Adding without using a + or - sign

R (24 characters)

length(sequence(scan()))

What this does:

• scan reads input from STDIN (or a file)
• sequence generates integer sequences starting from 1 and concatenates the sequences. For example, sequence(c(2, 3)) results in the vector 1 2 1 2 3
• length calculates the number of elements in the concatenated vector

Example 1:

> length(sequence(scan()))
1: 123
2: 468
3:
Read 2 items
[1] 591

Example 2:

> length(sequence(scan()))
1: 702
2: 720
3:
Read 2 items
[1] 1422

Perl (no +/-, no tie-breakers, 29 chars)

s!!xx!;s!x!$"x<>!eg;say y!!!c

As a bonus, you can make the code sum more than two numbers by adding more xs to the s!!xx!.

Alternatively, here are two 21-char solutions with 1 and 3 tie-breakers respectively

say length$"x<>.$"x<>

say log exp(<>)*exp<>

Note: These solutions use the say function, available since Perl 5.10.0 with the -E command line switch or with use 5.010. See the edit history of this answer for versions that work on older perls.

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How does the solution with no tie-breakers work?

• s!!xx! is a regexp replacement operator, operating by default on the $_ variable, which replaces the empty string with the string xx. (Usually / is used as the regexp delimiter in Perl, but really almost any character can be used. I chose ! since it's not a tie-breaker.) This is just a fancy way of prepending "xx" to $_ — or, since $_ starts out empty (undefined, actually), it's really a way to write $_ = "xx" without using the equals sign (and with one character less, too).

• s!x!$"x<>!eg is another regexp replacement, this time replacing each x in $_ with the value of the expression $" x <>. (The g switch specifies global replacement, e specifies that the replacement is to be evaluated as Perl code instead of being used as a literal string.) $" is a special variable whose default value happens to be a single space; using it instead of " " saves one char. (Any other variable known to have a one-character value, such as $& or $/, would work equally well here, except that using $/ would cost me a tie-breaker.)

  The <> line input operator, in scalar context, reads one line from standard input and returns it. The x before it is the Perl string repetition operator, and is really the core of this solution: it returns its left operand (a single space character) repeated the number of times given by its right operand (the line we just read as input).

• y!!!c is just an obscure way to (ab)use the transliteration operator to count the characters in a string ($_ by default, again). I could've just written say length, but the obfuscated version is one character shorter. :)

D

main(){
    int a,b;
    readf("%d %d",&a,&b);
    while(b){a^=b;b=((a^b)&b)<<1;}
    write(a);
}

bit twiddling for the win

as a bonus the compiled code doesn't contain a add operation (can't speak for the readf call though)