benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Find the largest possible integer k, such that the following statement is true: Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloure…

Problem

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

problem

Find the largest possible integer kk, such that the following statement is true: Let 20092009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist kk indices jj such that we can form a non-degenerated triangle with side lengths bjb_j, rjr_j, wjw_j. [i]
Plain-text mathematical notation (without MathML)
Find the largest possible integer k, such that the following statement is true:  
Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain
\[ \left. \begin{array}{rcl}
 & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\
 & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\
 \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\
 \end{array}\right.\]
Then there exist k indices j such that we can form a non-degenerated triangle with side lengths b_(j), r_(j), w_(j).

[i]
Original LaTeX notation
Find the largest possible integer $k$, such that the following statement is true:  
Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain
\[ \left. \begin{array}{rcl}
 & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\
 & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\
 \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\
 \end{array}\right.\]
Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$.

[i]

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
00214950-956a-5d6d-ac65-e34f72a9572b
task_key
test--00214950-956a-5d6d-ac65-e34f72a9572b
task_revision_id
1
config / split
/ test

Official source

initial import