1. (PLK 1997)计算 $29\frac{27}{28}\times 27\frac{14}{15}$
答：
$(30-\frac{1}{28})\times (28-\frac{1}{15})=840-1-2+\frac{1}{420})=837\frac{1}{420}$
2. (PLK 1997) 997-996-995+994+993-992+991-990-989+988+987-986+985-984-983+982+981-980+.....+7-6-5+4+3-2+1=?
答：
注意到规则 $+ - - + + -$ .所以 6 项为一组计算,每组的和为 1,共 $\frac{996}{6}=166$ 组. 得 $166+1=167$
3. (PLK 1997)计算
$1\times \left( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right) + 3\times \left( \frac{1}{2} + \frac{1}{3} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right)$
$ + 5\times \left( \frac{1}{3} + \frac{1}{4} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right) + 7\times \left( \frac{1}{4} + \frac{1}{5} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right)$
$+ 9\times \left( \frac{1}{5} + \frac{1}{6} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right) + 11\times \left( \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10}\right)$ +
$13\times \left( \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right)$ +
$15\times \left( \frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right)$ +
$17\times \left( \frac{1}{9} + \frac{1}{10} \right) +19\times \left( \frac{1}{10} \right)$
$ + 5\times \left( \frac{1}{3} + \frac{1}{4} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right) + 7\times \left( \frac{1}{4} + \frac{1}{5} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right)$
$+ 9\times \left( \frac{1}{5} + \frac{1}{6} + \cdot \cdot \cdot \cdot + \frac{1}{10} \right) + 11\times \left( \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10}\right)$ +
$13\times \left( \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right)$ +
$15\times \left( \frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right)$ +
$17\times \left( \frac{1}{9} + \frac{1}{10} \right) +19\times \left( \frac{1}{10} \right)$
答：
$1+(1+3)\times \frac{1}{2}+(1+3+5)\times \frac{1}{3}+(1+3+5+7)\times \frac{1}{4}+(1+3+5+7+9)\times \frac{1}{5}$
$+ (1+3+5+7+9+11)\times \frac{1}{6}+(1+3+5+\cdot \cdot \cdot \cdot +13)\times \frac{1}{7}+(1+3+5+\cdot \cdot \cdot \cdot +15)\times \frac{1}{8}$
$+ (1+3+5+\cdot \cdot \cdot \cdot +17)\times \frac{1}{9}+(1+3+5+\cdot \cdot \cdot \cdot +19)\times \frac{1}{10}$
$+ (1+3+5+7+9+11)\times \frac{1}{6}+(1+3+5+\cdot \cdot \cdot \cdot +13)\times \frac{1}{7}+(1+3+5+\cdot \cdot \cdot \cdot +15)\times \frac{1}{8}$
$+ (1+3+5+\cdot \cdot \cdot \cdot +17)\times \frac{1}{9}+(1+3+5+\cdot \cdot \cdot \cdot +19)\times \frac{1}{10}$
$ = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10$
$ = 55 $
4. (PLK 1998) 计算 $\frac{1\times 2\times 3+2\times 4\times 6+3\times 6\times 9+4\times 8\times 12+5\times 10\times 15}{1\times 3\times 5+2\times 6\times 10+3\times 9\times 15+4\times 12\times 20+5\times 15\times 25}$
答：
$\frac{(1\times 2\times 3)(1+2+3+4+5)}{(1\times 3\times 5)(1+2+3+4+5)}=\frac{2}{5}$
5. (PLK 1999) 化简 $\left( 1 - \frac{1}{3} \right) $ $\left( 1 - \frac{1}{4} \right) $$\left( 1 - \frac{1}{5} \right) $$\cdot \cdot \cdot \cdot \cdot \left( 1 - \frac{1}{1999} \right) $
答：
$\frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}\times \cdot \cdot \cdot \cdot \cdot \times \frac{1997}{1998}\times \frac{1998}{1999}=\frac{2}{1999}$
6. (PLK 1999) 分数 $\frac{44}{37}$ 可以寫成 $1 + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}$,其中 $x,y,z$ 是部同的整數.請問 $x + y + z$ 的値是多少?
答：
$\frac{44}{37} = \frac{37 + 7}{37} = 1 + \frac{7}{37} = 1 + \frac{1}{\frac{37}{7}}$
$\frac{44}{37} = \frac{37 + 7}{37} = 1 + \frac{7}{37} = 1 + \frac{1}{\frac{37}{7}}$
$= 1+ \frac{1}{\frac{7\times 5+2}{7}} $
$= 1+ \frac{1}{5+ \frac{2}{7}} $
$= 1+ \frac{1}{5+ \frac{1}{\frac{7}{2}}} $
$= 1+ \frac{1}{5+ \frac{1}{\frac{2\times 3+1}{2}}} $
$= 1+ \frac{1}{5+ \frac{2}{7}} $
$= 1+ \frac{1}{5+ \frac{1}{\frac{7}{2}}} $
$= 1+ \frac{1}{5+ \frac{1}{\frac{2\times 3+1}{2}}} $
$= 1 + \frac{1}{5 + \frac{1}{3 + \frac{1}{2}}}$
$ x = 5,y = 3,z = 2$. $x + y + z = 10$
7. (PLK 1999)计算
${1999}^{2} - \left( {1998}^{2} - \left( {1997}^{2} -\left( {1996}^{2} - \left( \cdot \cdot \cdot \cdot \cdot - \left({2}^{2} - {1}^{2} \right) \cdot \cdot \cdot \cdot \cdot \right)\right) \right) \right) $
${1999}^{2} - \left( {1998}^{2} - \left( {1997}^{2} -\left( {1996}^{2} - \left( \cdot \cdot \cdot \cdot \cdot - \left({2}^{2} - {1}^{2} \right) \cdot \cdot \cdot \cdot \cdot \right)\right) \right) \right) $
答：
${{1999}^{2}}-{{1998}^{2}}+{{1997}^{2}}-{{1996}^{2}}+{{1995}^{2}}-\cdot \cdot \cdot \cdot \cdot -{{2}^{2}}+{{1}^{2}}$
$ =\left( {{1999}^{2}}-{{1998}^{2}} \right)+\left( {{1997}^{2}}-{{1996}^{2}} \right)+\cdot \cdot \cdot \cdot \cdot +\left( {{3}^{2}}-{{2}^{2}} \right)+{{1}^{2}}$
$ =(1999+1998)+(1997+1996)+\cdot \cdot \cdot \cdot \cdot +3+2+1 $
$ =\frac{1}{2}(2000)(1999)$
$ =1999000 $
8. (PLK 2003)请问
$\frac{1}{2} + \left( \frac{1}{3} + \frac{2}{3} \right) + \left( \frac{1}{4} + \frac{2}{4} + \frac{3}{4} \right) + \left( \frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{4}{5} \right) \cdot \cdot \cdot \cdot \cdot + \left( \frac{1}{100} + \frac{2}{100} + \cdot \cdot \cdot \cdot \cdot + \frac{99}{100}\right)$
之値等于多少?
答：
$\frac{1}{2}+\frac{1+2}{3}+\frac{1+2+3}{4}+\frac{1+2+3+4}{5}+\cdot \cdot \cdot \cdot \cdot +\frac{1+2+3+4\cdot \cdot \cdot \cdot +99}{100}$
$=\frac{1}{2}+\frac{3}{3}+\frac{1}{4}\frac{3\times 4}{2}+\frac{1}{5}\times \frac{4\times 5}{2}+\cdot \cdot \cdot \cdot \cdot +\frac{1}{100}\times \frac{99\times 100}{2}$
$ =\frac{1}{2}+1+\frac{3}{2}+2+\frac{5}{2}+\cdot \cdot \cdot \cdot \cdot +\frac{99}{2} $
$ =\frac{1+2+3+\cdot \cdot \cdot \cdot +99}{2} $
$ =\frac{99\times 100}{2}=2475 $
9. (PLK 2008)计算
$\frac{\left(1\times 2\times 3 + 2\times 4 \times 6 + 3\times 6 \times 9 + \cdot \cdot \cdot \cdot \cdot + 2008\times 4016 \times 6024\right)}{3 \times 4 \times 5 + 6 \times 8 \times 10 + 9 \times 12 \times 15 + \cdot \cdot \cdot \cdot \cdot + 6024 \times 8032 \times 10040}$
答：
$\frac{(1\times 2\times 3)(1+2+3+\cdot \cdot \cdot \cdot \cdot +2008)}{(3\times 4\times 5)(1+2+3+\cdot \cdot \cdot \cdot +2008)}=\frac{1}{10}$
10. 计算 $(200001+20001+2001+201+21)\div 25$
答：8889
11. 计算
$3333 \times 2222 + 1111 \times 3333 - \frac{5555}{8888} \times 25 \% \times \frac{4}{25}$
答：1108879
12. 计算 $\frac{1}{2}+\frac{1}{2+4}+\frac{1}{2+4+6}+\cdot \cdot \cdot \cdot \cdot +\frac{1}{2+4+6+\cdot \cdot \cdot \cdot +98}$
答： $\frac{49}{50}$
13. 计算 $(\frac{1}{2}+\frac{1992}{1993}+\frac{1993}{1994})\times (\frac{1992}{1993}+\frac{1993}{1994}+\frac{1994}{1995})-(\frac{1992}{1993}+\frac{1993}{1994}+\frac{1994}{1995}+\frac{1}{2})\times (\frac{1992}{1993}+\frac{1993}{1994})$
答：
$A = \frac{1992}{1993}+\frac{1993}{1994},B = \frac{1992}{1993}+\frac{1993}{1994}+\frac{1994}{1995}$,
原式 $ = (\frac{1}{2}+A)\times B-(B+\frac{1}{2})\times B=\frac{1}{2}(B-A)=\frac{997}{1995}$
14. 计算 $ 3-5+7-9+11-13+\cdot \cdot \cdot \cdot +1995-1997+1999-2001+2003$
答：1003
15. 计算$(1-\frac{3}{2\times 4})\times (1-\frac{3}{3\times 5})\times (1-\frac{3}{4\times 6})\times \cdot \cdot \cdot \cdot \times (1-\frac{3}{9\times 11})$
答： $\frac{1}{3}$
16. 计算
$3000 + \frac{3000}{1 + 2} + \frac{3000}{1 + 2 + 3} + \frac{3000}{1 + 2 + 3 + 4} + \cdot \cdot \cdot \ cdot \ cdot + \frac{3000}{1 + 2 + 3 + \cdot \cdot \cdot \cdot \cdot + 2999}$
答： 5998
17. 计算 $\frac{12}{13}+\frac{129}{130}+\frac{1299}{1300}+\cdot \cdot \cdot \cdot +\frac{12999999999999}{13000000000000}$ 的整数部份是多少?
答： 12
$\frac{12}{13}+\frac{129}{130}+\frac{1299}{1300}+\cdot \cdot \cdot \cdot +\frac{12999999999999}{13000000000000}>\frac{12}{13}+\frac{12}{13}+\cdot \cdot \cdot \cdot +\frac{12}{13}=14\times \frac{12}{13}>12 $
$\frac{12}{13}+\frac{129}{130}+\frac{1299}{1300}+\cdot \cdot \cdot \cdot +\frac{12999999999999}{13000000000000}<1+1+1+\cdot \cdot \cdot \cdot +1=13$
18. 計算 $999999\frac{8}{9}+99999\frac{8}{9}+9999\frac{8}{9}+999\frac{8}{9}+99\frac{8}{9}+9\frac{8}{9}+\frac{2}{3}$
答： 1111110
19. 計算 $2011\times 2010-2010\times 2009+2009\times 2008-2008\times 2007+\cdot \cdot \cdot \cdot +5\times 4-4\times 3+3\times 2-2$
答：2022060
20. 計算 $(123456+234561+345612+456123+561234+612345)\div 7$
答：
令 $A=1+2+3+4+5+6=21$
原式 $ = ({{10}^{5}}\times A+{{10}^{4}}\times A+{{10}^{3}}\times A+{{10}^{2}}\times A+10\times A+A)\div 7$
$ = 3\times ({{10}^{5}}+{{10}^{4}}+{{10}^{3}}+{{10}^{2}}+10+1) $
$ = 3\times 111111 $
$ = 333333 $
21. 計算 $2010\times \frac{3}{8}-0.375\times 1949+3\frac{3}{4}\times 1.9$
答：
$ 2010\times \frac{3}{8}-\frac{3}{8}\times 1949+\frac{30}{8}\times \frac{19}{10} $
$ = 2010\times \frac{3}{8}-1949\times \frac{3}{8}+19\times \frac{3}{8} $
$ =\frac{3}{8}(2010-1949+19) $
$ =\frac{3}{8}(80) $
$ = 30 $
22. 計算 $1-(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024})$
答：如图:
原式 = 面积为1的正方形第十次等分后剩下的面积= $\frac{1}{1024}$
23. 计算 $\left( \frac{1}{30} + \frac{1}{41} + \frac{1}{43} + \frac{1}{47} + \frac{1}{53}\right) \times \left( \frac{1}{41} + \frac{1}{43} + \frac{1}{47} + \frac{1}{53} + \frac{1}{67} \right) -
\left( \frac{1}{30} + \frac{1}{41} + \frac{1}{47} + \frac{1}{53} + \frac{1}{67}\right)
\times \left( \frac{1}{41} + \frac{1}{43} + \frac{1}{47} + \frac{1}{53}\right)$
\left( \frac{1}{30} + \frac{1}{41} + \frac{1}{47} + \frac{1}{53} + \frac{1}{67}\right)
\times \left( \frac{1}{41} + \frac{1}{43} + \frac{1}{47} + \frac{1}{53}\right)$
答：
令 $A =\frac{1}{30}+\frac{1}{41}+\frac{1}{43}+\frac{1}{47}+\frac{1}{53}$,
$B =\frac{1}{41}+\frac{1}{43}+\frac{1}{47}+\frac{1}{53}$
原式 $A \times \left( B + \frac{1}{67} \right) - \left( A + \frac{1}{67} \right) \times B $
$ = (A-B)\times \frac{1}{67} $
$ = (A-B)\times \frac{1}{67} $
$ =\frac{1}{30}\times \frac{1}{67}=\frac{1}{2010} $
24. 一个正整数有 2020 各位数,每个位数上的数字都是 9.这个数字自已相乘,所得的积中各个数字上的数字和是多少?
答：
$x=\underbrace{9999............99}_{2010}$
${{\text{x}}^{\text{2}}}=\text{(1}\underbrace{\text{000}.............\text{0}}_{\text{2010}\text{0}}-1)\times (\underbrace{9999............99)}_{20109}$
$ =\underbrace{9999...........99}_{2010\text{9}}\underbrace{\text{0000}..............\text{00}}_{\text{2010}0}-\underbrace{9999...........99}_{20109} $
$ =\underbrace{9999...........99}_{20099}8\underbrace{0000.............00}_{2009\text{0}}\text{1} $
数字和为 $\text{9}\times \text{2009}+\text{8}+\text{1}=\text{18090}$ 
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