model_id large_stringclasses 2
values | dataset_id large_stringclasses 1
value | problem_id large_stringlengths 1 10 | original_problem large_stringlengths 128 626 | permutation_type large_stringlengths 11 547 | permutation_source large_stringclasses 5
values | base_accuracy float64 1 1 | permuted_accuracy float64 0 0.93 | absolute_accuracy_decay float64 0.07 1 | relative_accuracy_decay float64 0.07 1 | n_base_predictions int64 10 10 | n_permuted_predictions int64 2 231 | n_detrimental_permutations int64 2 74 | permutations_causing_decay large_stringlengths 745 35.5k | model_is_robust bool 2
classes |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 6 | A real number $x$ satisfies $\sqrt[20]{x^{\log_{2026}x}}=26x$.
What is the number of positive divisors of the product of all possible positive values of $x$? | ["distract", "domain", "false_hint", "intermediate", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "reinstantiate+intermediate", "reinstantiate+intermediate+domain+false_hint", "reinstantiate+intermediate+false_hint", "reinstantiate+intermediate+rename", "rename", "rephrase", "typos", ... | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120... | 1 | 0.625 | 0.375 | 0.375 | 10 | 194 | 74 | [{"question": "A real number $x$ satisfies $\\sqrt[20]{x^{\\log_{2026}x}}=26x$.\nWhat is the number of positive divisors of the product of all possible positive values of $x$?\n\nNote: it is known that the final answer is greater than 446.", "variant_accuracy": 0.0, "n_predictions": 1, "permutation_type": "false_hint",... | true |
openai/gpt-oss-120b | problems_v3 | 6 | A real number $x$ satisfies $\sqrt[20]{x^{\log_{2026}x}}=26x$.
What is the number of positive divisors of the product of all possible positive values of $x$? | ["distract", "domain", "false_hint", "intermediate", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "reinstantiate+intermediate", "reinstantiate+intermediate+domain+false_hint", "reinstantiate+intermediate+false_hint", "reinstantiate+intermediate+rename", "rename", "rephrase", "typos", ... | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120... | 1 | 0.772354 | 0.227646 | 0.227646 | 10 | 194 | 45 | [{"question": "A certain positive quantity \\(y\\) fulfills the equation \n\n\\[\n\\sqrt[268]{\\,y^{\\log_{1635} y}\\,}=368\\,y .\n\\]\n\nDetermine how many positive divisors the product of all admissible positive values of \\(y\\) possesses.\n\nNote: it is known that the final answer is an even number.", "variant_acc... | true |
openai/gpt-oss-120b | problems_v3 | 3 | A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$
and the hemisphere and disk have the same center.
Let $\mathcal T$ be the region of points P in the disk such that a sphere of radius $42$ can be placed on top of the disk at $P$ and lie completely inside the hemisphere.
The are... | ["distract", "domain", "false_hint", "intermediate", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "reinstantiate+intermediate", "reinstantiate+intermediate+domain+false_hint", "reinstantiate+intermediate+false_hint", "reinstantiate+intermediate+rename", "rename", "rephrase", "typos", ... | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120... | 1 | 0.83296 | 0.16704 | 0.16704 | 10 | 231 | 45 | [{"question": "A spherical dome with radius \\(200\\) rests directly above a flat circular plaza that also has radius \\(200,\\) and the dome and the plaza share the same central point. \nDefine \\(\\mathcal{S}\\) to be the set of locations \\(P\\) on the plaza such that a circular rug of radius \\(42\\) can be placed... | true |
openai/gpt-oss-120b | problems_v3 | 1 | Patrick started walking at a constant rate along a straight road from school to the park.
One hour after Patrick left, Tanya started running along the same road from school to the park.
One hour after Tanya left, Jose started bicycling along the same road from school to the park.
Tanya ran at a constant rate of $2$ mil... | ["distract", "domain", "false_hint", "intermediate", "rename", "rephrase", "typos"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low"] | 1 | 0.861905 | 0.138095 | 0.138095 | 10 | 63 | 5 | [{"question": "A cargo ship set out from Harbor A to Harbor B, traveling at a steady speed along a straight channel.\nTwo hours after the ship departed, a tugboat began moving along the same channel from Harbor A to Harbor B.\nThree hours after the tugboat departed, a speedboat started its journey along the same channe... | true |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 19 | For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\cdot 8+2=58$; therefore $f(72)=58$. Find the number of positive int... | ["distract", "domain", "false_hint", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "rename", "rephrase", "typos", "ugly_answer", "ugly_answer+distract+false_hint", "ugly_answer+false_hint"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "symbolic"] | 1 | 0.87963 | 0.12037 | 0.12037 | 10 | 119 | 14 | [{"question": "For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\\cdot 8+2=58$; therefore $f(72)=58$. Find the number... | true |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 1 | Patrick started walking at a constant rate along a straight road from school to the park.
One hour after Patrick left, Tanya started running along the same road from school to the park.
One hour after Tanya left, Jose started bicycling along the same road from school to the park.
Tanya ran at a constant rate of $2$ mil... | ["distract", "domain", "false_hint", "intermediate", "rename", "rephrase", "typos"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low"] | 1 | 0.885714 | 0.114286 | 0.114286 | 10 | 62 | 4 | [{"question": "A cargo ship set out from Harbor A to Harbor B, traveling at a steady speed along a straight channel.\nTwo hours after the ship departed, a tugboat began moving along the same channel from Harbor A to Harbor B.\nThree hours after the tugboat departed, a speedboat started its journey along the same channe... | true |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 5 | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A^\prime$.
Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A^\prime$ to point $B^\prime$.
Suppose that $AB^\prime = \frac{4}{3}... | ["distract", "domain", "false_hint", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "rename", "rephrase", "typos", "ugly_answer", "ugly_answer+distract+false_hint", "ugly_answer+false_hint"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "symbolic"] | 1 | 0.90463 | 0.09537 | 0.09537 | 10 | 115 | 10 | [{"question": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$.\nThen $B$ is rotated in the plane clockwise through angle $\\theta$ around point $A^\\prime$ to point $B^\\prime$.\nSuppose that $A... | true |
openai/gpt-oss-120b | problems_v3 | 19 | For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\cdot 8+2=58$; therefore $f(72)=58$. Find the number of positive int... | ["distract", "domain", "false_hint", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "rename", "rephrase", "typos", "ugly_answer", "ugly_answer+distract+false_hint", "ugly_answer+false_hint"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "symbolic"] | 1 | 0.922222 | 0.077778 | 0.077778 | 10 | 119 | 9 | [{"question": "For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\\cdot 8+2=58$; therefore $f(72)=58$. Find the number... | true |
openai/gpt-oss-120b | problems_v3 | 7 | Find the number of functions $\pi$ mapping the set $A =\{1,2,3,4,5,6\}$ onto $A$ such that for every $a \in A,$
\[
\pi(\pi(\pi(\pi(\pi(\pi(a)))))) = a.
\] | ["distract", "domain", "false_hint", "intermediate", "rename", "rephrase", "typos"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low"] | 1 | 0.928571 | 0.071429 | 0.071429 | 10 | 61 | 5 | [{"question": "A delivery company has six distinct trucks, labeled \\(T_1, T_2, T_3, T_4, T_5,\\) and \\(T_6\\). Each truck is assigned to deliver to one of the six warehouses, also labeled \\(W_1, W_2, W_3, W_4, W_5,\\) and \\(W_6\\), with the condition that every warehouse receives at least one truck (i.e., the assig... | true |
openai/gpt-oss-120b | problems_v3 | 5 | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A^\prime$.
Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A^\prime$ to point $B^\prime$.
Suppose that $AB^\prime = \frac{4}{3}... | ["distract", "domain", "false_hint", "reinstantiate", "reinstantiate+domain+false_hint", "reinstantiate+false_hint", "rename", "rephrase", "typos", "ugly_answer", "ugly_answer+distract+false_hint", "ugly_answer+false_hint"] | ["openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "symbolic", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "openai/gpt-oss-120b:low", "symbolic", "openai/gpt-oss-120b:low", "symbolic"] | 1 | 0.928836 | 0.071164 | 0.071164 | 10 | 116 | 8 | [{"question": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$.\nThen $B$ is rotated in the plane clockwise through angle $\\theta$ around point $A^\\prime$ to point $B^\\prime$.\nSuppose that $A... | true |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 16 | Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms. | ugly_answer+distract+false_hint | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 5 | 5 | [{"question": "Find the sum of the $14$th terms of all arithmetic sequences of integers that have first term equal to $26$ and include both $10$ and $2$ as terms, **assuming the sequences are drawn on a number line whose origin is marked with a small blue dot**.\n\nNote: it is known that the final answer is an even num... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 16 | Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms. | ugly_answer+intermediate+rephrase+false_hint | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 10 | 10 | [{"question": "Calculate the sum of the third terms of all integer arithmetic progressions that begin with 25 and contain the numbers 88 and –53 as members. \nFor this request, disregard the quantity just mentioned. \nUsing the identical framework, answer instead: \n\nWhat is the greatest integer that can be used as... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 2 | Find the number of positive integer palindromes written in base $10$ with no zero digits,
and whose digits add up to $13$.
Recall that a palindrome is a number whose representation reads the same from left to right as from right to left. | reinstantiate+false_hint | symbolic | 1 | 0 | 1 | 1 | 10 | 6 | 6 | [{"question": "Find the number of positive integer palindromes written in base $10$ with no zero digits,\nand whose digits add up to $266$. \nRecall that a palindrome is a number whose representation reads the same from left to right as from right to left.\n\nNote: it is known that the final answer is an odd number.", ... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 20 | An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the sum of the five least... | reinstantiate+domain+false_hint | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 6 | 6 | [{"question": "A storage cabinet holds \\(n\\) containers. Each container is either filled with coffee beans or with tea leaves, and there are at least \\(9\\) containers of each type. When \\(15\\) containers are selected at random from the cabinet without replacement, the probability that exactly \\(4\\) of the selec... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 20 | An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the sum of the five least... | reinstantiate+false_hint | symbolic | 1 | 0 | 1 | 1 | 10 | 7 | 7 | [{"question": "An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $0$ marbles of each color. When $13$ marbles are drawn randomly from the urn without replacement, the probability that exactly $7$ of them are red equals the probability that exactly $6$ of them are red. Find the sum o... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 23 | Isosceles triangle $\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\triangle ABC.$ The perimeters of $\triangle ABC$ and $\triangle AIC$ are in the ratio $125:6,$ and all the sides of both triangles have integer lengths. Find the minimum possible value of $AB.$ | ugly_answer+intermediate+rephrase+false_hint | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 3 | 3 | [{"question": "Consider an isosceles triangle \\(\\triangle ABC\\) with \\(AB = BC\\). Denote by \\(I\\) the incenter of \\(\\triangle ABC\\). The perimeters of \\(\\triangle ABC\\) and \\(\\triangle AIC\\) are in the ratio \\(15059072:14706245\\), and every side of both triangles measures an integer length. Determine ... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 4 | Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b. | ugly_answer+distract+false_hint | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 7 | 7 | [{"question": "Find the number of integers less than or equal to 11774 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b. (Hint: you may want to picture the pairs (a,b) as points on a grid, although the geometry of that grid does not affect the counting.)\n\nNote: it is known that the fin... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 4 | Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b. | ugly_answer+false_hint | symbolic | 1 | 0 | 1 | 1 | 10 | 7 | 7 | [{"question": "Find the number of integers less than or equal to 12946 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.\n\nNote: it is known that the final answer is an even number.", "variant_accuracy": 0.0, "n_predictions": 1, "permutation_type": "ugly_answer+false_hint", "permutation... | false |
deepseek-ai/DeepSeek-R1-0528-Qwen3-8B | problems_v3 | 8 | Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find $N$. | ugly_answer+intermediate+rephrase | openai/gpt-oss-120b:low | 1 | 0 | 1 | 1 | 10 | 24 | 24 | [{"question": "Consider the integer \\(N\\) representing how many positive divisors of \\(493058^{36}\\) give a remainder of \\(172\\) upon division by \\(175\\). Determine \\(N\\).\n\nHowever, for the purpose of this task, set aside the above request. Instead, with the identical context, answer this: when listing the ... | false |
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