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2020-I-6 | 2,020 | 6 | I | "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the (...TRUNCATED) | 173 | ["\n\nGiven a flat board with two circular holes of radii 1 and 2, and the distance between their ce(...TRUNCATED) | [
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] | 0.0625 | Geometry | "A Geometry problem involving two circular holes and two tangent spheres, where the sphere centers l(...TRUNCATED) | "Express each sphere’s center height as √(r² – (hole radius)²); this represents the vertic(...TRUNCATED) | "Set up a right triangle whose horizontal side is the given distance between the hole centers and wh(...TRUNCATED) | "Use the tangency condition—that the distance between the sphere centers equals 2r—to link the c(...TRUNCATED) | "Be cautious not to assume the sphere centers lie simply above the hole centers; a proper diagram he(...TRUNCATED) | "Double-check that all length dimensions are consistent when applying the Pythagorean theorem to der(...TRUNCATED) |
2020-I-7 | 2,020 | 7 | I | "A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so (...TRUNCATED) | 81 | ["\n\nLet's solve the problem step by step.\n\n**Problem Statement:**\nA club consisting of 11 men a(...TRUNCATED) | ["676455","= no"," something,...( which,,, wait","81","20","676041","3 + 5 + 7 + 11 = 26","1352726",(...TRUNCATED) | [
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] | 0.1875 | Combinatorics | "A combinatorics problem requiring counting committees from two groups with one group’s count exac(...TRUNCATED) | "Observe that a committee condition like \"one more of group A than group B\" naturally leads to sum(...TRUNCATED) | "Consider re‐indexing the sum to recognize a convolution identity that can collapse the sum into a(...TRUNCATED) | "Check that your index ranges respect the available counts in each group to avoid selecting more mem(...TRUNCATED) | "Once you simplify the expression, factorize the resulting binomial coefficient to reveal its underl(...TRUNCATED) | "Think about using generating functions or standard combinatorial identities (like Vandermonde’s i(...TRUNCATED) |
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