mohammad Jafari)
82) For all positive real numbers ๐ฅ,๐ฆ, ๐ง prove that:
๏ฟฝ๐ฅ2๐
๏ฟฝ๐ฅ20 + ๐ฆ20๏ฟฝ๏ฟฝ๐ฅ21 + ๐ฆ21๏ฟฝโฆ (๐ฅ2๐โ1 + ๐ฆ2๐โ1)โฅ๐ฅ + ๐ฆ + ๐ง
2๐
(Proposed by Mohammad Jafari)
83) For all positive real numbers ๐ฅ,๐ฆ, ๐ง such that ๐ฅ๐ฅ2+4
+ ๐ฆ๐ฆ2+4
+ ๐ง๐ง2+4
= 15 prove that:
๐ฅ๐ฅ + 6
+๐ฆ
๐ฆ + 6+
๐ง๐ง + 6
< 2.4
(Proposed by Mohammad Jafari)
![Page 15: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/15.jpg)
15
84) Find minimum real number ๐ such that for all real numbers ๐, ๐, ๐ :
๏ฟฝ๏ฟฝ2(๐2 + 1)(๐2 + 1) + ๐ โฅ 2๏ฟฝ๐ + ๏ฟฝ๐๐
(Proposed by Mohammad Jafari)
85) (IRAN TST 2011) Find minimum real number ๐ such that for all real numbers ๐, ๐, ๐,๐ :
๏ฟฝ๏ฟฝ(๐2 + 1)(๐2 + 1)(๐2 + 1) + ๐ โฅ 2(๐๐ + ๐๐ + ๐๐ + ๐๐ + ๐๐ + ๐๐)
(Proposed by Dr.Amir Jafari and Mohammad Jafari)
86) For all positive real numbers ๐, ๐, ๐ prove that:
1 +โ๐๐โ๐2
โค๏ฟฝ(๐ + ๐)2 + ๐2
2๐2 + 2๐2 + ๐2โค 2 +
โ๐๐โ๐2
(Proposed by Mohammad Jafari)
87) For all real numbers 1 โค ๐, ๐, ๐ prove that:
๏ฟฝ(๐ + ๐)2 + ๐2๐ + 2๐ + ๐
โค๏ฟฝ๐
(Proposed by Mohammad Jafari)
![Page 16: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/16.jpg)
16
Polynomial Problems:
88) Find all polynomials ๐(๐ฅ) and ๐(๐ฅ) with real coefficients such that:
๏ฟฝ [๐(๐ฅ โ ๐).๐(๐ฅ โ ๐ โ 1). (๐ฅ โ ๐ โ 3)]2010
๐=0
โฅ ๐(๐ฅ). ๐(๐ฅ โ 2010) ( โ๐ฅ โ โ)
(Proposed by Mohammad Jafari)
89) Find all polynomials ๐(๐ฅ) and ๐(๐ฅ) such that: i) ๐๏ฟฝ๐(๐ฅ)๏ฟฝ = ๐๏ฟฝ๐(๐ฅ)๏ฟฝ (โ๐ฅ โ โ) ii) ๐(๐ฅ) โฅ โ๐ฅ , ๐(๐ฅ) โค โ๐ฅ (โ๐ฅ โ โ)
(Proposed by Mohammad Jafari)
90) Find all polynomials ๐(๐ฅ) and ๐(๐ฅ) such that: i) โ๐ฅ โ โ โถ ๐(๐ฅ) > ๐(๐ฅ) ii) โ๐ฅ โ โ โถ ๐(๐ฅ). ๐(๐ฅ โ 1) = ๐(๐ฅ โ 1). ๐(๐ฅ)
(Proposed by Mohammad Jafari)
91) Find all polynomials ๐(๐ฅ) and ๐(๐ฅ) such that: ๐2(๐ฅ) + ๐2(๐ฅ) = (3๐ฅ โ ๐ฅ3). ๐(๐ฅ). ๐(๐ฅ) โ๐ฅ โ (0,โ3)
(Proposed by Mohammad Jafari)
92) The polynomial ๐(๐ฅ) is preserved with real and positive coefficients. If the sum of its coefficient's inverse equals 1, prove that :
๐(1).๐(๐ฅ) โฅ (๏ฟฝ๏ฟฝ๐ฅ๐44
๐=0
)4
(Proposed by Mohammad Jafari)
93) The polynomial ๐(๐ฅ) is preserved with real and positive coefficients and with degrees of ๐. If the sum of its coefficient's inverse equals 1 prove that :
๏ฟฝ๐(4) + 1 โฅ 2๐+1
(Proposed by Mohammad Jafari)
94) The polynomial ๐(๐ฅ) is increasing and the polynomial ๐(๐ฅ)is decreasing such that: 2๐๏ฟฝ๐(๐ฅ)๏ฟฝ = ๐๏ฟฝ๐(๐ฅ)๏ฟฝ + ๐(๐ฅ) โ๐ฅ โ โ Show that there is ๐ฅ0 โ โ such that:
๐(๐ฅ0) = ๐(๐ฅ0) = ๐ฅ0
![Page 17: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/17.jpg)
17
(Proposed by Mohammad Jafari)
95) Find all polynomials ๐(๐ฅ) such that for the increasing function ๐:โ+ โช {0} โ โ+ 2๐๏ฟฝ๐(๐ฅ)๏ฟฝ = ๐๏ฟฝ๐(๐ฅ)๏ฟฝ + ๐(๐ฅ) ,๐(0) = 0
(Proposed by Mohammad Jafari)
96) Find all polynomials ๐(๐ฅ) such that, for all non zero real numbers x,y,z that 1๐ฅ
+ 1๐ฆ
=1๐ง
๐ค๐ โ๐๐ฃ๐: 1
๐(๐ฅ) +1
๐(๐ฆ)=
1๐(๐ง)
(Proposed by Mohammad Jafari)
97) ๐(๐ฅ) is an even polynomial (๐(๐ฅ) = ๐(โ๐ฅ)) such that ๐(0) โ 0 .If we can write ๐(๐ฅ) 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 ๐(๐ฅ) such that: ๐(๐ฅ + 2)(๐ฅ โ 2) + ๐(๐ฅ โ 2)(๐ฅ + 2) = 2๐ฅ๐(๐ฅ) ( โ๐ฅ โ โ)
(Proposed by Mohammad Jafari)
99) If for polynomials ๐(๐ฅ) and ๐(๐ฅ) that all their roots are real: ๐ ๐๐๐๏ฟฝ๐(๐ฅ)๏ฟฝ = ๐ ๐๐๐(๐(๐ฅ))
Prove that there is polynomial ๐ป(๐ฅ) such that๐(๐ฅ)๐(๐ฅ) = ๐ป(๐ฅ)2.
(Proposed by Mohammad Jafari)
100) For polynomials ๐(๐ฅ) and ๐(๐ฅ) : [๐(๐ฅ2 + 1)] = [๐(๐ฅ2 + 1)] Prove that: ๐(๐ฅ) = ๐(๐ฅ).
(Proposed by Mohammad Jafari)
101) For polynomials ๐(๐ฅ) and ๐(๐ฅ) with the degree of more than the degree of the polynomial ๐(๐ฅ), we have :
๏ฟฝ๐(๐ฅ)๐(๐ฅ)๏ฟฝ
= ๏ฟฝ๐(๐ฅ)๐(๐ฅ)๏ฟฝ
(โ๐ฅ โ { ๐ฅ โฃโฃ ๐(๐ฅ) โ 0, ๐ฅ โ โ+ })
Prove that: ๐(๐ฅ) = ๐(๐ฅ).
(Proposed by Mohammad Jafari)
![Page 18: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/18.jpg)
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 ((โฆ ๐ฅ2 + โฏ๐ฅ + โฏ )).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 ๐(๐ฅ) and ๐(๐ฅ) have an amount in the interval [๐ โ 1,๐] (๐ โ โ) for๐ฅ โ [0,1].If ๐ is non-increasing such that:๐๏ฟฝ๐(๐๐ฅ)๏ฟฝ = ๐๐(๐(๐ฅ)) ,prove that there is ๐ฅ0 โ [0,1] such that ๐(๐(๐ฅ0) = ๐ฅ0.
(Proposed by Mohammad Jafari)
![Page 19: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/19.jpg)
19
Other Problems
104) Solve the following system in real numbers :
๏ฟฝ๐2 + ๐2 = 2๐1 + ๐2 = 2๐๐๐2 = ๐๐
(Proposed by Mohammad Jafari)
105) Solve the following system in real numbers :
โฉโชโชโจ
โชโชโง๐๐ =
๐2
1 + ๐2
๐๐ =๐2
1 + ๐2
๐๐ =๐2
1 + ๐2
(Proposed by Mohammad Jafari)
106) Solve the following system in positive real numbers : (๐,๐ โ โ)
๏ฟฝ(2 + ๐๐)(2 + ๐๐) = 9(2 + ๐๐)(2 โ ๐๐) = 3
(Proposed by Mohammad Jafari)
107) Solve the following system in real numbers :
๏ฟฝ๐ฅ๐ฆ2 = ๐ฆ4 โ ๐ฆ + 1๐ฆ๐ง2 = ๐ง4 โ ๐ง + 1๐ง๐ฅ2 = ๐ฅ4 โ ๐ฅ + 1
(Proposed by Mohammad Jafari)
108) Solve the following system in real numbers :
๏ฟฝ๐ฅ๐ฆ = ๐ฆ6 + ๐ฆ4 + ๐ฆ2 + 1๐ฆ๐ง3 = ๐ง6 + ๐ง4 + ๐ง2 + 1๐ง๐ฅ5 = ๐ฅ 6 + ๐ฅ4 + ๐ฅ2 + 1
(Proposed by Mohammad Jafari)
![Page 20: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/20.jpg)
20
109) Solve the following system in real numbers :
๏ฟฝ๐ฅ2. ๐ ๐๐2๐ฆ + ๐ฅ2 = ๐ ๐๐๐ฆ. ๐ ๐๐๐ง๐ฆ2. ๐ ๐๐2๐ง + ๐ฆ2 = ๐ ๐๐๐ง. ๐ ๐๐๐ฅ๐ง2. ๐ ๐๐2๐ฅ + ๐ง2 = ๐ ๐๐๐ฅ. ๐ ๐๐๐ฆ
(Proposed by Mohammad Jafari)
110) Solve the following system in real numbers :
๏ฟฝ(๐2 + 1). (๐2 + 1). (๐2 + 1) =
818
(๐4 + ๐2 + 1). (๐4 + ๐2 + 1). (๐4 + ๐2 + 1) =818
(๐๐๐)32
(Proposed by Mohammad Jafari)
111) Solve the following system in real numbers :
๏ฟฝ๐2 + ๐๐ = ๐2 + ๐๐๐2 + ๐๐ = ๐2 + ๐๐๐2 + ๐๐ = ๐2 + ๐๐
(Proposed by Mohammad Jafari)
112) Solve the following system in positive real numbers :
โฉโชโจ
โชโง๏ฟฝ๐ +
1๐๏ฟฝ ๏ฟฝ
๐ +1๐๏ฟฝ
= 2(๐ +1๐
)
๏ฟฝ๐ +1๐๏ฟฝ ๏ฟฝ
๐ +1๐๏ฟฝ
= 2(๐ +1๐
)
๏ฟฝ๐ +1๐๏ฟฝ ๏ฟฝ
๐ +1๐๏ฟฝ
= 2(๐ +1๐
)
(Proposed by Mohammad Jafari)
113) Solve the following equation for ๐ฅ โ (0, ๐2
): 2โ๐ฅ๐
+โ๐ ๐๐๐ฅ + โ๐ก๐๐๐ฅ = 12โ๐ฅ
+ โ๐๐๐ก๐ฅ + โ๐๐๐ ๐ฅ
(Proposed by Mohammad Jafari)
114) Find all functions ๐,๐: โ+ โ โ+ in the following system :
๏ฟฝ(๐(๐ฅ) + ๐ฆ โ 1)(๐(๐ฆ) + ๐ฅ โ 1) = (๐ฅ + ๐ฆ)2 โ๐ฅ, ๐ฆ, ๐ง โ โ+
(โ๐(๐ฅ) + ๐ฆ)(๐(๐ฆ) + ๐ฅ) = (๐ฅ + ๐ฆ + 1)(๐ฆ โ ๐ฅ โ 1) โ๐ฅ, ๐ฆ, ๐ง โ โ+
(Proposed by Mohammad Jafari)
![Page 21: 116 Problems](https://reader035.vdocuments.site/reader035/viewer/2022081718/54492c27b1af9f53618b4e69/html5/thumbnails/21.jpg)
21
115) Solve the following system in real positive numbers :
๏ฟฝโ๐4 + ๐3 + ๐2 = ๐ + ๐ + ๐โ๐4 + ๐3 + ๐2 = ๐ + ๐ + ๐โ๐4 + ๐3 + ๐2 = ๐ + ๐ + ๐โ๐4 + ๐3 + ๐2 = ๐ + ๐ + ๐
(Proposed by Mohammad Jafari)
116) Solve the following equation in real numbers : ๐ฅ๐ฆ
2๐ฅ๐ฆ + ๐ง2+
๐ฆ๐ง2๐ฆ๐ง + ๐ฅ2
+๐ง๐ฅ
2๐ง๐ฅ + ๐ฆ2= 1
(Proposed by Mohammad Jafari)