id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-b605dec1269503805a528267164ce2aeb162eb86e8af4e829d29a2848e5f516c | A car's tires have a diameter of 33 cm and are turning at a rate of 1 revolution every 4 seconds. Write a sine function for the height \( h \) in centimeters above the road of a piece of chewing gum that sticks to the tire, where \( t = 0 \) is the initial time. | [
"\\( h(t) = 16.5 \\left(1 + \\sin\\left(\\frac{\\pi}{2} (t - 1)\\right)\\right) = 33 \\sin^2\\left(\\frac{\\pi}{4} t\\right) \\)"
] | [
"A car's tires have a diameter of 33 cm and are turning at a rate of 1 revolution every 4 seconds. Write a sine function for the height \\( h \\) in centimeters above the road of a piece of chewing gum that sticks to the tire, where \\( t = 0 \\) is the initial time."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-26f22fd986a8655a57440d9b0b06846d1b97ad3c8da2fcc10ff1f94d027abfa9 | A car's wheels, each with radius \( a \) feet, have an angular velocity of \( \omega \) radians per second. A particle is thrown off from the tire, where it is assumed that \( a \omega^2 > g \). Neglecting air resistance, find the maximum height above the roadway which the particle can reach. Express your answer in ter... | [
"a + \\dfrac{a^2 \\omega^2}{2g}"
] | [
"A car's wheels, each with radius \\( a \\) feet, have an angular velocity of \\( \\omega \\) radians per second. A particle is thrown off from the tire, where it is assumed that \\( a \\omega^2 > g \\). Neglecting air resistance, find the maximum height above the roadway which the particle can reach. Express your ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-71d6f4c3419567ae1c10e890887876b538699edae366bbeef78dd830ca88aee7 | A car, a bike, and a truck travel from city A to city B. The bike reaches city B first, turns around, and starts traveling toward city A at the same speed. On its return trip, the bike meets the car 18 miles from city B and meets the truck 25 miles from city B. The car also turns around after reaching city B and meets ... | [
"60"
] | [
"A car, a bike, and a truck travel from city A to city B. The bike reaches city B first, turns around, and starts traveling toward city A at the same speed. On its return trip, the bike meets the car 18 miles from city B and meets the truck 25 miles from city B. The car also turns around after reaching city B and m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-af6a23d6e1adbc1b61242fa9aa82ad15271ec8bda572de79f9af2c852acdbf65 | A car, a van, a truck, and a bike are all traveling in the same direction on the same road at constant speeds. At 10am, the car overtakes the van; at noon it overtakes the bike. At 4pm the truck overtakes the bike and at 6pm the van overtakes the truck. The speed of the car is 120 km/h and the speed of the truck is 80 ... | [
"70"
] | [
"A car, a van, a truck, and a bike are all traveling in the same direction on the same road at constant speeds. At 10am, the car overtakes the van; at noon it overtakes the bike. At 4pm the truck overtakes the bike and at 6pm the van overtakes the truck. The speed of the car is 120 km/h and the speed of the truck i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7dd686ae9963ec58972812ffc219a766a3d99f329dd39f8ee72bd80b0196f552 | A car, a van, a truck, and a bike are all traveling in the same direction on the same road, each at its own constant speed. At 10am the car overtakes the van; at noon it overtakes the truck; at 2pm it overtakes the bike. At 4pm the truck overtakes the bike, and at 6pm the van overtakes the truck. i) Given that the spee... | [
"i) The speed of the van is 90 km/hr and the speed of the bike is 60 km/hr.\nii) The van overtakes the bike at 6pm."
] | [
"A car, a van, a truck, and a bike are all traveling in the same direction on the same road, each at its own constant speed. At 10am the car overtakes the van; at noon it overtakes the truck; at 2pm it overtakes the bike. At 4pm the truck overtakes the bike, and at 6pm the van overtakes the truck.\n\ni) Given that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3dab86e86a561d602016b6f8b556e2640eb42792166afacda365214448068ad6 | A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written such that no two sets coincide (one card is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if they can discard one card so that each of the ten digits... | [
"All moves except for taking the empty card."
] | [
"A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written such that no two sets coincide (one card is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if they can discard one card so that each of the ten d... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-dd00e13129893dbcba52d5b4ac8d4d550b59832e2c908f2ac34f2b065aed53cb | A card deck consists of 32 cards, each with a different subset of the set \( S = \{algebra, combinatorics, geometry, number theory, sequences\} \) written on it. Let \( N \) be the number of ways to burn 5 cards from this deck so that each element of \( S \) appears an odd number of times among the remaining cards. Fin... | [
"870"
] | [
"A card deck consists of 32 cards, each with a different subset of the set \\( S = \\{algebra, combinatorics, geometry, number theory, sequences\\} \\) written on it. Let \\( N \\) be the number of ways to burn 5 cards from this deck so that each element of \\( S \\) appears an odd number of times among the remaini... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75fa97e3fe8ed881e5230f49a11ef921c9417dc3b0387a8fdd993006e4d84636 | A card is an ordered 4-tuple $(a_1, a_2, a_3, a_4)$ where each $a_i$ is chosen from $\{0, 1, 2\}$. A line is an (unordered) set of three (distinct) cards $\{(a_1, a_2, a_3, a_4), (b_1, b_2, b_3, b_4), (c_1, c_2, c_3, c_4)\}$ such that for each $i$, the numbers $a_i, b_i, c_i$ are either all the same or all different. H... | [
"1080"
] | [
"A card is an ordered 4-tuple $(a_1, a_2, a_3, a_4)$ where each $a_i$ is chosen from $\\{0, 1, 2\\}$. A line is an (unordered) set of three (distinct) cards $\\{(a_1, a_2, a_3, a_4), (b_1, b_2, b_3, b_4), (c_1, c_2, c_3, c_4)\\}$ such that for each $i$, the numbers $a_i, b_i, c_i$ are either all the same or all dif... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30e9cf9be4b5ea6158b453016f4e7ff5d80b889300a816713296e75ce9f6f709 | A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | [
"\\frac{88}{169}"
] | [
"A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-511464bd7f17cabe763a54b149ea7adccc4bb1c5a700afc6f9bbcf7e306a4866 | A card is drawn at random from a Schafkopf deck. What is the probability that it is part of the "trump" suit? | [
"\\dfrac{7}{16}"
] | [
"A card is drawn at random from a Schafkopf deck. What is the probability that it is part of the \"trump\" suit?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-a9747d72c37096302c6cb552030e39e3b5e194f4072b38db384bbdb48469ad7c | A card is drawn at random from a standard deck of 52 cards. What is the probability that it is used in the game of Schafkopf? | [
"\\(\\frac{8}{13}\\)"
] | [
"A card is drawn at random from a standard deck of 52 cards. What is the probability that it is used in the game of Schafkopf?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-685c2d867fc6df88531baf109b28524629c598ca3e40fd96c7a2228adca769a4 | A card is drawn from a pack of 52 cards, the card is then replaced and the pack is reshuffled. This process is repeated six times. What is the probability that the cards drawn include two hearts, two diamonds, and two black cards? | [
"\\(\\frac{45}{512}\\)"
] | [
"A card is drawn from a pack of 52 cards, the card is then replaced and the pack is reshuffled. This process is repeated six times. What is the probability that the cards drawn include two hearts, two diamonds, and two black cards?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c08c500772cc6e9b63983f508a00e28ad4a6dce3ee9b0dddb561ce4313f17a0c | A card is randomly drawn from a standard deck of cards. After recording its value, the card is put back into the deck and the deck is reshuffled. This process is repeated three times. Supposing that aces are counted as 1, and jacks, queens, and kings are all counted as 10, what is the probability that the sum of the nu... | [
"\\dfrac{144}{2197}"
] | [
"A card is randomly drawn from a standard deck of cards. After recording its value, the card is put back into the deck and the deck is reshuffled. This process is repeated three times. Supposing that aces are counted as 1, and jacks, queens, and kings are all counted as 10, what is the probability that the sum of t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be7eecff7e729c70ef3c1b8bbab140a1a655b22422baaef33da30d217f2379c7 | A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top is 60 centimeters and its base has a radius of 30 centimeters, find the volume of the largest box that can be enclosed complet... | [
"64000"
] | [
"A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top is 60 centimeters and its base has a radius of 30 centimeters, find the volume of the largest box that can be enclosed co... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 30,
"genselect": 7
} | 37 | 0 | 0 |
omr-ws-9c040d3b3fdaf82296b7f29c8714f433734cf230f4e4eca10872db340d38a766 | A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has radius $30$ centimeters, find the volume of the largest box that can be comple... | [
"64000"
] | [
"A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has radius $30$ centimeters, find the volume of the largest box that can be c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4663e49c00e09a9e60f964be373e36cce1f5282d6ab201d8ec26d39c7b05f132 | A cardboard windmill with three blades is made from four equilateral triangles of side length 6 cm. Two triangles sharing a common vertex have the corresponding sides lying on the same straight line. The area of the circle swept out by the blades of the windmill is \( x \) cm\(^2\). What is the greatest integer less th... | [
"263"
] | [
"A cardboard windmill with three blades is made from four equilateral triangles of side length 6 cm. Two triangles sharing a common vertex have the corresponding sides lying on the same straight line. The area of the circle swept out by the blades of the windmill is \\( x \\) cm\\(^2\\). What is the greatest intege... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-f29089fc613282ebbc508b3fed3afe95df092ced43940db2342445053059d279 | A caretaker is giving candy to his two babies every minute, choosing one baby at random. Each baby can be in one of five moods: "upset," "sad," "okay," "happy," and "delighted." A baby's mood changes by one level: happier if they receive a candy, sadder if the other baby receives a candy. Both babies start at the "okay... | [
"337"
] | [
"A caretaker is giving candy to his two babies every minute, choosing one baby at random. Each baby can be in one of five moods: \"upset,\" \"sad,\" \"okay,\" \"happy,\" and \"delighted.\" A baby's mood changes by one level: happier if they receive a candy, sadder if the other baby receives a candy. Both babies sta... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-b38b5591f6af21bb74016fcc3410801667f3176f770873651297e9dc4f277b35 | A cargo plane leaves an airport traveling at 180 mph. Two hours later, a jet plane leaves the same airport traveling at 900 mph in the same direction. How long will it take for the jet plane to catch up to the cargo plane? | [
"0.5 hours"
] | [
"A cargo plane leaves an airport traveling at 180 mph. Two hours later, a jet plane leaves the same airport traveling at 900 mph in the same direction. How long will it take for the jet plane to catch up to the cargo plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-94ab5b0bc25bec2623e31cc8097aa236269be284841b0f237c91d8496c335c91 | A carnival game costs $1 to play and offers a $50 prize. The game involves picking cubes out of a bag. There are 6 cubes in the first bag, 4 green and 2 blue. If you pick a green cube, you lose. If you pick a blue cube, you get to pick from a second bag containing 4 cubes, 3 green and 1 blue. If you pick the blue cube ... | [
"\\(\\frac{25}{6}\\)"
] | [
"A carnival game costs $1 to play and offers a $50 prize. The game involves picking cubes out of a bag. There are 6 cubes in the first bag, 4 green and 2 blue. If you pick a green cube, you lose. If you pick a blue cube, you get to pick from a second bag containing 4 cubes, 3 green and 1 blue. If you pick the blue ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-8c1703ca088d4de20c0866bb17631fab0f69da8de0bbe4e306a84f1044676b12 | A carnival game has 1500 pinatas, each containing either 100 carnival tickets or 1 carnival ticket. A logician is told the number of pinatas with 100 tickets and deduces the following: 1) There will be some string of 3 consecutive pinatas that are all worth 100 tickets. 2) In at least one setup, every string of 15 cons... | [
"99"
] | [
"A carnival game has 1500 pinatas, each containing either 100 carnival tickets or 1 carnival ticket. A logician is told the number of pinatas with 100 tickets and deduces the following:\n\n1) There will be some string of 3 consecutive pinatas that are all worth 100 tickets.\n2) In at least one setup, every string o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-b04e4d50569d239ce4df3626a7fffbe1965b10bbcb4ef3fa03bf4a1c2b57c39d | A carnival man moves a pea among three shelves, A, B, and C. Whenever the pea is under A, he moves it with equal probability to A or B (but not C). When it is under B, he is sure to move it to C. When the pea is under C, it is equally likely to move it to A, B, or C. 1. Set up a three-state Markov chain by drawing the ... | [
"\\left( \\dfrac{5}{12}, \\dfrac{5}{12}, \\dfrac{1}{6} \\right)"
] | [
"A carnival man moves a pea among three shelves, A, B, and C. Whenever the pea is under A, he moves it with equal probability to A or B (but not C). When it is under B, he is sure to move it to C. When the pea is under C, it is equally likely to move it to A, B, or C.\n1. Set up a three-state Markov chain by drawin... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a9ea1aa0ef931c89c947b94cde91e712bb3bc975e87a3a3e2cee8414f40584b5 | A carpenter makes wooden crates open at the top by nailing together two rectangular plates $600$ cm$^2$ each, two rectangular plates of $1200$ cm$^2$ each, and a plate rectangular of $800$ cm$^2$. What is the volume, in liters, of the box? Note that $1$ liter $= 1000$ cm$^3$. | [
"24"
] | [
"A carpenter makes wooden crates open at the top by nailing together two rectangular plates $600$ cm$^2$ each, two rectangular plates of $1200$ cm$^2$ each, and a plate rectangular of $800$ cm$^2$. What is the volume, in liters, of the box? Note that $1$ liter $= 1000$ cm$^3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-ef5c35351bc5734c1c53bc2c0f6a6aa2f53bf8c32528893126c6c28b9e39f437 | A carpenter mixes a certain amount of shallots and vinegar to make mignonette sauce. Initially, the mixture has too little vinegar. He adds 9 kg more vinegar, and the new mixture is one-eighth shallot. Then, he adds 1.2 kg of shallots, and the resulting mixture is five-sixths vinegar. What was the proportion of vinegar... | [
"4 to 1"
] | [
"A carpenter mixes a certain amount of shallots and vinegar to make mignonette sauce. Initially, the mixture has too little vinegar. He adds 9 kg more vinegar, and the new mixture is one-eighth shallot. Then, he adds 1.2 kg of shallots, and the resulting mixture is five-sixths vinegar. What was the proportion of vi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-13334b66b810b7563f38e4441379fbbfb986ae1091232621b52b192a3cb799e6 | A carpet dealer starts with a carpet of dimensions \(a \cdot b\) where both \(a > 1\) and \(b > 1\). The dealer can exchange the carpet either with a carpet of dimensions \(\frac{1}{a} \cdot \frac{1}{b}\) or with two carpets of dimensions \(c \cdot b\) and \(\frac{a}{c} \cdot b\) (where \(c\) is chosen by the dealer). ... | [
"6"
] | [
"A carpet dealer starts with a carpet of dimensions \\(a \\cdot b\\) where both \\(a > 1\\) and \\(b > 1\\). The dealer can exchange the carpet either with a carpet of dimensions \\(\\frac{1}{a} \\cdot \\frac{1}{b}\\) or with two carpets of dimensions \\(c \\cdot b\\) and \\(\\frac{a}{c} \\cdot b\\) (where \\(c\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-3973d7b4ffa23d919eb8a53ca68ad403e580bdb066c12825854de0038dd50ffe | A carpet dealer starts with a carpet of dimensions \(a \cdot b\) where both \(a > 1\) and \(b > 1\). The dealer can exchange the carpet either with a carpet of dimensions \(\frac{1}{a} \cdot \frac{1}{b}\) or with two carpets of dimensions \(c \cdot b\) and \(\frac{a}{c} \cdot b\) (where \(c\) is chosen by the dealer). ... | [
"0"
] | [
"A carpet dealer starts with a carpet of dimensions \\(a \\cdot b\\) where both \\(a > 1\\) and \\(b > 1\\). The dealer can exchange the carpet either with a carpet of dimensions \\(\\frac{1}{a} \\cdot \\frac{1}{b}\\) or with two carpets of dimensions \\(c \\cdot b\\) and \\(\\frac{a}{c} \\cdot b\\) (where \\(c\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a6ff9316cdbf3f17af197987a77dd1e7d564006aa4540a1be5101ce96591ca5c | A carpet that is \(15\frac{1}{2}\) ft. long and \(12\frac{1}{2}\) ft. broad is placed centrally on the floor of a room whose length is \(17\) ft. \(3\) ins. and breadth is \(14\) ft. Find: (i) the widths of the sides and ends of the part left uncovered, (ii) the area left uncovered, (iii) the cost of staining this part... | [
"(i) 10.5 inches and 9 inches, (ii) 47.75 square feet, (iii) 0.8953125 £"
] | [
"A carpet that is \\(15\\frac{1}{2}\\) ft. long and \\(12\\frac{1}{2}\\) ft. broad is placed centrally on the floor of a room whose length is \\(17\\) ft. \\(3\\) ins. and breadth is \\(14\\) ft. Find:\n(i) the widths of the sides and ends of the part left uncovered,\n(ii) the area left uncovered,\n(iii) the cost o... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-bc124672d1f7b59f23a59c3175d8145b24e2e475c4bfc3757672d14d25c80339 | A carrom coin is stroked at an angle of $45^\circ$ from one corner of a rectangular board with length $a$ and breadth $b$. Determine the number of rebounds the coin will make before it is pocketed, expressed in terms of $a$ and $b$. | [
"\\(\\left(\\frac{a}{\\gcd(a, b)} + \\frac{b}{\\gcd(a, b)}\\right) - 2\\)"
] | [
"A carrom coin is stroked at an angle of $45^\\circ$ from one corner of a rectangular board with length $a$ and breadth $b$. Determine the number of rebounds the coin will make before it is pocketed, expressed in terms of $a$ and $b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-821eb6602ec6f99deec0e9b6938969aead4d5e27f38202a9116733be6a2dac17 | A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel? | [
"3195"
] | [
"A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-758a0bda09748b7743377cf495a75e9f39003bc6ed1ec67d98c89ca23a9b1f86 | A car’s present value is $20,000, and its value decreases by the same percentage every year. At the end of one year, it will be worth $18,000. What will it be worth at the end of 3 years? | [
"14850"
] | [
"A car’s present value is $20,000, and its value decreases by the same percentage every year. At the end of one year, it will be worth $18,000. What will it be worth at the end of 3 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a307dc92fb98f717b9e5dc59526ee17ffcc72814703d32fe7268ed62689b1bab | A cashier wants to work 5 days a week but also wants to have at least one of Saturday or Sunday off. In how many ways can he choose his work days? | [
"11"
] | [
"A cashier wants to work 5 days a week but also wants to have at least one of Saturday or Sunday off. In how many ways can he choose his work days?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-64344d6425b9c1d3a766d4a51f4b9178e9dc3871db617921827846ffbfe33d2a | A castaway drinks 1 oz from a 16oz bottle of tonic on the first day and refills it with water. On the second day, he drinks 2 oz and refills it with water. On the third day, he drinks 3 oz and refills it with water, and so on, until no tonic is left. How much water did he drink in total? | [
"120"
] | [
"A castaway drinks 1 oz from a 16oz bottle of tonic on the first day and refills it with water. On the second day, he drinks 2 oz and refills it with water. On the third day, he drinks 3 oz and refills it with water, and so on, until no tonic is left. How much water did he drink in total?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a31b537ad860de2f289db314edba0b025c19e6b95257e93d2fc0b12870d98d0a | A casting in the shape of a cube is seen to shrink $0.175$ on a side, with a reduction in volume of $2.48$. Find the original dimensions of the cube. | [
"2.26"
] | [
"A casting in the shape of a cube is seen to shrink $0.175$ on a side, with a reduction in volume of $2.48$. Find the original dimensions of the cube."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd6c72984c1d6151ff3bfa024a6a53e6ab81eaf14e1571c15a4783ac24e00416 | A castle has $k$ doors, each fitted with a different lock. There are 12 guards, each holding keys to certain doors, with a total of 1998 keys among them. Some keys are duplicates for the same door, and each key can only open one door. Given that any four guards can open all the doors, what is the maximum possible value... | [
"222"
] | [
"A castle has $k$ doors, each fitted with a different lock. There are 12 guards, each holding keys to certain doors, with a total of 1998 keys among them. Some keys are duplicates for the same door, and each key can only open one door. Given that any four guards can open all the doors, what is the maximum possible ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-b8d19cbd44700d2fd6c1bae759775207b86ab3c963412d4a18b0572164fd89d7 | A castle has a number of halls and $n$ doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no mo... | [
"$\\max\\{n-1, 2n-4\\}$"
] | [
"A castle has a number of halls and $n$ doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-e5d5d0364b7ae5d945233a3e010a3f6167c64049af6ec47798c5e3045ac3f749 | A cat and a mouse are on adjacent vertices of a rhombus ABCD, with the cat at A and the mouse at B. The mouse's safe house is at O, the intersection of the diagonals. There are holes at B, E (midpoint of BC), and C. The cat runs from A to B, and the mouse runs from B to C. If the mouse had taken route BO, he would have... | [
"4"
] | [
"A cat and a mouse are on adjacent vertices of a rhombus ABCD, with the cat at A and the mouse at B. The mouse's safe house is at O, the intersection of the diagonals. There are holes at B, E (midpoint of BC), and C. The cat runs from A to B, and the mouse runs from B to C. If the mouse had taken route BO, he would... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcf408200225f148d7c944ad30e33c9be256ac0b042a1ff0e0126ed8b7e77611 | A cat begins chasing a mouse at a distance $d_0 \, \text{m}$ in front of him. The mouse is always running in a straight line at a constant velocity $v_m \, \frac{\text{m}}{\text{s}}$ ($v_m > 0$), and the cat's instantaneous velocity $v_c(t)$ at time $t$ is always directly proportional to the distance between him and th... | [
"k > \\frac{v_m}{d_0}"
] | [
"A cat begins chasing a mouse at a distance $d_0 \\, \\text{m}$ in front of him. The mouse is always running in a straight line at a constant velocity $v_m \\, \\frac{\\text{m}}{\\text{s}}$ ($v_m > 0$), and the cat's instantaneous velocity $v_c(t)$ at time $t$ is always directly proportional to the distance between... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-b13159dee16e3810398bebd67937559bbfef6821041cafe00a653a6702bd88e7 | A cat can see 1 mile in any direction. The cat walks around the 13-mile perimeter of a triangle and sees every point inside the triangle. What is the largest possible area, in square miles, of the total region it sees? | [
"\\(\\frac{39}{2} + \\pi\\)"
] | [
"A cat can see 1 mile in any direction. The cat walks around the 13-mile perimeter of a triangle and sees every point inside the triangle. What is the largest possible area, in square miles, of the total region it sees?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 15
} | 27 | 0 | 0 |
omr-ws-78a9f12b5bf34f9144662a4dd499fe3f8f6a6863b3d17380662fa577fd527e47 | A cat catches 81 mice, arranges them in a circle, and numbers them from 1 to 81 in clockwise order. The cat counts "One, Two, Three!" in a clockwise direction, eating the mouse on the count of three and continuing with the next mouse. This process repeats until only two mice are left. If the mouse with the higher numbe... | [
"7"
] | [
"A cat catches 81 mice, arranges them in a circle, and numbers them from 1 to 81 in clockwise order. The cat counts \"One, Two, Three!\" in a clockwise direction, eating the mouse on the count of three and continuing with the next mouse. This process repeats until only two mice are left. If the mouse with the highe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-d4c3aba08f6bde78872d10edd14cc9e3d47eacc9d0ece864815de88b9eb9a141 | A cat has found $432_{9}$ methods in which to extend each of her nine lives. How many methods are there in base 10? | [
"353"
] | [
"A cat has found $432_{9}$ methods in which to extend each of her nine lives. How many methods are there in base 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e41f589990888242e60163ee898b7ab61f840bf44e7e72315104d205ac7b3b17 | A cat is located at point C, 60 meters directly west of a mouse located at point M. The mouse runs at 7 m/s in a direction 30º east of north. The cat runs at 13 m/s in a straight line to intercept the mouse as quickly as possible. Determine the time \( t \) in seconds it takes for the cat to catch the mouse. | [
"\\( \\sqrt{30} \\)"
] | [
"A cat is located at point C, 60 meters directly west of a mouse located at point M. The mouse runs at 7 m/s in a direction 30º east of north. The cat runs at 13 m/s in a straight line to intercept the mouse as quickly as possible. Determine the time \\( t \\) in seconds it takes for the cat to catch the mouse."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b90d673efaa5f1e1368abedf45f5b1c2c16763003d89c1e3381231c6325d4300 | A cat is tied to one corner of the base of a tower, where the base forms an equilateral triangle with side length $4$ m, and the cat is tied with a leash of length $8$ m. Let $A$ be the area of the region accessible to the cat. If $A = \frac{m}{n} k - \sqrt{\ell}$, where $m, n, k, \ell$ are positive integers such that ... | [
"201"
] | [
"A cat is tied to one corner of the base of a tower, where the base forms an equilateral triangle with side length $4$ m, and the cat is tied with a leash of length $8$ m. Let $A$ be the area of the region accessible to the cat. If $A = \\frac{m}{n} k - \\sqrt{\\ell}$, where $m, n, k, \\ell$ are positive integers s... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b0b8bb4f2cdfbb1780fd07119ed069c040c6d01b8c94a424d612f75f998ac97 | A cat is trying to catch a mouse in the non-negative quadrant \(N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}\). At time \(t=0\), the cat is at \((1,1)\) and the mouse is at \((0,0)\). The cat moves with speed \(\sqrt{2}\) and the mouse moves with speed \(1\). Both positions \(c(t)=(c_1(t),c_2(t))\) and \(m(t)=(m_1(t)... | [
"2 + \\sqrt{2}"
] | [
"A cat is trying to catch a mouse in the non-negative quadrant \\(N=\\{(x_1,x_2)\\in \\mathbb{R}^2: x_1,x_2\\geq 0\\}\\). At time \\(t=0\\), the cat is at \\((1,1)\\) and the mouse is at \\((0,0)\\). The cat moves with speed \\(\\sqrt{2}\\) and the mouse moves with speed \\(1\\). Both positions \\(c(t)=(c_1(t),c_2(... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-171b892d62db8d8f16ccc59cdd6b00449614e52cbbe37442e8f0b89b67702c9a | A category line $y=(e^x+e^{-x})/2$ rotates about the abscissa axis. The surface that results is a {\it catenoid}. Calculate the volume of the body formed by the catenoid and two planes which are perpendicular to the abscissa axis and are at the distances $a$ and $b$ from the origin. | [
"$\\frac{\\pi }{4}\\left[\\frac{e^{2b}-e^{2a}}{2}+\\frac{e^{-2a}-e^{-2b}}{2} +2(b-1)\\right]$"
] | [
"A category line $y=(e^x+e^{-x})/2$ rotates about the abscissa axis. The surface that results is a {\\it catenoid}. Calculate the volume of the body formed by the catenoid and two planes which are perpendicular to the abscissa axis and are at the distances $a$ and $b$ from the origin."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a5653457ef84c0e66372e19f0c5a8c5c2f20bf274047ed5933712b194ecc0bf | A cave contains 2 brown bats and 3 gray bats. The bats fly out of the cave one at a time. What is the probability that the fourth bat to fly out of the cave is gray? Express your answer as a common fraction. | [
"\\( \\frac{3}{5} \\)"
] | [
"A cave contains 2 brown bats and 3 gray bats. The bats fly out of the cave one at a time. What is the probability that the fourth bat to fly out of the cave is gray? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b8f78dbe0f253f5c96d35350b127202cfbfe194732a8c8d73a0a7ce0a59adb67 | A cell is cut from an $8 \times 8$ chessboard. An open broken line is built with vertices at the centers of the remaining cells, where each segment of the broken line has a length of $\sqrt{17}$ or $\sqrt{65}$. Determine when the number of such broken lines is larger: when the cut cell is $(1,2)$ or $(3,6)$. | [
"(1,2)"
] | [
"A cell is cut from an $8 \\times 8$ chessboard. An open broken line is built with vertices at the centers of the remaining cells, where each segment of the broken line has a length of $\\sqrt{17}$ or $\\sqrt{65}$. Determine when the number of such broken lines is larger: when the cut cell is $(1,2)$ or $(3,6)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-5eb41688abe7e181815946987a1442a235bd3da36ce2c4b7317ba5beaaa10097 | A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute over 30 hours. In January, Michelle sent 100 text messages and talked for 20.5 hours. How much did she have to pay? | [
"$28.00"
] | [
"A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute over 30 hours. In January, Michelle sent 100 text messages and talked for 20.5 hours. How much did she have to pay?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75cfb31fc31502218801846eb22a7d2033bc3ca0d5c8fc1ce56bac6a4168e74e | A census man is given the sums of the ages of every possible pair of ladies in a house: 30, 33, 41, 58, 66, 69. Calculate the individual ages of the ladies. | [
"11, 19, 22, 44"
] | [
"A census man is given the sums of the ages of every possible pair of ladies in a house: 30, 33, 41, 58, 66, 69. Calculate the individual ages of the ladies."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7509cfb200f046d35670e086ee18ef478edcc1f04790f9e537eea49729470b42 | A centrally symmetric convex polygon $F$ is given in the plane. Six polygons $F_1, \ldots, F_6$ are attached to it, each obtained from $F$ by parallel translations such that: 1) each of the pairs $F$ and $F_1$, $F$ and $F_2$, ..., $F$ and $F_6$ has a common point and does not overlap, 2) figures $F_1, \ldots, F_6$ do n... | [
"7A"
] | [
"A centrally symmetric convex polygon $F$ is given in the plane. Six polygons $F_1, \\ldots, F_6$ are attached to it, each obtained from $F$ by parallel translations such that:\n1) each of the pairs $F$ and $F_1$, $F$ and $F_2$, ..., $F$ and $F_6$ has a common point and does not overlap,\n2) figures $F_1, \\ldots, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4506aadc268fc5b4b6238efc6b32767fb5903f019c82f93ae152c37bae1c93b7 | A certain 8-digit number is divided by 11 and the remainder is \( r \), where \( 0 < r < 11 \). The digits of the number are reversed and then divided by 11. Find the remainder in terms of \( r \). | [
"11 - r"
] | [
"A certain 8-digit number is divided by 11 and the remainder is \\( r \\), where \\( 0 < r < 11 \\). The digits of the number are reversed and then divided by 11. Find the remainder in terms of \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e756de9e28b00a5cbd8ecdddaeff0eba864676d93ee916b233f21596439603d7 | A certain \( 4 \)-digit prime number has all prime digits. When any one of the digits is removed, the remaining three digits form a composite number in their initial order. What is the largest possible value of this number? | [
"7253"
] | [
"A certain \\( 4 \\)-digit prime number has all prime digits. When any one of the digits is removed, the remaining three digits form a composite number in their initial order. What is the largest possible value of this number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-a57cca96ead67d56adb96ca8023ff0c8d5419cac300cc56659c6ef8229b6ec08 | A certain alloy consists of two metals in a ratio of 1 to 2, another alloy contains the same metals in a ratio of 2 to 3. How many parts of both alloys are needed to produce a third alloy containing the metals in a ratio of 17 to 27? | [
"9 parts of the first alloy, and 35 parts of the second"
] | [
"A certain alloy consists of two metals in a ratio of 1 to 2, another alloy contains the same metals in a ratio of 2 to 3. How many parts of both alloys are needed to produce a third alloy containing the metals in a ratio of 17 to 27?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93df71428077c38f8c6f517c912798653e67dcafcf938610a9564af4493d2718 | A certain amount of wire is wound on several 800-m reels, though one reel is not completely filled. The same happens if 900-m reels are used, though the total number of reels is three less. If the wire is wound on 1100-m reels, then the number of reels needed will be a further six less, but all the reels will be full. ... | [
"25300m"
] | [
"A certain amount of wire is wound on several 800-m reels, though one reel is not completely filled. The same happens if 900-m reels are used, though the total number of reels is three less. If the wire is wound on 1100-m reels, then the number of reels needed will be a further six less, but all the reels will be f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c736d9a8e8db0a851f244e97f9c6f1f0aa65205533bf552c049425ec2e020a88 | A certain calculator has 4 functions. Each takes an input $n$ to an output: i) Adds 1 to the number. ii) Squares the number, subtracts 1, and divides by $n-1$. iii) Cubes the number, adds the number, divides by $n+1$, and multiplies by $\frac{2}{n}$. iv) Divides the number by 2 and subtracts 1. Starting from any number... | [
"3"
] | [
"A certain calculator has 4 functions. Each takes an input $n$ to an output:\ni) Adds 1 to the number.\nii) Squares the number, subtracts 1, and divides by $n-1$.\niii) Cubes the number, adds the number, divides by $n+1$, and multiplies by $\\frac{2}{n}$.\niv) Divides the number by 2 and subtracts 1.\nStarting from... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-1a56db33c8a3e2dd0199261c1ee620884e8e6a40ee6edd1283da8d00f7e7b91d | A certain circle's area is $x$ square units, and its circumference is $y$ units. The value of $x + y$ is $80\pi$. What is the radius of the circle, in units? | [
"8"
] | [
"A certain circle's area is $x$ square units, and its circumference is $y$ units. The value of $x + y$ is $80\\pi$. What is the radius of the circle, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4dc824583edaadd53849fdcb4704e6cb0ec993b422cf0bd151b6476f81d538f | A certain coin is weighted such that the chance of flipping heads is $\frac{1}{3}$ and the chance of flipping tails is $\frac{2}{3}$. Suppose that we win $\$3$ if we flip a heads on a coin toss, but lose $\$2$ if we flip tails. What is the expected value, in dollars, of our winnings after one flip? Express your answer ... | [
"-\\frac{1}{3}"
] | [
"A certain coin is weighted such that the chance of flipping heads is $\\frac{1}{3}$ and the chance of flipping tails is $\\frac{2}{3}$. Suppose that we win $\\$3$ if we flip a heads on a coin toss, but lose $\\$2$ if we flip tails. What is the expected value, in dollars, of our winnings after one flip? Express ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bfaadb2b8d424e6220a5e0858f535b24a2340d73f2cbc5c152b7a4d71af40548 | A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017. | [
"3025"
] | [
"A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9a4f535442db92184bdaa9e6b1a7d938ea0e4383a072192b612297b6703156f | A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 12 digits correct using the same rule? | [
"\\( 2 \\cdot 10^8 \\) seconds"
] | [
"A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 12 digits correct using the same rule?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-47ee94f1d8a0ec50579822f114dd1866b9f8afcbbc684569930c6d027f0c5a60 | A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 20 digits correct using the same rule? | [
"\\( 6.34 \\times 10^{8} \\) years"
] | [
"A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 20 digits correct using the same rule?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 29
} | 38 | 0 | 0 |
omr-ws-5fec66ed66e1ecbab8c281002ddc49a1096bd8f10664196a6e4865e901d2ae03 | A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 8 digits correct using the same rule? | [
"\\( 0.634 \\) years"
] | [
"A certain computer takes two seconds to compute a definite integral accurate to 4 digits to the right of the decimal point using the left rectangle rule. How long will it take to get 8 digits correct using the same rule?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-7384330e7aeeef33ceac117af8840a4b1add0274b2c56c8e7e85bbf410b5891a | A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three troublemakers are assigned to the same ... | [
"\\dfrac{105}{1081}"
] | [
"A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three troublemakers are assigned to the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e14e694b3f9ff1cb866e82ac041238c88b44121c7d790202a19d2b7b8058af7 | A certain ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(2, -3 + \sqrt{5})$ and $(2, -3 - \sqrt{5}).$ Find the length of the major axis. | [
"6"
] | [
"A certain ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(2, -3 + \\sqrt{5})$ and $(2, -3 - \\sqrt{5}).$ Find the length of the major axis."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43b264ee8a1b960c993ccfed5a0e2ec05bdf6a21d788bb56611b128982292536 | A certain farm has $b$ bulls and $c$ cows in the first generation. In each generation afterwards, each cow produces three offspring, and each bull produces two offspring. Given that there exists an integer $n > 1$ such that the $n$th generation has $2020$ animals in total, find the number of distinct ordered pairs $(b,... | [
"403"
] | [
"A certain farm has $b$ bulls and $c$ cows in the first generation. In each generation afterwards, each cow produces three offspring, and each bull produces two offspring. Given that there exists an integer $n > 1$ such that the $n$th generation has $2020$ animals in total, find the number of distinct ordered pairs... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-4b7b48e2528fc6603cf82ac94ab6f6d7c49fdc5033f81fb12b5feba59f75aada | A certain force must be applied to move a body of weight $P$ at rest on a horizontal plane. The friction force is proportional to the force pressing the body onto the plane and opposes the force acting to move the body; the proportionality factor (the coefficient of friction) is $k$. At what angle $\alpha $ to the hori... | [
"$\\varphi =\\arctan K$, $F=\\frac{KP}{\\sqrt {1+K^2}}$"
] | [
"A certain force must be applied to move a body of weight $P$ at rest on a horizontal plane. The friction force is proportional to the force pressing the body onto the plane and opposes the force acting to move the body; the proportionality factor (the coefficient of friction) is $k$. At what angle $\\alpha $ to th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2321519fc02a71f3cd7b65e6f4830942fa48a5ba5dc07a1732e5eb646395044c | A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$. | [
"429"
] | [
"A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f9386b4c5a0aebe6aacbee5f512d38710c49f497df83083b1392605bf4b21c57 | A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1-|x-2|$ for $1\le x \le 3$. Find the smallest $x$ for which $f(x) = f(2001)$. | [
"429"
] | [
"A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1-|x-2|$ for $1\\le x \\le 3$. Find the smallest $x$ for which $f(x) = f(2001)$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-8d133b1b5a440baad00fa68f4bd4a0b5c07e0923b9a1cef589d8a1d0235ebbd4 | A certain function \( f \) has the properties that \( f(3x) = 3f(x) \) for all positive real values of \( x \), and that \( f(x) = 1 - |x - 2| \) for \( 1 \le x \le 3 \). Find the smallest \( x \) for which \( f(x) = f(2001) \). | [
"429"
] | [
"A certain function \\( f \\) has the properties that \\( f(3x) = 3f(x) \\) for all positive real values of \\( x \\), and that \\( f(x) = 1 - |x - 2| \\) for \\( 1 \\le x \\le 3 \\). Find the smallest \\( x \\) for which \\( f(x) = f(2001) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ce05b9cd5cbec3c4a17092dd55598153baf8eab74e2e7b5c4a3c71fc22080577 | A certain function \( f \) has the properties that \( f(3x) = 3f(x) \) for all positive real values of \( x \), and that \( f(x) = 1 - |x - 2| \) for \( 1 \le x \le 3 \). Find the smallest \( x \) for which \( f(x) = f(77) \). | [
"13"
] | [
"A certain function \\( f \\) has the properties that \\( f(3x) = 3f(x) \\) for all positive real values of \\( x \\), and that \\( f(x) = 1 - |x - 2| \\) for \\( 1 \\le x \\le 3 \\). Find the smallest \\( x \\) for which \\( f(x) = f(77) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-bf633e6b5c6e3de4b83ac1d8472248e2470e086ec12164e586d037f86ea43f72 | A certain function \( f \) has the properties that \( f(3x) = 3f(x) \) for all positive real values of \( x \), and that \( f(x) = 1 - |x - 2| \) for \( 1 \leq x \leq 3 \). Find the smallest \( x \) for which \( f(x) = f(2001) \). | [
"1.26",
"429"
] | [
"A certain function \\( f \\) has the properties that \\( f(3x) = 3f(x) \\) for all positive real values of \\( x \\), and that \\( f(x) = 1 - |x - 2| \\) for \\( 1 \\leq x \\leq 3 \\). Find the smallest \\( x \\) for which \\( f(x) = f(2001) \\)."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-8b48602ffd764ed7eed3f33cc137512ae37c376d7c6b181a4f5657f6758f9e35 | A certain function has the properties that \( f(3x) = 3f(x) \) for all positive real values of \( x \), and that \( f(x) = 1 - |x - 2| \) for \( 1 \le x \le 3 \). Find the smallest \( x \) for which \( f(x) = f(2001) \). | [
"429"
] | [
"A certain function has the properties that \\( f(3x) = 3f(x) \\) for all positive real values of \\( x \\), and that \\( f(x) = 1 - |x - 2| \\) for \\( 1 \\le x \\le 3 \\). Find the smallest \\( x \\) for which \\( f(x) = f(2001) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-dbdcdb2ec4795006a69ac0d2f870eded4a75b2cca7943f8bb49e882c2ae512cf | A certain integer has $4$ digits when written in base $8$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$? | [
"33"
] | [
"A certain integer has $4$ digits when written in base $8$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-573cafee5b513acc0b8813fb1db1487b1964c4c213589eca19089de2358fd797 | A certain junior class has 1000 students and a certain senior class has 800 students. Among these students, there are 60 sibling pairs each consisting of 1 junior and 1 senior. If 1 student is to be selected at random from each class, what is the probability that the 2 students selected will be a sibling pair? | [
"\\dfrac{3}{40000}"
] | [
"A certain junior class has 1000 students and a certain senior class has 800 students. Among these students, there are 60 sibling pairs each consisting of 1 junior and 1 senior. If 1 student is to be selected at random from each class, what is the probability that the 2 students selected will be a sibling pair?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fa506d9f812111b89c3647d0b4b9029543524ae9aaa26aeeac9aa8220d80dbf | A certain number $M$ has a $3$ as its unit digit when written in the decimal expansion. A new number $N$ is formed from $M$ by moving the unit digit to the leftmost position. For example, if $M = 213$ then $N = 321$. What is the smallest number $M$ such that $N = 3M$? | [
"\\( \\frac{3(10^{28}-1)}{29} \\)"
] | [
"A certain number $M$ has a $3$ as its unit digit when written in the decimal expansion. A new number $N$ is formed from $M$ by moving the unit digit to the leftmost position. For example, if $M = 213$ then $N = 321$. What is the smallest number $M$ such that $N = 3M$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ba24d9eb653c7428ca2129100b0ad9a15230f52c23a8f4513d62117bdbaaaa59 | A certain number of days into the future is a Wednesday. What are the possibilities for how many days away this day is? Can you characterize all of them? | [
"\\(7n + x\\) where \\(n\\) is an integer and \\(x\\) is the number of days from the current day to the next Wednesday."
] | [
"A certain number of days into the future is a Wednesday. What are the possibilities for how many days away this day is? Can you characterize all of them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-2d7b6c8855611a9bfcc1f8f039994f10013ad86e21eafb0f2441dfce27b17510 | A certain number of persons were engaged to do a piece of work which would have taken them 24 hours if they had started at the same time. Instead, they started at equal intervals and continued to work until the whole was finished. The payment was proportional to the work done by each. The first person received 11 times... | [
"44"
] | [
"A certain number of persons were engaged to do a piece of work which would have taken them 24 hours if they had started at the same time. Instead, they started at equal intervals and continued to work until the whole was finished. The payment was proportional to the work done by each. The first person received 11 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-dc4835e17e1d92c9f1870161a5e788a8ab96dd3272014e4a39e858f1c36450b2 | A certain organism begins as three cells. Each cell splits and becomes two cells at the end of two days. At the end of another two days, every cell of the organism splits and becomes two cells. This process lasts for a total of 8 days, and no cells die during this time. How many cells are there at the end of the $8^\te... | [
"48"
] | [
"A certain organism begins as three cells. Each cell splits and becomes two cells at the end of two days. At the end of another two days, every cell of the organism splits and becomes two cells. This process lasts for a total of 8 days, and no cells die during this time. How many cells are there at the end of the $... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6ea4fea954d9a1dce16596953cd5f4e036c8556a901a3f7d64f081119e4da95 | A certain organization consists of five leaders and some number of regular members. Every year, the current leaders are kicked out of the organization. Next, each regular member must find two new people to join as regular members. Finally, five new people are elected from outside the organization to become leaders. In ... | [
"2435"
] | [
"A certain organization consists of five leaders and some number of regular members. Every year, the current leaders are kicked out of the organization. Next, each regular member must find two new people to join as regular members. Finally, five new people are elected from outside the organization to become leaders... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3dce62d2917d43186ed2585123b8f0ce09dad13a14674a5a68c9e4fe0900b3e2 | A certain parallelogram has sides of 10, 9, 10, and 9 units. The longest diagonal of this parallelogram is 17 units. If the other diagonal of this parallelogram is of length \( x \), what is \( \sqrt{x^2 + 289} \)? Express your answer in simplest radical form. | [
"\\(\\sqrt{362}\\)"
] | [
"A certain parallelogram has sides of 10, 9, 10, and 9 units. The longest diagonal of this parallelogram is 17 units. If the other diagonal of this parallelogram is of length \\( x \\), what is \\( \\sqrt{x^2 + 289} \\)? Express your answer in simplest radical form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-2dcdf3a2a6a295861d0bf3660f02515c12cbafe7f340bbb8ee42f419d25523e3 | A certain party of 2020 people has the property that, for any 4 people in the party, there is at least one person of those 4 that is friends with the other three (assume friendship is mutual). Call a person in the party a politician if they are friends with the other 2019 people in the party. If \( n \) is the number o... | [
"6055"
] | [
"A certain party of 2020 people has the property that, for any 4 people in the party, there is at least one person of those 4 that is friends with the other three (assume friendship is mutual). Call a person in the party a politician if they are friends with the other 2019 people in the party. If \\( n \\) is the n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-1d318f6b94c10249cf88741d6fa964c4cd8f59f12c1c6d26f52ca9fd08742437 | A certain point \( P \) is exactly 1, 2, and 3 units away from three corners of a square. Find the distance to the remaining corner and the side length of the square. | [
"Distance to the remaining corner: \\(\\sqrt{6}\\), Side length of the square: \\(\\sqrt{5+2\\sqrt{2}}\\)"
] | [
"A certain point \\( P \\) is exactly 1, 2, and 3 units away from three corners of a square. Find the distance to the remaining corner and the side length of the square."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-c4707ad1d135286399c41d8831ecb773f607584a44c19ebdd3968c52517530da | A certain point has rectangular coordinates $(10,3)$ and polar coordinates $(r, \theta).$ What are the rectangular coordinates of the point with polar coordinates $(r^2, 2 \theta)$? | [
"(91,60)"
] | [
"A certain point has rectangular coordinates $(10,3)$ and polar coordinates $(r, \\theta).$ What are the rectangular coordinates of the point with polar coordinates $(r^2, 2 \\theta)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d235304536f986a5a00c00dbb9f798eadec7e1f8b6eb23caf8cddca5b8cfbdb | A certain polygon has angles measuring only either 168 degrees or 132 degrees. Moreover, each vertex of a 168 degree angle in the polygon is between one vertex of a 168 degree angle and a vertex of a 132 degree angle, while each vertex of a 132 degree angle is between two vertices of 168 degree angles. How many sides d... | [
"15"
] | [
"A certain polygon has angles measuring only either 168 degrees or 132 degrees. Moreover, each vertex of a 168 degree angle in the polygon is between one vertex of a 168 degree angle and a vertex of a 132 degree angle, while each vertex of a 132 degree angle is between two vertices of 168 degree angles. How many si... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a4a0077766443923bebf3d2317b691b76ea1693e60f419a32078595dff44919 | A certain post office only uses rectangular prism-shaped packages that come in two sizes: small and large. All small packages have a surface area of 200 and a volume of 160. Doubling the height in a small package will result in a large package. If the surface area of each large package is equal to its volume, what is t... | [
"\\(\\sqrt{2} + 1\\)."
] | [
"A certain post office only uses rectangular prism-shaped packages that come in two sizes: small and large. All small packages have a surface area of 200 and a volume of 160. Doubling the height in a small package will result in a large package. If the surface area of each large package is equal to its volume, what... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2a7c35a3f933952fbd46dc575b1c896636ec65af9b201598c58ce71a1023856 | A certain rectangle has an area of $2(\sqrt{5} - 1)$ and has the property that when a square is removed from one end, the remaining rectangle is similar to the original rectangle. Find the length of the original rectangle. | [
"2"
] | [
"A certain rectangle has an area of $2(\\sqrt{5} - 1)$ and has the property that when a square is removed from one end, the remaining rectangle is similar to the original rectangle. Find the length of the original rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb9ffca5725fdfbcb20b9c70b8a1c2cfc4e203d8c52b69056dc68299ee8d8873 | A certain reduced fraction \(\frac{a}{b}\) (with \(a, b > 1\)) has the property that when 2 is subtracted from the numerator and added to the denominator, the resulting fraction has \(\frac{1}{6}\) of its original value. Find this fraction. | [
"\\(\\frac{3}{2}\\)"
] | [
"A certain reduced fraction \\(\\frac{a}{b}\\) (with \\(a, b > 1\\)) has the property that when 2 is subtracted from the numerator and added to the denominator, the resulting fraction has \\(\\frac{1}{6}\\) of its original value. Find this fraction."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-82bdabb55ee46e225a540c2e228bf8ef8c77024a8a2447660ee325d0a561f320 | A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers. | [
"(3,-2,2)"
] | [
"A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d844488b576b5fa8aec56c257a8105bada9a6b066fe010640ea5a31b97dc365a | A certain sporting goods store imported 100 basketballs and 80 volleyballs for $2800. When selling the products, the price for each basketball increased by 5% and for each volleyball by 10%. After all the products are sold, the store collected $3020. How much is the sum of the prices of one basketball and one volleybal... | [
"32"
] | [
"A certain sporting goods store imported 100 basketballs and 80 volleyballs for $2800. When selling the products, the price for each basketball increased by 5% and for each volleyball by 10%. After all the products are sold, the store collected $3020. How much is the sum of the prices of one basketball and one voll... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c8aa4c793e6c00b6fe66dd1aa997a479c43bc445e279775ab2aa90aa5edfceb9 | A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? | [
"3"
] | [
"A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60b0e0f69a1248ac43eda43ecc0fb69e42387df2073d3e55b92d4483c4e4dac2 | A certain sum of money was arranged in $n$ piles. An $n$th part of the money in the first pile was taken from it and put into the second pile. Then an $n$th part of the money in the enlarged second pile was taken from it and put into the third pile. The same operation was continued from the third to the fourth pile, an... | [
"$x_1=\\frac{n^2-2n}{(n-1)^2}A$; $x_2=\\frac{n^2-2n+2}{(n-1)^2}A$; $x_3=x_4=\\ldots =x_n=A$"
] | [
"A certain sum of money was arranged in $n$ piles. An $n$th part of the money in the first pile was taken from it and put into the second pile. Then an $n$th part of the money in the enlarged second pile was taken from it and put into the third pile. The same operation was continued from the third to the fourth pil... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0807beb7580937970558a11561d4b93680ba53c7f9115740e4b70264bdf2a83c | A certain sum was to be laid out in the funds, with the interest rates of 5 per cents, 4 per cents, and 3 per cents forming a decreasing arithmetic progression with a common difference of $\frac{1}{4}$. The money was divided between the 5 per cents and the 3 per cents. After 8 years, all the funds rose by 4 per cent an... | [
"£100,000, £39,062 10s., £60,937 10s., £4$\\tfrac{3}{4}$, £4$\\tfrac{1}{2}$, £4$\\tfrac{1}{4}$"
] | [
"A certain sum was to be laid out in the funds, with the interest rates of 5 per cents, 4 per cents, and 3 per cents forming a decreasing arithmetic progression with a common difference of $\\frac{1}{4}$. The money was divided between the 5 per cents and the 3 per cents. After 8 years, all the funds rose by 4 per c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-944105fad49316f71b43c2d4c3e1da3dfb2cfaf78dd922e578b4a240cad18e8d | A certain telescope contains both a parabolic mirror and a hyperbolic mirror. The parabola and hyperbola share a focus point F, which is 14 meters above the vertex of the parabola. The hyperbola's second focus is 2 meters above the parabola's vertex. The vertex of the hyperbolic mirror is 1 meter below F. Position a co... | [
"\\(\\frac{x^2}{11} - \\frac{y^2}{25} = -1\\)"
] | [
"A certain telescope contains both a parabolic mirror and a hyperbolic mirror. The parabola and hyperbola share a focus point F, which is 14 meters above the vertex of the parabola. The hyperbola's second focus is 2 meters above the parabola's vertex. The vertex of the hyperbolic mirror is 1 meter below F. Position... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-514f2f60ae238ef269b51205095bb7c070531358eca14d8f22b20a7fb5567468 | A certain type of cereal boxes come with different collectible cards: red, blue, and green. Each box contains at most one card. The probabilities of getting no card, a red card, a blue card, and a green card are $\frac{1}{2}, \frac{1}{3}, \frac{1}{8}, \frac{1}{24}$, respectively. What is the expected number of cereal b... | [
"17.5"
] | [
"A certain type of cereal boxes come with different collectible cards: red, blue, and green. Each box contains at most one card. The probabilities of getting no card, a red card, a blue card, and a green card are $\\frac{1}{2}, \\frac{1}{3}, \\frac{1}{8}, \\frac{1}{24}$, respectively. What is the expected number of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d7d8cab2ef6a8ca6a8010402bd388cf97a383099f5791672b67af0ba03bb87e | A chain consists of 2000 links. What is the least number of the links of the chain that should be cut so that it is possible to take any number of links ranging from 1 to 2000 by using the parts of the chain thus obtained? | [
"One link"
] | [
"A chain consists of 2000 links. What is the least number of the links of the chain that should be cut so that it is possible to take any number of links ranging from 1 to 2000 by using the parts of the chain thus obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6d902cf6543988bb579ef2ae3a740d6731e13dc9d1d68d3c504599c5fe88c37 | A chain has \( n > 3 \) numbered links. A customer asks for the order of the links to be changed to a new order. The jeweller opens the smallest possible number of links, but the customer chooses the new order to maximize this number. How many links have to be opened? | [
"\\( \\lceil \\frac{n}{2} \\rceil \\)"
] | [
"A chain has \\( n > 3 \\) numbered links. A customer asks for the order of the links to be changed to a new order. The jeweller opens the smallest possible number of links, but the customer chooses the new order to maximize this number. How many links have to be opened?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-373df394a4ca8188d3634098b8410d7322495b3ee2bf2545028f01984cf071b1 | A chain has a density that starts at a unit value and then becomes a linear function of distance from one end. When the chain is hung over a one-unit table, one hanging piece is twice as long as the other hanging piece. Determine the density function of the chain. | [
"1 - \\dfrac{2}{7}x"
] | [
"A chain has a density that starts at a unit value and then becomes a linear function of distance from one end. When the chain is hung over a one-unit table, one hanging piece is twice as long as the other hanging piece. Determine the density function of the chain."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-af4d4ff31c9b749b59c16f46c2b0041632e95196cd82ffecfec363c67fde4cc9 | A chain of 17 electrical bulbs is circular, with each bulb initially on. An operation consists of sending a signal from one bulb to the next: if the bulb the signal leaves is on, the next bulb changes status; if off, the next bulb does not change status. How many operations does it take, at least, for all the bulbs to ... | [
"34"
] | [
"A chain of 17 electrical bulbs is circular, with each bulb initially on. An operation consists of sending a signal from one bulb to the next: if the bulb the signal leaves is on, the next bulb changes status; if off, the next bulb does not change status. How many operations does it take, at least, for all the bulb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-ffca4014dedff96d09f18af0c2a7a4e4224a1f775846cfce70be04f1ae814610 | A changing point of a sequence is defined as a term (other than the first term) that differs from the previous term. Find the average number of changing points for all possible binary sequences of \( p \) zeros and \( q \) ones, where \( p \leq q \). | [
"\\(\\frac{2pq}{p+q}\\)"
] | [
"A changing point of a sequence is defined as a term (other than the first term) that differs from the previous term. Find the average number of changing points for all possible binary sequences of \\( p \\) zeros and \\( q \\) ones, where \\( p \\leq q \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-063e4d599da299dd37b09fb9b6eaff0dbb48bcfca28b38c3b88fe09e3bcf718b | A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of $ \\$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-ee2b94ed04f4f89f3ccedf8ac186ac6ce4c5861fd5b3684c61303fce5f611288 | A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest for half price. How much money is raised by the full-price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest for half price. How much money is raised by the full-price tickets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-bbbedcc129f693612fdb6071b5738123873412f0e90be7bae06af2d50ae72d15 | A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full price tickets?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 18
} | 50 | 0 | 0 |
omr-ws-38434364e18a87711630687a6fb72088a1978ae6f6cd2962ac87461e881bc5a2 | A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money (in dollars) is raised by the full-price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money (in dollars) is raised by the full-price tickets?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.