You may also use decimal points.
 
Here are the numbers   1   to   20:
 
1   =   (9 + 9) / (9 + 9)
2   =   (9 + √9) / (9 – √9)
3   =   (9 + 9 + 9) / 9
4   =   (√9 * √9) / 9   +   √9
5   =   (9 – √9)   –   (9 / 9)
6   =   (9 * √9) / 9   +   √9
7   =   (9 / 9)   +   √9   +   √9
8   =   (99) / 9   –   √9
9   =   (√9 * √9)   –   (9 – 9)

10   =   (√9 * √9)   +   (9 / 9)
11   =   99 / (√9 * √9)
12   =   (9 * 9) / 9   +   √9
13   =   9   +   √9   +   (9 / 9)
14   =   (99) / 9   +   √9
15   =   (√9 + √9) * √9   –   √9
16   =   (9 / .9)   +   √9   +   √9
17   =   (9 + 9)   –   (9 / 9)
18   =   ((√9 + √9) * 9) / √9
19   =   (9 + 9)   +   (9 / 9)
20   =   (99 / 9)   +   9

 
Please extend the list
 
 

Perimeter   =   20
Area   =   10 √3   ~   17.3205

one of its angles measures 60 degrees:

Applying the Law of cosines

7^2   =   8^2   +   5^2   –   (2*8*5) cos(60)

The triangle has a 60 degrees angle between the sides of length   5   and   8.

How many non-similar triangles with sides of integer length and angle 60 degrees can you find?
 
 
 

And   f(p) = n(p + n),     where  p  is an odd prime

and find  n  so that  f(p)  is a square

1(3 + 1)  =  4  =  2^2                                                        4(5 + 4)  =  36  =  6^2
9(7 + 9)  =  144  =  12^2

25(11 + 25)  =  900  =  30^2                                 36(13 + 36)  =  1764  =  42^2
64(17 + 64)  =  5184  =  72^2                                81(19 + 81)  =  8100  =  90^2
121(23 + 121)  =  17424  =  132^2                  196(29 + 196)  =  44100  =  210^2
225(31 + 225)  =  57600  =  240^2              324(37 + 324)  =  116964  =  342^2

These results indicate that there exists a positive integer   n   such that   n(p + n)   is a perfect square

Could you show that for any odd prime p there’s a unique positive integer   n   such that  
n(p + n)   is a perfect square?

Using  1, 2, 3, 4, 5  only once and the basic operations (+, -, * , / , !) to make all the numbers from  1  to  100
 

0 = 5 + 3 – (4*2/1)                                                5 = 4*3/(5 – 1) + 2
1 = (4 + 5)/3 – 2*1                                                 6 = 4*3/(5 – (1 + 2))
2 = (3 + 5 – 4)/2*1                                                7 = (3*2/1) + 5 – 4
3 = 4 + 5 – (3*2/1)                                                 8 = 4*2/(5 – (3 + 1))
4 = (3 + 5)/( 4- 2*1)                                              9 = 4*3/1 – 5 + 2

10 = 5*(3 – 4/2 + 1)                                             15 = 5*(4 – 3 + 2)/1
11 = 5*2 + 4/2 – 1                                                 16 = (3 + 5) * (4 – 2)/1
12 = 5*2 + 4/(3 – 1)                                             17 = 5*3 + 4 – 2/1
13 = 5*3/1 – 4 + 2                                                18 = 3 * (5 + 4/2 – 1)
14 = 4*(5 + 3 – 1)/2                                             19 = 5*4 – 3/(2 + 1)

20 = (3 + 4/2 – 1) * 5                                          25 = (4 + 3 – 2/1)*5
21 = (4 + 5 – 2) * 3/1                                           26 = 4! + 2/(3! – 5*1)
22 = (3! + 5) * (4 – 2/1)                                      27 = (4 + 2)*5 – 3/1
23 = (5*4 + 3)/(2 – 1)                                          28 = 4!/(5 – 2)!*(3! + 1)
24 = (3 + 5 – 2/1)*4                                             29 = 5*3! – 4/2 + 1

30 = (5 + 3)*4 – 2/1                                             35 = 5*(4!/3 – 2 + 1)
31 = 5*3! + 4/2 – 1                                                36 = 3!*(5 + 4/2 – 1)
32 = 5*3! + 4/1 – 2                                               37 = 5*(4!/3 – 1) + 2
33 = 5*(4 + 3) – 2/1                                              38 =  5!/3 – 4 + 2*1
34 = 5*3! + 4/(2 – 1)                                            39 = 2*(5! – 4 + 1)/3!

40 = 5!/(4 – 3 + 2) * 1                                         45 = 5*(4!/3 + 2 – 1)
41 = 2*(5! + 4 – 1)/3!                                           46 =  5*(4!/2 – 3) + 1
42 = 5!/3 + 4 – 2*1                                               47 =  5!/3 + 4*2 – 1
43 = (4! – 2*5)*3 + 1                                            48 =  (1 + (5 – 3)/2)*4!
44 = 4 * (3! + 5/(2 – 1))                                       49 = (3! + 1)*(4!/2 – 5)

50 = (5!/4 – 3! + 1) * 2                                        55 = 5!*2/4 – 3! + 1
51 = 1 * (5! + 3! – 4!)/2                                       56 = (5 + 1)!/(2*3!) – 4
52 = 4!*(2 + 1) – 5!/3!                                         57 = (5!/(2 + 4) – 1)*3
53 = 5!*2/4 – (3! + 1)                                           58 = (5 + 1)!/(4*3) – 2
54 = 1*(5! – 4!)/2 + 3!                                         59 = 5!/2 + 3 – 4*1

60 = 5!*(3 – 2 + 1)/4                                            65 = 2*(5! – 4!)/3 + 1
61 = 5!*(4 – 3)/2 + 1                                             66 = 4!*(5 + 1)/2 – 3!
62 = 5!/3 + 4! – 2*1                                              67 = 3*4! + 5/(1 – 2)
63 = 5!/(4 – 2) + 3*1                                            68 = 5!/(3 – 1) + 4*2
64 = 1*(5! – 4)/2 + 3!                                           69 = 5!/2 + (4 – 1)*3

70 = 3*4! + (1 – 5)/2                                            75 = 2*5!/3 – (4 + 1)
71 = 5!/2 + 4*3 – 1                                                76 = 4!*3!/2 + 5 – 1
72 = (2 + 1) * (5!/4 – 3!)                                     77 = 3*4!/(2 – 1) + 5
73 = 3*(4! + 2) – 5/1                                            78 = 3*(4! + (5 – 1)/2)
74 = 4*5!/3! – (2 + 1)!                                         79 = 4*5!/3! – 2 + 1

80 = 5!*(3! – 4)/(1 + 2)                                       85 = 3*(4! + 5) – 2/1
81 = 4*5!/3! + 2 – 1                                               86 = 3!*(4! + 5)/2 – 1
82 = 4*(5!/3! + 1) – 2                                           87 = 4*(5!/3! + 2) – 1
83 = 2*5!/3 + 4 – 1                                               88 = (5 + 3) * (4!/2 – 1)
84 = 3*5!/4 – (2 + 1)!                                          89 = (5!*3)/4 – 2 + 1

90 = (4 + 1) * (5!/3! – 2)                                     95 = (5!*4)/(2 + 3) – 1
91 = (5!*3)/4 + 2 – 1                                             96 = (5! – 4!) * 3/(2 + 1)
92 = 5*(4! – 3!) + 2/1                                         97 = (3!)!/5 + 1 – 2*4!
93 = 5! – (4! + 1*3!/2)                                        98 = (5!*4)/(3! – 1) + 2
94 = 5! – (4*3! + 2/1)                                          99 =  5! + 3!/2 – 4!*1

100 = 5*(4! – 3! + 2)/1

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

and find numbers that repeat the digits of the chosen prime and divisible by this prime
 

For example,

19,   1919

11191   is divisible by 19,        11191   =   19 * 19 * 31

17,   1717

22333   =   23 * 971
44333   =   43 * 1031
555533333   =   53 * 10481761

13,   1131,   1313,   3133

Divisible by   11:
11,   1111,   111111,   11111111,   1111111111, …

N.B.   numbers of the form   abababab…   are not the examples I’m looking for.

E.g.   1313,   131313…   1717,   171717…,   191919…

What I consider to be good examples are:

11191,   1131,   3133,   22333,   44333,   555533333

Find larger ones (if possible) and find other numbers using the remaining 2-digit numbers.

Find integers   x,   y   and   z   such that

                                                 (x – y + z)^3   =   x || y || z

                                                 (-x + y + z)^3   =   x || y || z

                                                 (x + y + z)^3   =   x || y || z

                                                 (x + y – z)^3   =   x || y || z

For example,

(9 + 11 + 25)^3    =    91125    =    9 || 11 || 25    =    45^3

(786 – 330 + 467)^3   =   786330467   =   786 || 330 || 467

(188 – 132 + 517)^3    =    188132517    =    188 || 132 || 517

(258 – 474 + 853)^3    =    258474853    =    258 || 474 || 853

(-360 + 944 + 128)^3    =    360944128    =    360 || 944 || 128

 
 
And,

(13 – 3 + 1)^3   =   1331
(103 – 03 + 01)^3   =   1030301
(1003 – 003 + 001)^3   =   1003003001
(10003 – 0003 + 0001)^3   =   1000300030001
(100003 – 00003 + 00001)^3   =   1000030000300001
(1000003 – 000003 + 000001)^3   =   1000003000003000001

(106 – 12 + 08)^3   =   1061208
(1006 – 012 + 008)^3   =   1006012008
(10006 – 0012 + 0008)^3   =   1000600120008
(100006 – 00012 + 00008)^3   =   1000060001200008
(1000006 – 000012 + 000008)^3   =   1000006000012000008

 
 
@InfinitelyManic found:

(5 + 1 + 2)^3   =   512   =   5 || 1 || 2
(3 – 723 + 875)^3   =   3723875   =   3 || 723 || 875
(9 + 73 – 36)^3   =   97336   =   9 || 73 || 36
(9 – 129 + 329)^3   =   9129329   =   9 || 129 || 329
(19 + 51 – 12)^3   =   195112   =   19 || 51 || 12
(20 – 570 + 824)^3   =   20570824   =   20 || 570 || 824
(23 – 393 + 656)^3   =   23393656   =   23 || 393 || 656
(23 – 639 + 903)^3   =   23639903   =   23 || 639 || 903
(43 – 243 + 551)^3   =   43243551   =   43|| 243 || 551
(45 + 65 – 33)^3   =   456533   =   45 || 65 || 33
(48 – 228 + 544)^3   =   48228544   =   48 || 228 || 544
(78 – 402 + 752)^3   =   78402752   =   78 || 402 || 752
(-123 + 263 + 91)^3   =   12326391   =   123 || 263 || 91
 
 
 

You can use the primes   2, 3, 5, 7   and the basic operations   +, -, *, /   and   !   (factorial) only once to make numbers from   1   to   100.
 

N.B.    you may use decimal points
 

For example,

0   =   (7 + 3)/5 – 2
1   =   (7 – 5) – (3 – 2)     or     1   =   (3 * 5) – (2 * 7)
2   =   7 – 3! + (3 – 2)
3   =   (7 – 2) – (5 – 3)
4   =   2*3 – (7 – 5)

 
How many other numbers can you express?
 
 
 
 
 
 

288   is a non-palindrome and a non-square number.

Its reverse is:     Rev (288)   =   882

and,

288   *   882   =   254016   =   504^2   is a perfect square.

 
882 * 288   =   504^2
80802 * 20808   =   41004^2
8008002 * 2008008   =   4010004^2
800080002 * 200080008   =   400100004^2
80000800002 * 20000800008   =   40001000004^2
8000008000002 * 2000008000008   =   4000010000004^2

and so on.

                                                 ——————————————
 
David @InfinitelyManic found :

528 * 825   =   660^2
768 * 867   =   816^2
1584 * 4851   =   2772^2
2178 * 8712   =   4356^2
10989 * 98901   =   32967^2
13104 * 40131   =   22932^2
14544 * 44541   =   25452^2
15984 * 48951   =   27972^2
20808 * 80802   =   41004^2
21978 * 87912   =   43956^2
26208 * 80262   =   45864^2
27648 * 84672   =   48384^2
27848 * 84872   =   48616^2
36828 * 82863   =   55242^2
48139 * 93184   =   66976^2
48951 * 15984   =   27972^2
49686 * 68694   =   58422^2
57399 * 99375   =   75525^2
 
 
 
 
 

Integers such that the sum of prime factors equal the product of the digits of those integers:
 

The prime factors of   18   are:                           2 * 3 * 3
The sum is:                                                                 2  +  3  +  3  =  8
DigitProduct (18)  =                                                1 * 8  =  8

The prime factors of  25  are:                             5 * 5
The sum is:                                                                  5  +  5  =  10
DigitProduct (25)  =                                                2 * 5  =  10

Find other examples.
 

                                                 ——————————————
 

(2)

Integers such that the sum of prime factors and the product of the digits are perfect powers:
 
The prime factors of  14  are:                              2 * 7
The sum is:                                                                  2  +  7  =  9  =  3^2
DigitProduct (14) =                                                 1 * 4  =  2^2

The prime factors of  24  are:                             2 * 2 * 2 * 3
The sum is:                                                                  2  +  2  +  2  +  3  =  9  =  3^2
DigitProduct (24) =                                                 2 * 4  =  8  =  2^3

The prime factors of  39  are:                             3 * 13
The sum is:                                                                  3  +  13  =  16  =  4^2
DigitProduct (39) =                                                 3 * 9  =  27  =  3^3

The prime factors of  55  are:                             5 * 11
The sum is:                                                                  5  +  11  =  16  =  4^2
DigitProduct (55) =                                                 5 * 5  =  5^2

The prime factors of  66  are:                             2 * 3 * 11
The sum is:                                                                  2  +  3  +  11  =  4^2
DigitProduct (66) =                                                 6 * 6  =  6^2

The prime factors of  94  are:                             2 * 47
The sum is:                                                                  2  +  47  =  49  =  7^2
DigitProduct (94) =                                                 9 * 4  =  36  =  6^2

 
Find other such integers.
 

                                                 ——————————————
 

(3)

The product of the digits of   236   is the reverse of the sum of its prime factors:
 

The prime factors of  236  are:                          2 * 2 * 59
The sum is:                                                                  2  +  2  +  59  =  63
DigitProduct (236) =                                              2 * 3 * 6  =  36

 
Find other integers with the same property
 

                                                 ——————————————
 
(4)

Digit Permutations:

The product of the digits of   2537   is a digit permutation of the sum of its prime factors

2537   =   43 * 59

Sum of prime factors:     43 + 59   =   102

DigitProduct (2537)   =   2 * 5 * 3 * 7   =   210

2537   is also a multiple of a prime containing only prime digits   (2, 3, 5, 7)

(123 + 132 + 213 + 231 + 312 + 321)/2 = 666
(123 + 132 + 213 + 231 + 312 + 321)/3 = 444
(123 + 132 + 213 + 231 + 312 + 321)/4 = 333
(123 + 132 + 213 + 231 + 312 + 321)/6 = 222
(123 + 132 + 213 + 231 + 312 + 321)/9 = 148

234 + 243 + 324 + 342 + 423 + 432 = 1998

(234 + 243 + 324 + 342 + 423 + 432)/2 = 999
(234 + 243 + 324 + 342 + 423 + 432)/3 = 666
(234 + 243 + 324 + 342 + 423 + 432)/6 = 333
(234 + 243 + 324 + 342 + 423 + 432)/9 = 222

345 + 354 + 435 + 453 + 534 + 543 = 2664

(345 + 354 + 435 + 453 + 534 + 543)/2 = 1332
(345 + 354 + 435 + 453 + 534 + 543)/3 = 888
(345 + 354 + 435 + 453 + 534 + 543)/4 = 666
(345 + 354 + 435 + 453 + 534 + 543)/6 = 444
(345 + 354 + 435 + 453 + 534 + 543)/8 = 333
(345 + 354 + 435 + 453 + 534 + 543)/9 = 296

456 + 465 + 546 + 564 + 645 + 654 = 3330

(456 + 465 + 546 + 564 + 645 + 654)/2 = 1665
(456 + 465 + 546 + 564 + 645 + 654)/3 = 1110
(456 + 465 + 546 + 564 + 645 + 654)/5 = 666
(456 + 465 + 546 + 564 + 645 + 654)/6 = 555
(456 + 465 + 546 + 564 + 645 + 654)/9 = 370

567 + 576 + 657 + 675 + 756 + 765 = 3996

(567 + 576 + 657 + 675 + 756 + 765)/2 = 1998
(567 + 576 + 657 + 675 + 756 + 765)/3 = 1332
(567 + 576 + 657 + 675 + 756 + 765)/4 = 999
(567 + 576 + 657 + 675 + 756 + 765)/6 = 666
(567 + 576 + 657 + 675 + 756 + 765)/9 = 444

678 + 687 + 768 + 786 + 867 + 876 = 4662

(678 + 687 + 768 + 786 + 867 + 876)/2 = 2331
(678 + 687 + 768 + 786 + 867 + 876)/3 = 1554
(678 + 687 + 768 + 786 + 867 + 876)/6 = 777
(678 + 687 + 768 + 786 + 867 + 876)/7 = 666
(678 + 687 + 768 + 786 + 867 + 876)/9 = 518

789 + 798 + 879 + 897 + 978 + 987 = 5328

(789 + 798 + 879 + 897 + 978 + 987)/2 = 2664
(789 + 798 + 879 + 897 + 978 + 987)/3 = 1776
(789 + 798 + 879 + 897 + 978 + 987)/4 = 1332
(789 + 798 + 879 + 897 + 978 + 987)/6 = 888
(789 + 798 + 879 + 897 + 978 + 987)/8 = 666
(789 + 798 + 879 + 897 + 978 + 987)/9 = 592

135 + 153 + 315 + 351 + 513 + 531 = 1998

(135 + 153 + 315 + 351 + 513 + 531)/2 = 999
(135 + 153 + 315 + 351 + 513 + 531)/3 = 666
(135 + 153 + 315 + 351 + 513 + 531)/6 = 333
(135 + 153 + 315 + 351 + 513 + 531)/9 = 222

357 + 375 + 537 + 573 + 735 + 753 = 3330

(357 + 375 + 537 + 573 + 735 + 753)/2 = 1665
(357 + 375 + 537 + 573 + 735 + 753)/3 = 1110
(357 + 375 + 537 + 573 + 735 + 753)/5 = 666
(357 + 375 + 537 + 573 + 735 + 753)/6 = 555
(357 + 375 + 537 + 573 + 735 + 753)/9 = 370

579 + 597 + 759 + 795 + 957 + 975 = 4662

(579 + 597 + 759 + 795 + 957 + 975)/2 = 2331
(579 + 597 + 759 + 795 + 957 + 975)/3 = 1554
(579 + 597 + 759 + 795 + 957 + 975)/6 = 777
(579 + 597 + 759 + 795 + 957 + 975)/7 = 666
(579 + 597 + 759 + 795 + 957 + 975)/9 = 518

024 + 042 + 204 + 240 + 402 + 420 = 1332

(024 + 042 + 204 + 240 + 402 + 420)/2 = 666
(024 + 042 + 204 + 240 + 402 + 420)/3 = 444
(024 + 042 + 204 + 240 + 402 + 420)/4 = 333
(024 + 042 + 204 + 240 + 402 + 420)/6 = 222
(024 + 042 + 204 + 240 + 402 + 420)/9 = 148

246 + 264 + 426 + 462 + 624 + 642 = 2664

(246 + 264 + 426 + 462 + 624 + 642)/2 = 1332
(246 + 264 + 426 + 462 + 624 + 642)/3 = 888
(246 + 264 + 426 + 462 + 624 + 642)/4 = 666
(246 + 264 + 426 + 462 + 624 + 642)/6 = 444
(246 + 264 + 426 + 462 + 624 + 642)/8 = 333
(246 + 264 + 426 + 462 + 624 + 642)/9 = 296

468 + 486 + 648 + 684 + 846 + 864 = 3996

(468 + 486 + 648 + 684 + 846 + 864)/2 = 1998
(468 + 486 + 648 + 684 + 846 + 864)/3 = 1332
(468 + 486 + 648 + 684 + 846 + 864)/4 = 999
(468 + 486 + 648 + 684 + 846 + 864)/6 = 666
(468 + 486 + 648 + 684 + 846 + 864)/9 = 444