1990 AHSME Problems/Problem 13

Problem

If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$?

1. START $X$ AT $3$ AND $S$ AT $0$.  
2. INCREASE THE VALUE OF $X$ BY $2$.   
3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$. 
4. IF $S$ IS AT LEAST $10000$,   
       THEN GO TO INSTRUCTION $5$;  
       OTHERWISE, GO TO INSTRUCTION $2$.  
       AND PROCEED FROM THERE.  
5. PRINT THE VALUE OF $X$.  
6. STOP. 

$\text{(A) } 19\quad \text{(B) } 21\quad \text{(C) } 23\quad \text{(D) } 199\quad \text{(E) } 201$

Solution

Looking at the first few values, it becomes clear that the program stops when \[5+7+9+11+13+\ldots+(x-2)+x\ge 10000\] which is to say \[1+3+5+7+9+11+13+\ldots+(x-2)+x\ge 10004\] However, the left hand side is now simply the square $\frac{(x+1)^2}4$. Multiplying out, we get \[x+1\ge \sqrt{40016}\approx 200.039996...\]

So the correct answer is $201$, which is $\fbox{E}$

See also

1990 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
All AHSME Problems and Solutions


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