merican Computer cience eague 1996í97 ll-tar Contest olutions Round è1 1. The answer shown is the only correct answer. 2*è*B-CèDè 2. The expression is

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1 Name: chool: Div: ID: 1996í97 ll-tar Contest hort Problems Round è1 eave this box blank! 1. ëjrë BIC Expressions Write a BIC expression that is an exact translation of the following: Twice the diæerence of the product of and B and the quotient of C and D Use the fewest number of parentheses as possible. 2. ëjr, Int, rë Boolean lgebra ist all of the following choices that are equivalent to the expression è æ Bèè + Bèè + Bè èaè æ B èbè + B ècè B èdè + B èeè æ B 3. ëint, rë What Does This Program Do If the following program prints the number 9 when it is run, list all possible integer inputs to the program. 10 input : B=3 : C=1 20 if èé=bè and èècéè or èbécèè then = if èébè and èbécè then = +C 40 if èèécè and èébèè or èbécè then = -C 50 if é5 then =+2 else =+5 60 print 70 end 4. ëint, rë PreæxèPostæxèInæx Notation et è represent the unary operation ëgreatest integer less than or equal to the value of its operand." For example, the postæx expression 26 3 è è has a value of 8. Evaluate the following preæx expression: + è è * è è è è - æ

2 merican Computer cience eague 1996í97 ll-tar Contest olutions Round è1 1. The answer shown is the only correct answer. 2*è*B-CèDè 2. The expression is simpliæed by applying DeMorgan's aw twice and then the deænition of XOR: è æ Bèè + Bèè + Bè =è æ Bèèè + Bè+è + Bèè =èæbèèè æ Bè+èæ Bèè =èæbèèèbè+èbèè =èæbèè æ Bè =èæbè èeè The only choice equivalent to æ B is choice èeè. 3. We can solve this program by trying small values, say between,10 and 10, looking for inputs that cause the program to print 9. If you follow this approach, you need to be able to convince yourself that all values less than,10 and greater than 10 won't work. lternatively, work backwards, one line at a time: ine 50: fter this line is executed, must have avalue of 9. If the then were executed, must have started out equal to 7. If the else were executed, must have started out with a value of 4. ine 40: fter this line is executed, must have avalue of 4 or 7. The then is always executed because BéC. Thus, must have started out with avalue of 5 or 8. 3, 5, and 8 ine 30: fter this line is executed, must have avalue of 5 or 8. If the then were executed, then must have started out equal to 4 or 7. But these are both impossible, because must be less than B. Therefore, the then was not executed, and started out with a value of 5 or 8. ine 20: fter this line is executed, must have a value of 5 or 8. Consider a ænal value of 8. If the then were executed, started out with a value of 6. But this is impossible, since must be less than B; thus started out equal to 8. Consider a ænal value of 5. If the then were executed, then started out with a value of 3. This is OK. If the then were not executed, then start out with a value of 5. This is also OK. 4. The evaluation is as follows èa box is put around the part of the expression that is about to be evaluatedè: + è è * è è è è - æ è è 6 4 è è -2 è è è 1.5 è -2 è è è -2 è the fraction 3è5 is also acceptable + 1 è

3 Name: chool: Div: ID: 1996í97 ll-tar Contest hort Problems Round è2 eave this box blank! 5. ëjrë What Does This Program Do When the following program is run, the string ëhi mom!" is printed three times. Find all integer values that could be given as input. 10 input x : y = 7 20 if èx é 2è and èy é xè then print "hi mom!" 30 if èx é yè or èx é 5è then print "hi mom!" 40 if èx éé 0è and èx*x é 5è then print "hi mom!" 50 end 6. ëjr, Int, rë Digital Electronics Find all ordered triples that make the circuit below true. B C 7. ëint, rë Pascal fter the following program is run, what is the ænal value of a program lltar97; var a, b, c: integer; procedure two èvar c, b, a: integerè; begin a:=a-2; b:=b-1; c:=a+b; end; procedure one èvar a, b, c: integerè; begin a:=a-1; b:=b-1; c:=a+b; twoèa,b,cè; end; begin a:=20; b:=10; c:=0; while a é 0 do begin oneèa,b,cè; c:=c+1; end; end. 8. ëint, rë isp Evaluate the following expression: ècr èrevere ècdr ècdr 'èè1 2è è3 4è è5 6 7èèèèèè

4 merican Computer cience eague 1996í97 ll-tar Contest olutions Round è2 5. In order for the string to be printed three times, it must be printed in lines 20, 30, and 40. From line 20, x =1; 0;,1;,2;::: will cause the string to be printed. ll of these values will cause line 30 to print the string. In line 40, both clauses of the if must be true for the string to be printed. The clause xéé0 eliminates 0 from the possible values of x. The clause x*xé5 is true only when x=1,,1, and,2. 1,,1, and,2 6. The circuit can be represented by the Boolean lgebra expression èè æ BèèB + Cèè We could construct a truth table for this expression, but because it's an ND of three terms, it's worth trying to reason about it. The term means that must be true. This gives us 4 possible triples to consider: è1,0,0è, è1,0,1è, è1,1,0è, and è1,1,1è. The term æ B means that and B must be diæerent. This leaves just è1,0,0è and è1,0,1è. The term B + C means than either B or C must be true. This leaves just è1,0,1è. è1,0,1è 7. The table at the right shows the values of variables just after c is incremented in the main loop. a b c , , ,6 27 2,8 11,20,10,9,20 8. The evaluation goes from the inside to the outside: ècdr 'èè1 2è è3 4è è5 6 7èèè è èè3 4èè5 6 7èè ècdr 'èè3 4è è5 6 7èèè è èè5 6 7èè èrevere 'èè5 6 7èèè è èè5 6 7èè ècr 'èè5 6 7èèè è è5 6 7è è5 6 7è

5 Name: chool: Div: ID: 1996í97 ll-tar Contest hort Problems Round è3 eave this box blank! 9. ëjr, Intë Computer Number ystems Express the square of in base ëjr, Int, rë Bit tring Flicking Find all values of x, 5 bits long, that make the following expression equal to èhift-2 è10101 OR xèè XOR èrcirc-2 è11011 ND ècirc-2 xèèè 11. ëint, rë Recursive Functions Evaluate f è22è given the following: fèxè = 8 é é: x + fèx +3è whenever x is prime and larger than 2 x æ fèx, 5è whenever x is even 2 x +1 otherwise 12. ërë Data tructures If you built a binary search tree with the keystkeèin this orderè, you'd end up with a tree whose internal path length is 18. uppose that you could swap up to 2 pairs of letters èe.g., the with the ærst and the T with the Kè to insert the letters in a diæerent order èe.g., KTE,using the 2 swaps aboveè. What is the minimum internal path length a resulting tree could have

6 merican Computer cience eague 1996í97 ll-tar Contest olutions Round è3 9. We could convert the number to base 10, square it, and then convert back to base 16. lternatively, we could do all the multiplication in base 16. We'll do the latter, but take advantage of the fact that to make the multiplication simple. èa + bè 2 = a 2 +2ab + b = è è 2 = æ 800 æ = æ = = B =è70+8è 2 = æ70 æ = = 3840 Finally, = 47B Write x as abcde and simplify each side: eft side of XOR: èhift-2 è10101 OR abcdeèè èhift-2 1b1d1è 1d100 Right side of XOR: èrcirc-2 è11011 ND ècirc-2 abcdeèèè èrcirc-2 è11011 ND cdeabèè èrcirc-2 cd0abè abcd0 Now, these two values XOR'd together must equal That is, XOR abcd0 = ook at each bit, from left-to-right: 1d and the term 0000* is also acceptable 1 XOR a = 1 implies a=0 d XOR b = 0 implies b=d 1 XOR c = 1 implies c=0 0 XOR d = 0 implies d=0, and thus b=0 11. The evaluation is as follows: fè22è = 22=2 æ fè17è = 11 æ fè17è fè17è = 17 + fè20è fè20è = 20=2 æ fè15è = 10 æ fè15è fè15è = Working backwards, we have f è20è = 10æ16 = 160, f è17è = = 177, and fè22è = 11 æ 177 = The shortest internal path length occurs when the binary tree is a complete tree. With 8 nodes, the shortest internal path length is 13. uch a tree can be formed with just one swap: the K and the. èother exchanges will also produce a tree whose internal path length is 13.è 13

7 Name: chool: Div: ID: 1996í97 ll-tar Contest hort Problems Round è4 eave this box blank! 13. ëint, rë ssembly anguage What is the ænal value of C when the following program is run DC 2 B DC 50 C DC 0 TOP OD C DD =1 TORE C OD UB B TORE B UB BG END C TOP 14. ëint, rë Fs and Regular Expressions How many diæerent strings of length 5 are generated by the following regular expression: èww+è æ C æ 15. ëint, rë Data tructures Insert the nodes T KEinto a minimal heap. That is, a heap where each node is less than or equal to its two children. èthis is diæerent than the heap described in the C Category Description booklet, where nodes were greater than their children.è Draw the resulting heap.

8 merican Computer cience eague 1996í97 ll-tar Contest olutions Round è4 13. The high-level program equivalent to the program is as follows: = 2 B = 50 C = 0 TOP: C = C + 1 B = B - IF B-C é 0 THEN GOTO TOP END The following table shows the values of B and C just before the BG is executed: B C ll strings must end with an and must contain a C. Between the C and, there are zero or more 's. Before the C, there's either nothing, or some combination of and WW, for instance WW,, or WW. The complete list of 5-letter strings are as follows: C, C, C, C, WWC, WWC, and WWC Here's what the tree looks like after each letter is added: T E K T T T T K