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24 ÊÔ§ËÒ¤Á 2005, 07:17
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˹ѧÊ×ͪ×èÍ Problems in mathematics with hints and solutions ¢Í§ V.Govorov áÅФ³Ð àÅèÁ¹Õé¹èÒ¨ÐËÒÂ×Áä´éµÒÁËéͧÊÁØ´ÁËÒÇÔ·ÂÒÅÑ ÀÒÂã¹àÅèÁ¨ÐºÃÃ¨Ø problems äÇé 200 ¡ÇèÒ¢éÍ µèÍàÃ×èͧæ˹Öè§
ËÑÇ¢éÍ·ÕèºÃèØã¹Ë¹Ñ§Ê×Í àªè¹ log ,expo ,domain range, sequence ,trigonometry ,geometry,introduction to calculus etc.
¢ÍµÑ´ºÒ§ÊèǹÁÒ ãËéÅͧ·Óà¾ÅÔ¹æáÅéǡѹ¹Ð¤ÃѺ
trigonometry
1. sin2x = cos2x-sin2x+1
2. sinx+sin2x+cos2x=(sin4x)(sin5x) +cos24x
3. (sin7x+cos7x)2=2sin211x+sin30x
4. (sin2x)(tanx) +(cos2x)(cotx)+2(sinx)(cosx)=4[:sqrt]3/3
exponential &logarithm equations and inequalities
\huge \begin{array}{lcr} 5. \quad 3^{2x+1}=3^{x+2}+\sqrt{1-6\cdot 3^{x}+3^{2(x+1)}} \\
6.\quad \sqrt{9^{x}-3^{x+2}}>3^{x}-9 \\
7. \quad x^{2}\cdot 2^{2x}+9(x+2)(2^{x})+8x^{2}\leq (x+2)\cdot 2^{2x}+9x^{2}\cdot 2^{x}+8x+16 \\
8. \quad log_{3x+7}(9+12x+4x^{2}) +log_{2x+3}(6x^{2}+23x+21) =4 \\
9. \quad \frac{x-1}{log_{3}(9-3^{x})-3} \leq 1 \\
10.\quad log _{5}x+log_{x}(\frac{x}{3})< \frac{(log_{5}x)(2-log_{3}x)}{log_{3}x} \\
11. \quad 3^{log_{x}2}=y^{log_{5}y} , 2^{log_{y}3}=x^{log_{7}x} \end{array}
1. n[:pi]+tan-1(-1[:plusminus][:sqrt]3)
2. n[:pi]/3 ,2[:pi]k , [:pi]((2m+1)/11)
3. [:pi]((2n+1)/44) , [:pi]((6k+(-1)k)/48)
4. [:pi]((3n+(-1)n)/6)
5. log3(2+[:sqrt]11/3)
6. (2,[:infinity])
7. [-1,0][:union] [2,3]
8. -1/4
9. [log30.9,2)
10. (1,3)[:union] (0,1/[:sqrt]5)
11. \large x= 2^{\sqrt[3]{\frac{(log_{2}7)^{2}}{log_{3}5}}} \quad y= 3^{\sqrt[3]{\frac{(log_{3}5)^{2}}{log_{2}7}}}
à»ç¹Ë¹Ñ§Ê×ͤè͹¢éÒ§¨Ðà¡èҫѡ˹èÍ áµè¹èÒ¨ÐÁÕ»ÃÐ⪹ìÊÓËÃѺ ¹éͧæ Á.»ÅÒÂÍÂèÒ§áç â´Â੾ÒФ¹·ÕèäÁèÁÕ⨷Âì·Ó
˹ѧÊ×ͪ×èÍ Problems in mathematics with hints and solutions ¢Í§ V.Govorov áÅФ³Ð àÅèÁ¹Õé¹èÒ¨ÐËÒÂ×Áä´éµÒÁËéͧÊÁØ´ÁËÒÇÔ·ÂÒÅÑ ÀÒÂã¹àÅèÁ¨ÐºÃÃ¨Ø problems äÇé 200 ¡ÇèÒ¢éÍ µèÍàÃ×èͧæ˹Öè§
ËÑÇ¢éÍ·ÕèºÃèØã¹Ë¹Ñ§Ê×Í àªè¹ log ,expo ,domain range, sequence ,trigonometry ,geometry,introduction to calculus etc.
¢ÍµÑ´ºÒ§ÊèǹÁÒ ãËéÅͧ·Óà¾ÅÔ¹æáÅéǡѹ¹Ð¤ÃѺ
trigonometry
1. sin2x = cos2x-sin2x+1
2. sinx+sin2x+cos2x=(sin4x)(sin5x) +cos24x
3. (sin7x+cos7x)2=2sin211x+sin30x
4. (sin2x)(tanx) +(cos2x)(cotx)+2(sinx)(cosx)=4[:sqrt]3/3
exponential &logarithm equations and inequalities
\huge \begin{array}{lcr} 5. \quad 3^{2x+1}=3^{x+2}+\sqrt{1-6\cdot 3^{x}+3^{2(x+1)}} \\
6.\quad \sqrt{9^{x}-3^{x+2}}>3^{x}-9 \\
7. \quad x^{2}\cdot 2^{2x}+9(x+2)(2^{x})+8x^{2}\leq (x+2)\cdot 2^{2x}+9x^{2}\cdot 2^{x}+8x+16 \\
8. \quad log_{3x+7}(9+12x+4x^{2}) +log_{2x+3}(6x^{2}+23x+21) =4 \\
9. \quad \frac{x-1}{log_{3}(9-3^{x})-3} \leq 1 \\
10.\quad log _{5}x+log_{x}(\frac{x}{3})< \frac{(log_{5}x)(2-log_{3}x)}{log_{3}x} \\
11. \quad 3^{log_{x}2}=y^{log_{5}y} , 2^{log_{y}3}=x^{log_{7}x} \end{array}
1. n[:pi]+tan-1(-1[:plusminus][:sqrt]3)
2. n[:pi]/3 ,2[:pi]k , [:pi]((2m+1)/11)
3. [:pi]((2n+1)/44) , [:pi]((6k+(-1)k)/48)
4. [:pi]((3n+(-1)n)/6)
5. log3(2+[:sqrt]11/3)
6. (2,[:infinity])
7. [-1,0][:union] [2,3]
8. -1/4
9. [log30.9,2)
10. (1,3)[:union] (0,1/[:sqrt]5)
11. \large x= 2^{\sqrt[3]{\frac{(log_{2}7)^{2}}{log_{3}5}}} \quad y= 3^{\sqrt[3]{\frac{(log_{3}5)^{2}}{log_{2}7}}}