# omnimath / omnimath_490 - taskset: [omnimath](https://harnessreport.com/tasks/omnimath.md) - difficulty: hard - category: math - language: - runnable from the site: no - agent timeout: 600s ## Results by harness _none yet_ ## Instruction ``` # Mathematical Problem (a) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set? (b) Let $ABC$ be a triangle and $P$ be a point. The isogonal conjugate of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate? (c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$? (d) Is it possible to cut an equilateral triangle into 2017 pieces, and rearrange the pieces into a square? (e) Let $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D, E, F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle BPC, PE$ bisects $\angle CPA$, and $PF$ bisects $\angle APB$. Is it necessarily true that $AP+BP+CP \geq 2(PD+PE+PF)$? (f) Let $P_{2018}$ be the surface area of the 2018-dimensional unit sphere, and let $P_{2017}$ be the surface area of the 2017-dimensional unit sphere. Is $P_{2018}>P_{2017}$? ## Instructions Solve the mathematical problem above and write your final answer to `/workspace/answer.txt`. **Important**: Write only your final answer to the file, not the full solution process. ### Guidelines - Provide your final numerical answer or mathematical expression - Write the answer as plain text (no special formatting needed) - Be precise and clear in your answer - The answer should directly respond to what the problem asks for ### Example If the problem asks "What is 2 + 2?", your answer file should contain: ``` 4 ``` Your answer will be evaluated against the correct solution using an automated grading system. ``` --- Harness Report runs agent harnesses from their GitHub repos on Harbor tasks and records every model call. Every page is also `.md` and `.json`; index: https://harnessreport.com/llms.txt · MCP: https://harnessreport.com/mcp