Question
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:
Move Rule: After a move which adds (x,y) to the grasshopper's coordinates, her next move adds either (x,y-1) or (x-1,y) to her coordinates.
Thus, her second move adds either (3,3) or (2,4) to her position, landing her at (6,7) or (5,8) accordingly.
If the grasshopper's x and y coordinates never decrease, and if she ends at the point (a,a) where a is as large as possible, then what is the value of a?
Could I have an explanation with the answer too? Thanks!!
Move Rule: After a move which adds (x,y) to the grasshopper's coordinates, her next move adds either (x,y-1) or (x-1,y) to her coordinates.
Thus, her second move adds either (3,3) or (2,4) to her position, landing her at (6,7) or (5,8) accordingly.
If the grasshopper's x and y coordinates never decrease, and if she ends at the point (a,a) where a is as large as possible, then what is the value of a?
Could I have an explanation with the answer too? Thanks!!
Answers
Steve
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)
Similarly for y, which can add a max of 4+3+2+1+0 = 10
So, she has to end up at (6,6)
Tan. Os
14
AoPS Admin
Students...why do you cheat so much?
1232312213
its 14
1232312213
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: $(0,0)\rightarrow (3,4) \rightarrow (6,7) \rightarrow (9,9) \rightarrow (11,11) \rightarrow (13,12) \rightarrow (14,13) \rightarrow (14,14)$.
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: $(0,0)\rightarrow (3,4) \rightarrow (6,7) \rightarrow (9,9) \rightarrow (11,11) \rightarrow (13,12) \rightarrow (14,13) \rightarrow (14,14)$.
1232312213
from AOPS website
yeyejuiceboy
14
Hithere
14 is the answer
AoPS Admin
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Asking this question is against the rules and the law itself. Pls stop cheating
Asking this question is against the rules and the law itself. Pls stop cheating
hi!
The answer is 14.
D
What grade are you in?