奧林匹克數學題(小學)

1) 2/15 + 2/60 + 2/48 + 2/144 + 2/99 + 2/264 + 2/168 + 2/420
=(2/15 + 2/60) + (2/48 + 2/144) + (2/99 + 2/264) +( 2/168 + 2/420)
=1/6+1/18+1/36+1/60
=(10+1/3*10+1/6*10+1)/60
=(11+5)/60
=4/15

2) 11,211 x 11,011
= 11,111 x 11,011 + 1,101,100
=11,111 x 11,111+1,101,100-1,111,100
=123,454,321-10,000
=123,444,321

3) similarly,
111,121,111 × 111,101,111=12,345,678,887,654,321
therefore, 111,121,111 × 111,101,111 ÷ 11,111,111
= 12,345,678,887,654,321 ÷ 11,111,111
=1,111,111,111

4)
since, 1/2=0.5
1/3+2/3=1
1/4+2/4+3/4=1.5
1/5+2/5+3/5+4/5=2

then, when follow sequence, we have
1/2+1/3+2/3+1/4+2/4+3/4+1/5+2/5+3/5+4/5+...+1/100+2/100+...+99/100
=1/2+(1/3+2/3)+(1/4+2/4+3/4)............+(1/100+2/100+...+99/100)
=0.5+1+1.5+2........+49.5
=(1+2+3.....+99)/2
=4950/2
=2475

5) Let 19991999=Y
19991999 × 19991998-19992000 × 19991997
=[Y x (Y-1)] -[(Y+1) x (Y-2)]
=Y^2-Y-Y^2+Y+2
=2

111121111 = 11111 x 10001
111101111 = 1111 x 100001
111121111 x 111101111 = 11111 x 10001 x 1111 x 100001
= 11111 x 100001 x 1111 x 10001
= 1111111111 x 11111111