id stringlengths 9 15 | solution stringlengths 48 2.92k | answer stringlengths 1 90 ⌀ | metadata dict | problem stringlengths 47 710 | tags listlengths 1 19 | idea stringlengths 57 181 | original_index int64 2.72k 279k | candidates listlengths 150 150 | bma_scores listlengths 150 150 | jaccard_scores listlengths 150 150 | domains listlengths 1 1 | relevance_deepseek listlengths 150 150 | relevance_scores listlengths 150 150 | relevance_scores_full listlengths 150 150 ⌀ | ordinal_scores listlengths 150 150 ⌀ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
aops_546838 | Danica has a mansion, with floors 1 to 9. Danica wants to paint the wall of each floor with either pink, yellow, or sky blue. In how many ways can this be done if :
(a) At least one floor is painted pink, at least one floor is painted yellow, and at least one floor is painted sky blue.
(a) Exactly four floors ar... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Danica has a mansion, with floors 1 to 9. Danica wants to paint the wall of each floor with either pink, yellow, or sky blue. In how many ways can this be done if : \n\n(a) At least one floor is painte... | Danica has a mansion with floors \(1\) to \(9\). Danica paints the wall of each floor either pink, yellow, or sky blue. In how many ways can this be done if:
(a) At least one floor is painted pink, at least one floor is painted yellow, and at least one floor is painted sky blue?
(b) Exactly four floors are painted ye... | [
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"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/Di... | Apply inclusion–exclusion to count colorings using all three colors and use the multinomial coefficient for a fixed color distribution. | 268,787 | [
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aops_1859371 | [hide="Answer"]$N=75582$. So sum of digits of $N$ is $\boxed{27}$ [/hide] | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "There are $N$ numbers of $11$ digit positive integers such that the digits from left to right are non-decreasing. (For example, $12345678999, 55555555555,23345557889$). Find the sum of the digits of $N$.",... | There are \(N\) numbers of 11-digit positive integers such that the digits from left to right are non-decreasing (for example, \(12345678999\), \(55555555555\), \(23345557889\)). Find the sum of the digits of \(N\). | [
"/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients",
"/Mathematics/DiscreteMathematics/Combinatorics/Enumeration",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathem... | Convert the non‑decreasing digit condition into a stars‑and‑bars count of multisets of size 11 from digits 1 to 9. | 183,265 | [
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aops_2143452 | If $a+b=k,$ then $0 \le c+d+e \le 4-k,$ so there are $k+1$ choices for $a,b$ and $\binom{6-k}{2}$ choices for $c,d,e.$ The cardinality is $\binom{6}{2} + 2 \cdot \binom{5}{2} + 3 \cdot \binom{4}{2} = 15+20+18=53.$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $N = \\{0, 1, 2, 3, . . .\\}$. Find the cardinality of the set \n$$\\{(a, b, c, d, e) \\in N^5 \\colon 0 \\leq a+b \\leq 2, 0 \\leq a+b+c+d+e \\leq 4 \\}.$$",
"content_html": "Let <span style=\"w... | Let \(N=\{0,1,2,3,\dots\}\). Find the cardinality of the set
\[
\{(a,b,c,d,e)\in N^5 \mid 0\le a+b\le 2,\ 0\le a+b+c+d+e\le 4\}.
\] | [
"/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients",
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"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics",
"/Mathematics/Di... | Fix k = a+b, then count (a,b) and the remaining variables using stars‑and‑bars under the reduced sum constraint. | 194,655 | [
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numina_10115729 | Solution: Let $I$ be the set of all paths from $O$ to $P$;
$A_{1}$ be the set of all paths from $O$ to $P$ passing through $A B$;
$A_{2}$ be the set of all paths from $O$ to $P$ passing through $C D$;
$A_{3}$ be the set of all paths from $O$ to $P$ passing through $E F$;
$A_{4}$ be the set of all paths from $O$ to $P$ ... | \mathrm{C}_{15}^{5}-\mathrm{C}_{4}^{2}\mathrm{C}_{10}^{3}-\mathrm{C}_{6}^{2}\mathrm{C}_{8} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Example 2.3.6 Find the number of shortest paths from point $O(0,0)$ to point $P(10,5)$ in a coordinate grid that do not pass through any of the segments $AB, CD, EF, GH$, where the coordinates of $A, B, C, D, E, F, G, H$ are $A(2,2), B(3,2)$,
$$
\begin{array}{l}
C(4,2), D(5,2), E(6,2), F(6,3), \\
G(7,2), H(7,3) .
\end{... | [
"/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients",
"/Mathematics/DiscreteMathematics/Combinatorics/Enumeration",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/Combinatorics/LatticePathsandPolygons",
"/Mathematics/DiscreteMathemat... | Use inclusion‑exclusion to count total lattice paths and subtract those that go through each forbidden segment, adding back intersections. | 91,180 | [
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numina_10092095 | In the numbers between 1000 and 9999, all are ordered arrangements of 4 digits. Therefore, to find the number of certain arrangements, we need to make 4 choices: the units, tens, hundreds, and thousands digits. Since the number we want is odd, the units digit can be any one of 1, 3, 5, 7, 9. The tens and hundreds digit... | 2240 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Example 2: How many odd numbers with all different digits are there between 1000 and 9999? | [
"/Mathematics/DiscreteMathematics/Combinatorics/Enumeration",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/Combinatorics/Permutations",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/Ge... | Apply the multiplication principle to count digit arrangements with the odd-unit and nonzero-thousands restrictions and distinct digits. | 26,722 | [
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0.20... | null | null |
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