Starting from the point (0,0), a grasshopper makes a series of moves on the coordinate plane. The grasshopper's first move takes her to (3,4). She then proceeds via this rule:

since the coordinates never decrease, and she started by adding (3,4), she can only make 3 more moves which decrease the x amount added, making her maximum x value 3+2+1+0 = 6

Similarly for y, which can add a max of 4+3+2+1+0 = 10

So, she has to end up at (6,6)

14

Students...why do you cheat so much?

its 14

Here, we use a bit of imagination and let $s$ stand for the sum of the $x$ and $y$ coordinates of the grasshopper's current location.

Initially, $s = 0$. After the first move, the grasshopper is at $(3,4)$, so $s = 7$. The next move adds either $(3,3)$ or $(2,4)$. In either case, $s$ must increase by $6$, up to $13$. In the next moves, $s$ must increase by $5$, then $4$, then $3$, then $2$, then $1$. Finally, the grasshopper stops because she cannot continue moving without decreasing at least one of her coordinates.

After these moves, $s$ is equal to $7 + 6 + 5 + 4 + 3 + 2 + 1 = 28$. Since the trip ends at the point $(a,a)$, we have $28= s = 2a$. Therefore, $a = \boxed{14}$.

Notice that it is possible to get to $(14,14)$ by following the path: $(0,0)\rightarrow (3,4) \rightarrow (6,7) \rightarrow (9,9) \rightarrow (11,11) \rightarrow (13,12) \rightarrow (14,13) \rightarrow (14,14)$.

from AOPS website

14

14 is the answer

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Asking this question is against the rules and the law itself. Pls stop cheating

The answer is 14.

What grade are you in?