Maths olympiad

1)6x = 8ˣ
2(3x) = 2^(3x)
3x = 2^(3x - 1)Let 3x - 1 be y , then
y+1 = 2ʸFor y > 1 , y+1 < 2ʸ , no solutions.For y = 1 , it is a solution.
i.e. 3x - 1 = 1 , x = 2/3For 0 < y < 1 , y+1 > 2ʸ , no solutions.For y = 0 , it is a solution.
i.e. 3x - 1 = 0 , x = 1/3So x = 1/3 or x = 2/3 .

2)54321 ^ 43210
= (54321)^(149 x 290)54321 ^ 149
=...+ 149C4 * 54320⁴+ 149C3 * 54320³ + 149C2 * 54320² + 149 * 54320 + 1149C4 * 54320⁴(mod 100000)
= 149C4 * 5432⁴(mod 10) * 10000
= 19720001 * 2⁴(mod 10) * 10000
= 1 * 16 (mod 10) * 10000
= 60000149C3 * 54320³ (mod 100000)
= 540274 * 5432³ (mod 100) * 1000
= 74 * 32³ (mod 100) * 1000
= 2424832 (mod 100) * 1000
= 32000149C2 * 54320² (mod 100000)
= 11026 * 5432² (mod 1000) * 100
= 26 * 432² (mod 1000) * 100
= 4852224 (mod 1000) * 100
= 22400149 * 54320 (mod 100000)
= 93680So 54321^149
= 60000 + 32000 + 22400 + 93680 + 1
= 208081
= 8081 (mod 100000) So
54321 ^ 43210 (mod 100000)
= 8081 ^ 290 (mod 100000)
= ... + 290C3 * 8080³ + 290C2 * 8080² + 290 * 8080 + 1 (mod 100000)290C3 * 8080³ (mod 100000)
= 4022880 * 808³ (mod 100) * 1000
= 80 * 8³ (mod 100) * 1000
= 40960 (mod 100) * 1000
= 60000 290C2 * 8080² (mod 100000)
= 41905 * 808² (mod 1000) * 100
= 905 * 652864 (mod 1000) * 100
= 905 * 864 (mod 1000) * 100
= 781920 (mod 1000) * 100
= 92000290 * 8080 (mod 100000)
= 2343200 (mod 100000)
= 43200Hence 54321 ^ 43210 = 60000 + 92000 + 43200 + 1 = 195201
= 95201 (mod 100000)

3)17x⁴ + 34ax³ + (25a²+10)x² + (8a³ +10a)x + (a⁴+25) = 0
(16x⁴ + 32ax³ + 24a²x² + 8a³x + a⁴) + (x⁴+ 2ax³ + (a²+10)x² + 10ax + 25) = 0
(2x + a)⁴+ (x⁴ + 2ax³ + 10x² + a²x² + 10ax + 25) = 0
(2x + a)⁴+ (x⁴ + 2x² (ax + 5) + (ax + 5)²) = 0
(2x + a)⁴+ (x² + ax + 5)² = 0
==>
2x + a = 0 and x² + ax + 5 = 0x = - a/2 , sub. into x² + ax + 5 = 0 :a²/4 - a²/2 + 5 = 0
a² = 20∴ The product of all possible values of a = - 20

4)Let ABCD be the four-digit positive integer. (A , B , C ,D = 1 to 9)Case 1 :
OOOO type , for example : 8888
9 ways
Case 2 :
OOXX type , for example : 2288 , 6116 , 9595.
9C2 * 4!/(2! 2!) = 216 ways

2012-05-20 19:30:25 補充：
Case 3 :
OOOX type
If O is not a perfect square , then O , X = 2 , 8
2P2 * 4C1 = 8 ways

If O is a perfect square , then X must be another perfect square ,
for example , 4449 , 9994 , 1999.
3P2 * 4 = 24 ways

Total = 8 + 24 = 32 ways

2012-05-20 19:30:38 補充：
Case 4 :
OOXY type
Then X , Y = 1 , 4 , 9 or X , Y = 2 , 8

7 * 3C2 * 4! / 2! + 7 * 2C2 * 4! / 2!
= 252 + 84 = 336 ways

2012-05-20 19:30:51 補充：
Case 5 :
OXYZ type

Type 1 : For example , when O*X , Y*Z are both perfect square ,
then O*X , Y*Z = 1*4 , 2*8 or 4*9 , 2*8 or 1*9 , 2*8
i.e. O , X , Y , Z = 1 , 4 , 2 , 8 or 4 , 9 , 2 , 8 or 1 , 9 , 2 , 8
4P4 * 3 = 72 ways

2012-05-20 19:31:02 補充：
Type 2 : For example , when O*X*Y , Z are both perfect square ,
then O*X*Y = 2*3*6 , Z = 1 , 4 , 9 or O*X*Y = 2*4*8 , Z = 1 , 9
3C1 * 4P4 + 2C1 * 4P4 = 120 ways

2012-05-20 19:31:08 補充：
Type 3 : For example , when O*X*Y*Z is a perfect square ,
then O*X*Y*Z = 1 * 2 * 3 * 6
4P4 = 24 ways

Total 72 + 120 + 24 = 216 ways

∴ There are 9 + 216 + 32 + 336 + 216 = 809 four-digit positive integers have their product of digits being a positive square number.

2012-05-20 19:33:31 補充：
自由自在( 知識長 ) :

You help me a lot ! Thankyou very much!

2012-05-20 19:41:24 補充：
起初我以為第4題最易 , 怎料原來它才是最難的!

2012-05-21 16:44:46 補充：
6169 is included in case4 OOXY ==> 6619.

第２題，由 Euler-Carmichael 定理，對任何 (a, 10^5) = 1, a^5000 ≡ 1 (mod 10^5)。

輕易可得：
54321^1250 ≡ 1 (mod 10^5)
4321^625 ≡ 1 (mod 10^5)

因此，
54321^43210
≡ 4321^85
≡ (4320+1)^85
≡ 1 + (85)(4320) + (85)(42)(4320)^2 + (85)(83)(14)(4320)^3
≡ 1 + 67200 + 68000 + 60000
≡ 95201 (mod 10^5)

2012-05-21 17:52:15 補充：
也可用 321^625 ≡ 1 (mod 10^5)：

54321^43210
≡ (321^85) + 40000
≡ (320 + 1)^85 + 40000
≡ 40001 + (85)(320) + (85)(42)(320)^2 + (85)(83)(14)(320)^3
≡ 40001 + 27200 + 68000 + 60000
≡ 95201 (mod 10^5)。

Alternate method for question 2:

rewrite 54321 ^ 43210

= (50000 + 4000 + 300 + 20 + 1) ^ (43210)

= (1 ^ 43210)
+ (20 ^ 1) x (1 ^ 43209 ) x 43210 C 1
+ (20 ^ 2) x (1 ^ 43208 ) x 43210 C 2
+ (20 ^ 3) x (1 ^ 43207 ) x 43210 C 3
+ (20 ^ 4) x (1 ^ 43206 ) x 43210 C 4
+ (300 ^ 1) x (1 ^ 43209 ) x 43210 C 1
+ (300 ^ 2) x (1 ^ 43208 ) x 43210 C 2
+ (4000 ^ 1) x (1 ^ 43209 ) x 43210 C 1
+ (20 ^ 1) x (300 ^ 1) x (1 ^ 43208 ) x 43210 C 1 x 43209 C 1
+ (20 ^ 2) x (300 ^ 1) x (1 ^ 43207 ) x 43210 C 2 x 43208 C 1
+ terms divisible by 100000

= 1
+ 20 x 43210
+ 400 x (43210 x 43209 / 2)
+ 8000 x (43210 x 43209 x 43208 / 6)
+ 160000 x (43210 x 43209 x 43208 x 43207/ 24)
+ 300 x 43210
+ 90000 x (43210 x 43209 / 2)
+ 4000 x 43210
+ 6000 x (43210 x 43209)
+ 120000 x (43210 x 43209 x 43208 / 2)
+ terms divisible by 100000

= 1 + 64200 + 78000 + 60000 + 63000 + 50000 + 40000 + 40000 + 10A

= 95201 + 10A (where A is an integer)

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