# 1990 AHSME Problems/Problem 13

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## 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 $\fbox{E}$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 