problem stringclasses 6 values | responses listlengths 16 16 | grading_scheme listlengths 4 7 |
|---|---|---|
"A line in the plane is called \\\\emph{sunny} if it is not parallel to any of the $x$-axis, the $y$(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, let's make sure I understand the probl(...TRUNCATED) | [{"desc":"1 point should be given for just finding the answer $k=0,1,3$","points":1,"title":"Reachin(...TRUNCATED) |
"Let $\\Omega$ and $\\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, since there's no previous attempt summ(...TRUNCATED) | [{"desc":"Explicitly redefining the point of tangency $X$, either as the point of intersection of $M(...TRUNCATED) |
"Let $\\mathbb{N}$ denote the set of positive integers. A function $f\\colon\\mathbb{N}\\to\\mathbb{(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, we need to understand what a bonza fun(...TRUNCATED) | [{"desc":"Showing that there are only a finite number of primes $p$, for which $f(p)\\neq 1$","point(...TRUNCATED) |
"A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.\nThe(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, let's make sure I understand the probl(...TRUNCATED) | [{"desc":"Providing final answer with verification.","points":1,"title":"Final answer"},{"desc":"Rul(...TRUNCATED) |
"Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, I need to understand the game rules cl(...TRUNCATED) | [{"desc":"For describing a strategy that works for Bazza for $\\lambda < \\frac{\\sqrt{2}}{2}$ (no p(...TRUNCATED) |
"Consider a $2025 \\times 2025$ grid of unit squares. Matilda wishes to place on the grid some recta(...TRUNCATED) | ["Okay, let's try to tackle this problem step by step. First, let's make sure I understand the probl(...TRUNCATED) | [{"desc":"Explicitly providing a construction and verifying that the desired property holds.","point(...TRUNCATED) |
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