| [ |
| { |
| "name": "inspect_evals/livebench - Sample ef6c6830ebf0e71d74e7ceb45a242f2b87228933eebfe47db8d798d5738cdaeb", |
| "description": "Task from inspect_evals/livebench dataset, sample ef6c6830ebf0e71d74e7ceb45a242f2b87228933eebfe47db8d798d5738cdaeb", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nIn an acute triangle $ABC$, let $M$ be the midpoint of $\\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\\overline{AQ}$. Prove that $NB=NC$.\n\n\nThe solution is:\nLet $X$ be the foot from $A$ to $\\overline{BC}$ . By definition, <missing 1> . Thus, <missing 2> , and <missing 3> .\n\nFrom this, we have <missing 4> , as <missing 5> . Thus, $M$ is also the midpoint of $XQ$ .\n\nNow, <missing 6> if $N$ lies on the perpendicular bisector of $\\overline{BC}$ . As $N$ lies on the perpendicular bisector of <missing 7> , which is also the perpendicular bisector of $\\overline{BC}$ (as $M$ is also the midpoint of $XQ$ ), we are done.\n\nThe formulae are:\n<expression 1> \\angle AXM = \\angle MPC = 90^{\\circ} \n<expression 2> \\frac{MP}{MX} = \\frac{MC}{MA} = \\frac{MP}{MQ} = \\frac{MA}{MB} \n<expression 3> MC=MB \n<expression 4> NB = NC \n<expression 5> \\overline{XQ} \n<expression 6> \\triangle AXM \\sim \\triangle MPC \n<expression 7> \\triangle BMP \\sim \\triangle AMQ \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "ef6c6830ebf0e71d74e7ceb45a242f2b87228933eebfe47db8d798d5738cdaeb", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 6e12ab2219a4051c396de5afc4228be52c2d8617c7f5621be8c364f50016c80d", |
| "description": "Task from inspect_evals/livebench dataset, sample 6e12ab2219a4051c396de5afc4228be52c2d8617c7f5621be8c364f50016c80d", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet $\\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\\mathbb{R}^{+}\\rightarrow\\mathbb{R}^{+}$ such that, for all $x, y \\in \\mathbb{R}^{+}$, \\[ f(xy + f(x)) = xf(y) + 2 \\]\n\n\nThe solution is:\nMake the following substitutions to the equation:\n\n1. <missing 1> 2. <missing 2> 3. <missing 3> It then follows from (2) and (3) that <missing 4> , so we know that this function is linear for $x > 1$ . Substitute <missing 5> and solve for $a$ and $b$ in the functional equation; we find that <missing 6> .\n\nNow, we can let $x > 1$ and <missing 7> . Since $f(x) = x + 1$ , <missing 8> , so <missing 9> . It becomes clear then that <missing 10> as well, so $f(x) = x + 1$ is the only solution to the functional equation.\n\nThe formulae are:\n<expression 1> f(y) = y + 1 \n<expression 2> f(xy + f(x)) = xy + x + 2 = xf(y) + 2 \n<expression 3> f(x) = ax+b \n<expression 4> xy + f(x) > x > 1 \n<expression 5> y \\le 1 \n<expression 6> (x, 1 + \\frac{f(1)}{x}) \\rightarrow f(x + f(x) + f(1)) = xf\\biggl(1 + \\frac{f(1)}{x}\\biggr) + 2 \n<expression 7> (1, x + f(x)) \\rightarrow f(x + f(x) + f(1)) = f(x + f(x)) + 2 = xf(1) + 4 \n<expression 8> f(x) = x + 1 \\forall x > 1 \n<expression 9> (x, 1) \\rightarrow f(x + f(x)) = xf(1) + 2 \n<expression 10> f(1 + \\frac{f(1)}{x}) = f(1) + \\frac{2}{x} \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "6e12ab2219a4051c396de5afc4228be52c2d8617c7f5621be8c364f50016c80d", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample c03217423ccb2181ef9e3106b4c116aaf1bcdbd45b3a97e04110665690177eca", |
| "description": "Task from inspect_evals/livebench dataset, sample c03217423ccb2181ef9e3106b4c116aaf1bcdbd45b3a97e04110665690177eca", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nConsider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.\n\n\nThe solution is:\nWe claim the answer is $(\\frac{n+1}{2})^2$ .\n\nFirst, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from $1$ to $n$ from left to right and each row from $1$ to $n$ from top to bottom. We color a tile R if its row and column are odd, a tile G is its row is even but its column is odd, a tile B if its row and column is even, and a tile Y if its row is odd but its column is even.\n\nLemma 1: Throughout our moves, the color of the uncolored tile stays an invariant.\n\nConsider that a domino can either only change rows or can only change columns. Therefore, sliding a domino into the hole and creating a new one has two possible colors. Of these, note that the new hole will always trivially be two tiles away from the old hole, meaning that the parity of both the row and column number stays the same. Thus, the lemma holds.\n\nLemma 2: There are more red tiles than any other color. Because each color is uniquely defined by the parity of a pair of column and row number, it satisfies to show that given an odd integer $n$ , there are more odd positive integers less than or equal to $n$ than even ones. Obviously, this is true, and so red will have more tiles than any other color.\n\nLemma 3: For any starting configuration $C$ and any blank tile $B$ such that the blank tile's color matches the blank tile's color of $C$ , there is no more than one unique configuration $C'$ that can be produced from $C$ using valid moves.\n\nWe will use proof by contradiction. Assume there exists two different $C'$ . We can get from one of these $C'$ to another using moves. However, we have to finish off with the same hole as before. Before the last move, the hole must be two tiles away from the starting hole. However, because the domino we used to move into the blank tile's spot is in the way, that hole must be congruent to the hole produced after the first move. We can induct this logic, and because there is a limited amount of tiles with the same color, eventually we will run out of tiles to apply this to. Therefore, having two distinct $C'$ with the same starting hole $B$ is impossible with some $C$ .\n\nWe will now prove that $(\\frac{n+1}{2})^2$ is the answer. There are $\\frac{n+1}{2}$ rows and $\\frac{n+1}{2}$ columns that are odd, and thus there are $(\\frac{n+1}{2})^2$ red tiles. Given lemma 3, this is our upper bound for a maximum. To establish that $(\\frac{n+1}{2})^2$ is indeed possible, we construct such a $C$ :\n\nIn the first column, leave the first tile up blank. Then, continuously fill in vertically oriented dominos in that column until it reaches the bottom. In the next <missing 1> columns, place <missing 2> vertically oriented dominos in a row starting from the top. At the bottom row, starting with the first unfilled tile on the left, place horizontally aligned dominos in a row until you reach the right.\n\nObviously, the top left tile is red. It suffices to show that any red tile may be uncovered. For the first column, one may slide some dominos on the first column until the desired tile is uncovered. For the bottom row, all the first dominos may be slid up, and then the bottom dominos may be slid to the left until the desired red tile is uncovered. Finally, for the rest of the red tiles, the bottom red tile in the same color may be revealed, and then vertically aligned dominos in the same column may be slid down until the desired tile is revealed. Therefore, this configuration may produce $(\\frac{n+1}{2})^2$ different configurations with moves.\n\nHence, we have proved that $(\\frac{n+1}{2})^2$ is the maximum, and we are done.\n\nThe formulae are:\n<expression 1> n-1 \n<expression 2> \\frac{n-1}{2} \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "c03217423ccb2181ef9e3106b4c116aaf1bcdbd45b3a97e04110665690177eca", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 698d9bd9ad134096358dad82317dae610d9f265255aeee52c644871b83eb7a85", |
| "description": "Task from inspect_evals/livebench dataset, sample 698d9bd9ad134096358dad82317dae610d9f265255aeee52c644871b83eb7a85", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nA positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.\n\nAfter analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.\n\n\nThe solution is:\nThe contrapositive of the claim is somewhat easier to conceptualize: If it is not guaranteed that the game will end (i.e. the game could potentially last forever), then Bob is not able to force the game to end (i.e. Alice can force it to last forever). So we want to prove that, if the game can potentially last indefinitely, then Alice can force it to last indefinitely. Clearly, if there is $1$ number on the board initially, all moves are forced. This means the claim is true in this specific case, because if the game \"potentially\" can last forever, this means it must last forever, since the game can only be played in one way. Ergo Alice can \"force\" this to occur because it is guaranteed to occur. Now we look at all cases where there is more than $1$ number on the board.\n\n\nCase 1: $v_2 (a)=0$ The game lasts forever here no matter what. This is true because, if the game ends, it means the board was in some position $P$ , Alice added $a$ to some number on the board, and all the numbers now on the board are odd. If there are only odd numbers on the board in position $P$ , Alice will add $a$ to some number on the board, making it even, meaning this cannot have been the state $P$ of the board. If at least one number n on the board is even, Alice can add $a$ to a number other than $n$ , meaning there is still at least 1 even number on the board, meaning this also cannot have been the state $P$ of the board. This covers all possible boards when $v_2 (a)=0$ , so we're done.\n\n\nCase 2: <missing 1> If there is at least one number n on the board that is even, the game can also last forever. On any move, Alice will add $a$ to this number $n$ if and only if $v_2 (n)=1$ . This way, the new number <missing 2> satisfies $v_2 (n') \\geq 2$ . If Bob does divides $n'$ until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in $v_2 (n') \\geq 2$ . This means that Alice can always keep an even number on the board for Bob to divide no matter how Bob plays.\n\nIf there is no even number on the board, then the game can clearly not last forever. No matter what Alice does, Bob will have no even number to divide after her move.\n\n\nGeneral Case: <missing 3> In general, it seems to be the case that the game can last indefinitely for some given $a$ if and only if there exists some number $n$ on the board such that $v_2 (n) \\geq v_2 (a)$ , so we should aim to prove this.\n\n1. \"If\"\n\nAlice can apply a similar strategy to the strategy used in case 2. On any move, Alice will add $a$ to $n$ if and only if $v_2 (n)=v_2 (a)$ . If she does this addition, then $v_2 (n') \\geq v_2 (a)+1$ , keeping an even number on the board. Even if Bob divides $n'$ until $v_2 (n)=v_2 (a)$ , Alice will apply the same strategy and keep $v_2 (n') \\geq v_2 (a)+1$ . Alice's use of this strategy ensures that there always exists some number n on the board such that $v_2 (n) \\geq v_2 (a)$ , ensuring there always exists an even number n on the board.\n\n2.\"Only If\"\n\nIf <missing 4> for all n on the board, this means that Alice can never change the value of $v_2 (n)$ for any $n$ on the board. Only Bob can do this, and Bob will subtract $1$ from each $v_2 (n)$ until they are all equal to $0$ (all odd), ending the game.\n\nWe've shown that the game can last indefinitely iff there exists some number $n$ on the board such that $v_2 (n) \\geq v_2 (a)$ , and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy. This proves the contrapositive, proving the claim.\n\nThe formulae are:\n<expression 1> n'= n+a \n<expression 2> v_2 (a)=x \n<expression 3> v_2 (n) < v_2 (a) \n<expression 4> v_2 (a)=1 \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "698d9bd9ad134096358dad82317dae610d9f265255aeee52c644871b83eb7a85", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 56155e5b8c1dc430cc262691aea0a75d71f7dee78b6e15304607caf0b56ecfbd", |
| "description": "Task from inspect_evals/livebench dataset, sample 56155e5b8c1dc430cc262691aea0a75d71f7dee78b6e15304607caf0b56ecfbd", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet $n\\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\\dots$, $n^2$ in a $n \\times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?\n\n\nThe solution is:\nThe answer is all <missing 1> .\n\nProof that primes work\nSuppose <missing 2> is prime. Then, let the arithmetic progressions in the $i$ th row have least term <missing 3> and common difference <missing 4> . For each cell with integer $k$ , assign a monomial <missing 5> . The sum of the monomials is\\[x(1+x+\\ldots+x^{n^2-1}) = \\sum_{i=1}^n x^{a_i}(1+x^{d_i}+\\ldots+x^{(n-1)d_i}),\\]where the LHS is obtained by summing over all cells and the RHS is obtained by summing over all rows. Let $S$ be the set of $p$ th roots of unity that are not $1$ ; then, the LHS of the above equivalence vanishes over $S$ and the RHS is\\[\\sum_{p \\mid d_i} x^{a_i}.\\]Reducing the exponents (mod $p$ ) in the above expression yields\\[f(x) := \\sum_{p \\mid d_i} x^{a_i \\pmod{p}} = 0\\]when <missing 6> . Note that <missing 7> is a factor of <missing 8> , and as $f$ has degree less than $p$ , $f$ is either identically 0 or <missing 9> .\n\nThe formulae are:\n<expression 1> d_i \n<expression 2> a_i \n<expression 3> f(x) \n<expression 4> x^k \n<expression 5> f(x)=1+x+\\ldots+x^{p-1} \n<expression 6> \\boxed{\\text{prime } n} \n<expression 7> n=p \n<expression 8> x \\in S \n<expression 9> \\prod_{s \\in S} (x-s)=1+x+\\ldots+x^{p-1} \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "56155e5b8c1dc430cc262691aea0a75d71f7dee78b6e15304607caf0b56ecfbd", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 0d3a48049dc5da31531654f59e7866832579f812b892610bd1b178fc4d010d0c", |
| "description": "Task from inspect_evals/livebench dataset, sample 0d3a48049dc5da31531654f59e7866832579f812b892610bd1b178fc4d010d0c", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet $n \\geq 3$ be an integer. We say that an arrangement of the numbers $1, 2, \\dots, n^2$ in a $n \\times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?\n\nThe solution is:\nThe answer is all <missing 1> . Proof that primes work. Suppose <missing 2> is prime. Then, let the arithmetic progressions in the $i$ th row have least term <missing 3> and common difference $d_i$ . For each cell with integer $k$ , assign a monomial <missing 4> . The sum of the monomials is. <missing 5> , \n where the LHS is obtained by summing over all cells and the RHS is obtained by summing over all rows. Let $S$ be the set of $p$ th roots of unity that are not $1$ ; then, the LHS of the above equivalence vanishes over $S$ and the RHS is <missing 6> . \n Reducing the exponents (mod $p$ ) in the above expression yields <missing 7> when <missing 8> . Note that <missing 9> is a factor of <missing 10> , and as $f$ has degree less than $p$ , $f$ is either identically 0 or $f(x)=1+x+\\ldots+x^{p-1}$ .\n - If $f$ is identically 0, then $p$ never divides $d_i$ . Thus, no two elements in each row are congruent $\\pmod{p}$ , so all residues are represented in each row. Now we can rearrange the grid so that column $i$ consists of all numbers <missing 11> , which works.\n - If $f(x)=1+x+\\ldots+x^{p-1}$ , then $p$ always divides $d_i$ . It is clear that each $d_i$ must be $p$ , so each row represents a single residue $\\pmod{p}$ . Thus, we can rearrange the grid so that column $i$ contains all consecutive numbers from <missing 12> to $ip$ , which works.\n All in all, any prime $n$ satisfies the hypotheses of the problem.\n Proof that composites do not work\n Let <missing 13> . Look at the term <missing 14> ; we claim it cannot be part of a column that has cells forming an arithmetic sequence after any appropriate rearranging. After such a rearrangement, if the column it is in has common difference <missing 15> , then <missing 16> must also be in its column, which is impossible. If the column has difference <missing 17> , then no element in the next row can be in its column. If the common difference is <missing 18> , then <missing 19> and <missing 20> , which are both in the row above it, must both be in the same column, which is impossible. Therefore, the grid is not column-valid after any rearrangement, which completes the proof.\n\nThe formulae are:\n<expression 1> x \\in S \n<expression 2> a^2b+ab-d \n<expression 3> n=p \n<expression 4> x^k \n<expression 5> f(x) := \\sum_{p \\mid d_i} x^{a_i \\pmod{p}} = 0 \n<expression 6> x(1+x+\\ldots+x^{n^2-1}) = \\sum_{i=1}^n x^{a_i}(1+x^{d_i}+\\ldots+x^{(n-1)d_i}) \n<expression 7> \\sum_{p \\mid d_i} x^{a_i} \n<expression 8> d<ab=n \n<expression 9> \\prod_{s \\in S} (x-s)=1+x+\\ldots+x^{p-1} \n<expression 10> n=ab \n<expression 11> d > ab = n \n<expression 12> a^2b + ab - 2d = a^2b - ab \n<expression 13> a^2b + ab - d = a^2b \n<expression 14> a_i \n<expression 15> \\boxed{\\text{prime } n} \n<expression 16> i \\pmod{p} \n<expression 17> d = ab = n \n<expression 18> 1 + (i-1)p \n<expression 19> a^2b+ab \n<expression 20> f(x) \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "0d3a48049dc5da31531654f59e7866832579f812b892610bd1b178fc4d010d0c", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 3f55db366e970e1965bbba20e9958a4c0ea52e4224d11d875d390983485dd32b", |
| "description": "Task from inspect_evals/livebench dataset, sample 3f55db366e970e1965bbba20e9958a4c0ea52e4224d11d875d390983485dd32b", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet ABC be a triangle with incenter $I$ and excenters $I_a$, $I_b$, $I_c$ opposite $A$, $B$, and $C$, respectively. Given an arbitrary point $D$ on the circumcircle of $\\triangle ABC$ that does not lie on any of the lines $II_{a}$, $I_{b}I_{c}$, or $BC$, suppose the circumcircles of $\\triangle DIIa$ and $\\triangle DI_bI_c$ intersect at two distinct points $D$ and $F$. If $E$ is the intersection of lines $DF$ and $BC$, prove that $\\angle BAD = \\angle EAC$.\n\n\nThe solution is:\nSet <missing 1> as the reference triangle, and let <missing 2> with homogenized coordinates. To find the equation of circle <missing 3> , we note that <missing 4> (not homogenized) and <missing 5> . Thus, for this circle with equation <missing 6> , we compute that \\[ u_1=bc, \\] \\[ v_1=\\frac{bc^2x_0}{bz_0-cy_0}, \\] \\[ w_1=\\frac{b^2cx_0}{cy_0-bz_0}. \\]. For circle <missing 7> with equation <missing 8> ,, we find that \\[ u_2=-bc, \\] \\[ v_2=\\frac{bc^2x_0}{bz_0+cy_0}, \\] \\[ w_2=\\frac{b^2cx_0}{cy_0+bz_0}. \\] The equation of the radical axis is <missing 9> with <missing 10> , <missing 11> , and <missing 12> . We want to consider the intersection of this line with line <missing 13> , so set <missing 14> . The equation reduces to <missing 15> . We see that <missing 16> and <missing 17> , so \\[ \\frac{v}{w}=\\frac{b^2z_0}{c^2y_0}, \\] which is the required condition for <missing 18> and <missing 19> to be isogonal.\n\nThe formulae are:\n<expression 1> D=(x_0,y_0,z_0) \n<expression 2> w=\\frac{2b^3cx_0z_0}{c^2y_0^2-b^2z_0^2} \n<expression 3> x=0 \n<expression 4> vy+wz=0 \n<expression 5> DI_BI_C \n<expression 6> \\overline{BC} \n<expression 7> a^2yz+b^2xz+c^2xy=(x+y+z)(u_1x+v_1y+w_1z) \n<expression 8> u=u_1-u_2 \n<expression 9> I=(a:b:c) \n<expression 10> ux+vy+wz=0 \n<expression 11> DII_A \n<expression 12> \\overline{AD} \n<expression 13> w=w_1-w_2 \n<expression 14> a^2yz+b^2xz+c^2xy=(x+y+z)(u_2x+v_2y+w_2z) \n<expression 15> I_A=(-a:b:c) \n<expression 16> \\overline{AE} \n<expression 17> v=v_1-v_2 \n<expression 18> v=\\frac{2bc^3x_0y_0}{b^2z_0^2-c^2y_0^2} \n<expression 19> \\triangle ABC \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "3f55db366e970e1965bbba20e9958a4c0ea52e4224d11d875d390983485dd32b", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample 3649fb05ed87c2f3ed8a933e7c8f032b54a5d6f0d23504cdd578541dd60c0f08", |
| "description": "Task from inspect_evals/livebench dataset, sample 3649fb05ed87c2f3ed8a933e7c8f032b54a5d6f0d23504cdd578541dd60c0f08", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nFind all integers $n \\geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\\cdots<d_k=n!$, then we have \n $ d_2-d_1 \\leq d_3-d_2 \\leq \\cdots \\leq d_k-d_{k-1} .$ \n \n\n\nThe solution is:\nWe can start by verifying that $n=3$ and $n=4$ work by listing out the factors of <missing 1> and <missing 2> . We can also see that <missing 3> does not work because the terms <missing 4> , and $24$ are consecutive factors of <missing 5> . Also, <missing 6> does not work because the terms <missing 7> , and $9$ appear consecutively in the factors of <missing 8> .\n\nNote that if we have a prime number <missing 9> and an integer <missing 10> such that both $k$ and <missing 11> are factors of $n!$ , then the condition cannot be satisfied.\n\nIf $n\\geq7$ is odd, then <missing 12> is a factor of $n!$ . Also, <missing 13> is a factor of $n!$ . Since <missing 14> for all $n\\geq7$ , we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 15> . Since we have two consecutive factors of $n!$ and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all odd $n\\geq7$ .\n\nIf $n\\geq8$ is even, then <missing 16> is a factor of $n!$ . Also, <missing 17> is a factor of $n!$ . Since <missing 18> for all $n\\geq8$ , we can use Bertrand's Postulate again to show that there is at least one prime number $p$ such that <missing 19> . Since we have two consecutive factors of $n!$ and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all even $n\\geq8$ .\n\nTherefore, the only numbers that work are $n=3$ and $n=4$ .\n\nThe formulae are:\n<expression 1> p>n \n<expression 2> n<p<n^2-4n+3 \n<expression 3> k>p \n<expression 4> k+1 \n<expression 5> 2n<n^2-4n+3 \n<expression 6> 5! \n<expression 7> n=5 \n<expression 8> 6! \n<expression 9> 6, 8 \n<expression 10> 2n<n^2-2n \n<expression 11> 4! \n<expression 12> 15, 20 \n<expression 13> 3! \n<expression 14> (2)(\\frac{n-2}{2})(n-2)=n^2-4n+4 \n<expression 15> n=6 \n<expression 16> n<p<n^2-2n \n<expression 17> (n-3)(n-1)=n^2-4n+3 \n<expression 18> (2)(\\frac{n-1}{2})(n-1)=n^2-2n+1 \n<expression 19> (n-2)(n)=n^2-2n \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "3649fb05ed87c2f3ed8a933e7c8f032b54a5d6f0d23504cdd578541dd60c0f08", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample e5b72d9cdb39c6e5f8798a557f119fff35d98fd165f55659e71bf6ac38b5f178", |
| "description": "Task from inspect_evals/livebench dataset, sample e5b72d9cdb39c6e5f8798a557f119fff35d98fd165f55659e71bf6ac38b5f178", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet $S_1, S_2, \\dots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \\subseteq \\{S_1, S_2, \\dots, S_{100}\\}$, the size of the intersection of the sets in $T$ is a multiple of $|T|$. What is the smallest possible number of elements which are in at least 50 sets?\n\nThe solution is:\nThe answer is <missing 1> . We encode with binary strings <missing 2> of length 100. Write <missing 3> if $w$ has 1's in every component $v$ does, and let $|v|$ denote the number of 1's in $v$ . Then for each $v$ , we let $f(v)$ denote the number of elements <missing 4> such that <missing 5> . So the problem condition means that $f(v)$ translates to the statement <missing 6> for any <missing 7> , plus one extra condition <missing 8> . And the objective function is to minimize the quantity <missing 9> . So the problem is transformed into a system of equations over <missing 10> . Note already that it suffices to assign $f(v)$ for $|v| \\geq 50$ , then we can always arbitrarily assign values of $f(v)$ for $|v| < 50$ by downwards induction on $|v|$ to satisfy the divisibility condition (without changing $M$ ). Thus for the rest of the solution, we altogether ignore $f(v)$ for $|v| < 50$ and only consider $P(u)$ for <missing 11> . Now consider the construction: <missing 12> This construction is valid since if <missing 13> for <missing 14> then <missing 15> <missing 16> is indeed a multiple of <missing 17> , hence $P(u)$ is true. In that case, the objective function is <missing 18> as needed.\n\n\u00b6 Proof of bound. We are going to use a smoothing argument: if we have a general working assignment $f$ , we will mold it into <missing 19> . We define a push-down on $v$ as the following operation: Pick any $v$ such that $|v| \\geq 50$ and <missing 20> . Decrease $f(v)$ by $|v|$ . For every $w$ such that <missing 21> and <missing 22> , increase <missing 23> by 1. Claim \u2014 Apply a push-down preserves the main divisibility condition. Moreover, it doesn't change $A$ unless $|v| = 50$ , where it decreases $A$ by 50 instead. Proof. The statement $P(u)$ is only affected when <missing 24> : to be precise, one term on the right-hand side of $P(u)$ decreases by $|v|$ , while <missing 25> terms increase by 1, for a net change of <missing 26> . So $P(u)$ still holds. To see $A$ doesn't change for <missing 27> , note $|v|$ terms increase by 1 while one term decreases by <missing 28> . When $|v| = 50$ , only $f(v)$ decreases by 50.\u25a1\nNow, given a valid assignment, we can modify it as follows: First apply pushdowns on <missing 29> until $f(1\\ldots1) = 100$ ; Then we may apply pushdowns on each $v$ with <missing 30> until <missing 31> ; Then we may apply pushdowns on each $v$ with <missing 32> until <missing 33> ; ...and so on, until we have <missing 34> for $|v| = 50$ . Hence we get $f(1\\ldots1) = 100$ and <missing 35> for all <missing 36> . However, by downwards induction on <missing 37> , we also have <missing 38> since <missing 39> and $f(v)$ are both strictly less than $|v|$ . So in fact <missing 40> , and we're done.\n\nThe formulae are:\n<expression 1> f = f_0 \n<expression 2> P(u): |u| \\texttt{ divides } \\sum_{v \\supseteq u} f(v) \n<expression 3> |w| = |v| - 1 \n<expression 4> f_0 \n<expression 5> x\\in S_i \\iff v_i = 1 \n<expression 6> v \\in \\mathbb{F}_2^{100} \n<expression 7> f_0(v) = 2|v| - 100. \n<expression 8> = (100 - k) \\cdot 2^k = |u| \\cdot 2^k \n<expression 9> v \\subseteq w \n<expression 10> u \\neq 0, \\dots 0 \n<expression 11> 50 \\leq |v| \\leq 100 \n<expression 12> 1\\ldots1 \n<expression 13> f_0(v) \n<expression 14> |v| = 99, 98, \\ldots, 50 \n<expression 15> |v| - |u| \n<expression 16> u \\subseteq v \n<expression 17> A = \\sum_{i=50}^{100} \\binom{100}{i}(2i - 100) = 50\\binom{100}{50} \n<expression 18> f(w) \n<expression 19> |v| > 50 \n<expression 20> -|v| \n<expression 21> \\sum_{v \\supseteq u} f_0(v) = \\binom{k}{0} \\cdot 100 + \\binom{k}{1} \\cdot 98 + \\binom{k}{2} \\cdot 96 + \\cdots + \\binom{k}{k} \\cdot (100 - 2k) \n<expression 22> f(v) < 99 \n<expression 23> A := \\sum_{|v| \\geq 50} f(v) \n<expression 24> |u| \\geq 50 \n<expression 25> w \\subseteq v \n<expression 26> f(v) < 98 \n<expression 27> 50\\binom{100}{50} \n<expression 28> f(1\\dots 1) > 0 \n<expression 29> f(v) \\equiv f_0(v) \\pmod{|v|} \\implies f(v) = f_0(v) \n<expression 30> x\\in \\bigcup S_i \n<expression 31> |v| = 99 \n<expression 32> 0 \\leq f(v) < |v| \n<expression 33> |u| = 100 - k \n<expression 34> k \\leq 50 \n<expression 35> |u| \n<expression 36> |v| = 98 \n<expression 37> f(v) \\geq |v| \n<expression 38> mathbb{Z}_{\\geq 0} \n<expression 39> f(v) < 50 \n<expression 40> -|u| \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "e5b72d9cdb39c6e5f8798a557f119fff35d98fd165f55659e71bf6ac38b5f178", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| }, |
| { |
| "name": "inspect_evals/livebench - Sample bc3551b5f643d8920f428909104fe2640a50f6b8a17fff6196aa78f25df28bb6", |
| "description": "Task from inspect_evals/livebench dataset, sample bc3551b5f643d8920f428909104fe2640a50f6b8a17fff6196aa78f25df28bb6", |
| "prompt": "You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format \"<expression Y> = $<latex code>$\" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>. Remember, if you are not sure, give your best guess. \n\nThe question is:\nLet $(m,n)$ be positive integers with $n \\geq 3$ and draw a regular $n$-gon. We wish to triangulate this $n$-gon into $n-2$ triangles each colored one of $m$ colors so that each color has an equal sum of areas. For which $(m,n)$ is such a triangulation and coloring possible?\n\nThe solution is:\nThe answer is if and only if $m$ is a proper divisor of $n$ . Throughout this solution, we let <missing 1> and let the regular $n$ -gon have vertices <missing 2> . We cache the following frequent calculation: Lemma - The triangle with vertices <missing 3> has signed area <missing 4> . Proof. Rotate by <missing 5> to assume WLOG that <missing 6> . Apply complex shoelace to the triangles with vertices <missing 7> to get <missing 8> which equals the above. \u25a1\n\nConstruction: It suffices to actually just take all the diagonals from the vertex 1, and then color the triangles with the $m$ colors in cyclic order. To see this works one can just do the shoelace calculation: for a given residue <missing 9> , we get an area <missing 10> <missing 11> <missing 12> <missing 13> . (We allow degenerate triangles where <missing 14> with area zero.) However, if $m$ is a proper divisor of $n$ , then <missing 15> . Similarly, <missing 16> . So the inner sum vanishes, and the total area of the $m$ -th color equals <missing 17> which does not depend on the residue $r$ , proving the coloring works. \u00b6 Proof of necessity. It's obvious that <missing 18> (in fact <missing 19> ). So we focus on just showing <missing 20> . Repeating the same calculation as above, we find that if there was a valid triangulation and coloring, the total area of each color would equal <missing 21> . However: Claim \u2014 The number <missing 22> is not an algebraic integer when <missing 23> . Proof. This is easiest to see if one knows the advanced result that <missing 24> is a number field whose ring of integers is known to be <missing 25> . Hence if one takes <missing 26> as a <missing 27> -basis of $K$ , then <missing 28> is the subset where each coefficient is integer. \u25a1 However, each of the quantities <missing 29> is <missing 30> times an algebraic integer. Since a finite sum of algebraic integers is also an algebraic integer, such areas can never sum to $S$ .\n\nThe formulae are:\n<expression 1> K := \\mathbb{Q}(\\omega) \n<expression 2> (\\omega^{-1}, \\omega^0, \\omega^1, \\ldots, \\omega^{n-2}) \n<expression 3> 1, \\omega^a, \\omega^b \n<expression 4> \\frac{n}{m} \\frac{(\\omega - 1)(\\omega^{-1} + 1)}{2i} \n<expression 5> = \\frac{\\omega - 1}{2i} \\left(\\frac{n}{m}(1 + \\omega^{-1}) + \\sum_{j \\equiv r \\mod m} (\\omega^{-j} - \\omega^j)\\right) \n<expression 6> T(a,b) \n<expression 7> 1, \\omega, \\ldots, \\omega^{n-1} \n<expression 8> \\frac{1}{2i} \\det \\begin{bmatrix} 1 & 1 & 1 \\ \\omega^a & \\omega^{-a} & 1 \\ \\omega^b & \\omega^{-b} & 1 \\end{bmatrix} = \\frac{1}{2i} \\det \\begin{bmatrix} 0 & 0 & 1 \\ \\omega^a - 1 & \\omega^{-a} - 1 & 1 \\ \\omega^b - 1 & \\omega^{-b} - 1 & 1 \\end{bmatrix} \n<expression 9> = \\sum_{j \\equiv r \\mod m} \\frac{(\\omega^{-j} - 1)(\\omega^1 - 1)(\\omega^j - \\omega^{-1})}{2i} \n<expression 10> \\sum_{j \\equiv r \\mod m} \\omega^j = \\omega^r(1 + \\omega^m + \\omega^{2m} + \\cdots + \\omega^{n-m}) = 0 \n<expression 11> = \\frac{\\omega - 1}{2i} \\sum_{j \\equiv r \\mod m} (\\omega^{-j} - 1)(\\omega^j - \\omega^{-1}) \n<expression 12> m | n \n<expression 13> m < n \n<expression 14> \\omega^{-k} \n<expression 15> \\omega^k, \\omega^{k+a}, \\omega^{k+b} \n<expression 16> \\sum_{j \\equiv r \\mod m} \\text{Area}(\\omega^j, \\omega^0, \\omega^{j+1}) = \\sum_{j \\equiv r \\mod m} T(-j, 1) \n<expression 17> \\mathcal{O}_K \n<expression 18> m \\leq n - 2 \n<expression 19> \\mathbb{Q} \n<expression 20> S := \\frac{n}{m} \\frac{(\\omega - 1)(\\omega^{-1} + 1)}{2i} \n<expression 21> \\sum_{j \\equiv r \\mod m} \\omega^{-j} = 0 \n<expression 22> m \\nmid n \n<expression 23> \\mathcal{O}_K = \\mathbb{Z}[\\omega] \n<expression 24> T(a,b) := \\frac{(\\omega^a - 1)(\\omega^b - 1)(\\omega^{-a} - \\omega^{-b})}{2i} \n<expression 25> 2i \\cdot S \n<expression 26> r \\mod m \n<expression 27> k = 0 \n<expression 28> j \\in {-1,0} \n<expression 29> \\frac{1}{2i} \n<expression 30> \\omega = \\exp(2\\pi i/n) \n\n\nYour final answer should be STRICTLY in the format:\n\n<Detailed reasoning>\n\nAnswer: <comma separated list of numbers representing expression identifiers>", |
| "mcp_config": { |
| "inspect": { |
| "command": "docker", |
| "args": [ |
| "run", |
| "--rm", |
| "-i", |
| "--init", |
| "-v", |
| "/var/run/docker.sock:/var/run/docker.sock", |
| "-v", |
| "/home/genteki/.cache/inspect_evals:/root/.cache/inspect_evals", |
| "hud-inspect:latest" |
| ] |
| } |
| }, |
| "setup_tool": { |
| "name": "load_task", |
| "arguments": { |
| "task_name": "inspect_evals/livebench", |
| "sample_id": "bc3551b5f643d8920f428909104fe2640a50f6b8a17fff6196aa78f25df28bb6", |
| "model": "claude-sonnet-4-20250514" |
| } |
| }, |
| "evaluate_tool": { |
| "name": "score_task", |
| "arguments": {} |
| }, |
| "agent_config": { |
| "agent_class": "server.agents.InspectAgent", |
| "disallowed_tools": [ |
| "_store_agent_message", |
| "load_task", |
| "score_task" |
| ], |
| "allowed_tools": [], |
| "initial_screenshot": false |
| }, |
| "system_prompt": "You are a helpful assistant." |
| } |
| ] |
