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Omni-MATH / Let C be the hyperbola y²−x²=1. Given a point P₀ on the x-axis, we construct a sequence of points (P_(n)) on the x-axis in the following manner: let ℓ_(n) be the line with slope 1 …

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problem

Let C\mathcal{C} be the hyperbola y2−x2=1y^{2}-x^{2}=1. Given a point P0P_{0} on the xx-axis, we construct a sequence of points (Pn)\left(P_{n}\right) on the xx-axis in the following manner: let ℓn\ell_{n} be the line with slope 1 passing through PnP_{n}, then Pn+1P_{n+1} is the orthogonal projection of the point of intersection of ℓn\ell_{n} and C\mathcal{C} onto the xx-axis. (If Pn=0P_{n}=0, then the sequence simply terminates.) Let NN be the number of starting positions P0P_{0} on the xx-axis such that P0=P2008P_{0}=P_{2008}. Determine the remainder of NN when divided by 2008.
Plain-text mathematical notation (without MathML)
Let C be the hyperbola y²−x²=1. Given a point P₀ on the x-axis, we construct a sequence of points (P_(n)) on the x-axis in the following manner: let ℓ_(n) be the line with slope 1 passing through P_(n), then P_(n+1) is the orthogonal projection of the point of intersection of ℓ_(n) and C onto the x-axis. (If P_(n)=0, then the sequence simply terminates.) Let N be the number of starting positions P₀ on the x-axis such that P₀=P₂₀₀₈. Determine the remainder of N when divided by 2008.
Original LaTeX notation
Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$. Given a point $P_{0}$ on the $x$-axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$-axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing through $P_{n}$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_{n}$ and $\mathcal{C}$ onto the $x$-axis. (If $P_{n}=0$, then the sequence simply terminates.) Let $N$ be the number of starting positions $P_{0}$ on the $x$-axis such that $P_{0}=P_{2008}$. Determine the remainder of $N$ when divided by 2008.

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Source and history

task_id
ec909461-7393-593c-93af-bc2bc01b7dc9
task_key
test--ec909461-7393-593c-93af-bc2bc01b7dc9
task_revision_id
1
config / split
/ test

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