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OlympiadBench / 2265 / A goat starts at the origin (0,0) and then makes several moves. On move 1 , it…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

A goat starts at the origin (0,0)(0,0) and then makes several moves. On move 1 , it travels 1 unit up to (0,1)(0,1). On move 2 , it travels 2 units right to (2,1)(2,1). On move 3 , it travels 3 units down to (2,−2)(2,-2). On move 4 , it travels 4 units to (−2,−2)(-2,-2). It continues in this fashion, so that on move nn, it turns 90∘90^{\circ} in a clockwise direction from its previous heading and travels nn units in this new direction. After nn moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.
Plain-text mathematical notation (without MathML)
A goat starts at the origin (0,0) and then makes several moves. On move 1 , it travels 1 unit up to (0,1). On move 2 , it travels 2 units right to (2,1). On move 3 , it travels 3 units down to (2,−2). On move 4 , it travels 4 units to (−2,−2). It continues in this fashion, so that on move n, it turns 90^(∘) in a clockwise direction from its previous heading and travels n units in this new direction. After n moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.
Original LaTeX notation
A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\circ}$ in a clockwise direction from its previous heading and travels $n$ units in this new direction. After $n$ moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.

answer type

Tuple

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
2265
task_id
4037d698-987d-575b-b8aa-20526067a4f5
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--2265
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import