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-a1c8c11efc9dd8ccb00bb1f64ce4a35f34fafed2805e7d038599ca334aab3c39 | Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points $P$ and $Q$. A line $l$ is drawn through $P$ and intersects the circle ω1 at $X \neq P$ and ω2 at $Y \neq P$. Find the maximum value of $PX \cdot PY$. | [
"120"
] | [
"Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points $P$ and $Q$. A line $l$ is drawn through $P$ and intersects the circle ω1 at $X \\neq P$ and ω2 at $Y \\neq P$. Find the maximum value of $PX \\cdot PY$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-5b3bcd236eece6116eb1f3c17fae143119a9b0b7723562008f237d357bdc7e98 | Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points P and Q. A line L is drawn through P and intersects circle ω1 at $X \neq P$ and circle ω2 at $Y \neq P$. Find the maximum value of PX·PY. | [
"120"
] | [
"Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points P and Q. A line L is drawn through P and intersects circle ω1 at $X \\neq P$ and circle ω2 at $Y \\neq P$. Find the maximum value of PX·PY."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-9aa99cc86aa5af79184a5323c66a6ba1b684ec9b508bc0a2e1ce079ffd77b3d2 | Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points P and Q. A line l is drawn through P and intersects the circle ω1 at X (distinct from P) and ω2 at Y (distinct from P). Find the maximum value of PX·PY. | [
"130"
] | [
"Two circles, ω1 and ω2, have radii of 5 and 12 respectively, and their centers are 13 units apart. The circles intersect at two different points P and Q. A line l is drawn through P and intersects the circle ω1 at X (distinct from P) and ω2 at Y (distinct from P). Find the maximum value of PX·PY."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-415f8b9ff1e5274de327550efece16f98148091fc0a6f026247fe13a9d0b49c9 | Two circular mirrors of radius 1, centered at points $(0, 5)$ and $(0, -5)$ respectively, are given. A light source is located at $(-3, 1)$. An emitted particle of light bounces off of the first mirror, then the second mirror, and finally reaches a receptor at $(3, -1)$. Calculate the total distance traveled by the par... | [
"\\(4\\sqrt{26+3\\sqrt{10}}\\)"
] | [
"Two circular mirrors of radius 1, centered at points $(0, 5)$ and $(0, -5)$ respectively, are given. A light source is located at $(-3, 1)$. An emitted particle of light bounces off of the first mirror, then the second mirror, and finally reaches a receptor at $(3, -1)$. Calculate the total distance traveled by th... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-1df5053c9fd240b53a8cd508d2e3d7205263c07a864055cd4ddb79b068f8d4b4 | Two circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger pulley, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches? | [
"\\( 7\\pi + 8\\sqrt{3} \\)"
] | [
"Two circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger pulley, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-29cb46c79f113e3216282513731731b9fc4bc55e285747d2614e05b7a3b4d536 | Two circular pulleys with radii 40 cm and 10 cm are connected by a well-tensioned belt. The centers of the pulleys are 60 cm apart. What is the length of the belt? | [
"\\( 60(\\pi + \\sqrt{3}) \\)"
] | [
"Two circular pulleys with radii 40 cm and 10 cm are connected by a well-tensioned belt. The centers of the pulleys are 60 cm apart. What is the length of the belt?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-3b3aaea0ff142e8f3f905bbd1bdcd75089b030d164a340114a35c8bbda0c59d7 | Two circumferences \( C_{1} \) and \( C_{2} \) intersect at two points A and B. Construct a segment PQ by point B with one end on \( C_{1} \) and the other on \( C_{2} \) such that PQ is the greatest possible. | [
"\\( 2 O_1 O_2 \\)"
] | [
"Two circumferences \\( C_{1} \\) and \\( C_{2} \\) intersect at two points A and B. Construct a segment PQ by point B with one end on \\( C_{1} \\) and the other on \\( C_{2} \\) such that PQ is the greatest possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 7,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-568d40151d4521af14d49ab1f9b5c72be863a85655a3053efd5bb5d9f5887c68 | Two circumferences \(\omega_1\) and \(\omega_2\) with centers \(O_1\) and \(O_2\) meet at points \(J\) and \(M\). The line \(BC\) is a common tangent such that \(B\) belongs to \(\omega_1\) and \(C\) belongs to \(\omega_2\). The line \(BC\) meets \(O_1O_2\) at \(F\). The line \(FM\) meets \(\omega_1\) and \(\omega_2\) ... | [
"90"
] | [
"Two circumferences \\(\\omega_1\\) and \\(\\omega_2\\) with centers \\(O_1\\) and \\(O_2\\) meet at points \\(J\\) and \\(M\\). The line \\(BC\\) is a common tangent such that \\(B\\) belongs to \\(\\omega_1\\) and \\(C\\) belongs to \\(\\omega_2\\). The line \\(BC\\) meets \\(O_1O_2\\) at \\(F\\). The line \\(FM\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7904ff3b9d3d9b7f1fcbf4d23f2266261adcd9c3f0820b954cadda3f9045ca6 | Two circumferences intersect at two points, one of which is $A$. From an arbitrary point on the extension of the common chord, two tangents are drawn to one of the circumferences, with contact points $M$ and $N$. Let $P$ and $Q$ be the intersection points of the lines $MA$ and $NA$ with the second circumference. If the... | [
"10"
] | [
"Two circumferences intersect at two points, one of which is $A$. From an arbitrary point on the extension of the common chord, two tangents are drawn to one of the circumferences, with contact points $M$ and $N$. Let $P$ and $Q$ be the intersection points of the lines $MA$ and $NA$ with the second circumference. I... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bba6b952886f6618c1b158d5e9a976e66f5de32e3d689afef84ca4e857392ea0 | Two circumferences of radii \( R_1 \) and \( R_2 \) are tangent externally to each other and both are tangent to a semicircle with radius 1. If \( A_1 \) and \( A_2 \) are the tangency points of the two circumferences with the diameter of the semicircle, find the length of \( \overline{A_1 A_2} \). | [
"\\( 2 \\times \\sqrt{R_1 \\times R_2} \\)"
] | [
"Two circumferences of radii \\( R_1 \\) and \\( R_2 \\) are tangent externally to each other and both are tangent to a semicircle with radius 1. If \\( A_1 \\) and \\( A_2 \\) are the tangency points of the two circumferences with the diameter of the semicircle, find the length of \\( \\overline{A_1 A_2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 7,
"tir": 1
} | 41 | 0 | 0 |
omr-ws-ab19027a711693c38701a7ac7a62648182ab1dc095fd1e3b9feff756df76c2a8 | Two circumferences of radii \( r \) and \( R \) (\( R > r \)) are tangent internally. Find the radius of the circumference that is tangent to these two circumferences and the line that connects the two centers. | [
"\\( \\frac{4Rr(R - r)}{(R + r)^2} \\)"
] | [
"Two circumferences of radii \\( r \\) and \\( R \\) (\\( R > r \\)) are tangent internally. Find the radius of the circumference that is tangent to these two circumferences and the line that connects the two centers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 4
} | 54 | 0 | 0 |
omr-ws-cfffe0723f0af439d372be76af2125b81fa21607e37f57b5aaef8e95304d5aae | Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the ... | [
"4"
] | [
"Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f9b5149f6d90fad6c8b971d6e1f3adc7ff8393d32c3246f782eb6241ea195c4 | Two circumferences of radius $1$, $c_1$ and $c_2$, are placed inside an angle with vertex $O$. $c_1$ is tangent to one ray of the angle, and $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and ... | [
"2",
"4"
] | [
"Two circumferences of radius $1$, $c_1$ and $c_2$, are placed inside an angle with vertex $O$. $c_1$ is tangent to one ray of the angle, and $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 4
} | 27 | 0 | 0 |
omr-ws-fab36e47bc7febac17a9491ea19608901551fffeee1ca4bf6da8c064ed49ca4e | Two circumferences with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). A line is tangent to the circumferences at points \( C \) and \( D \), with \( CD \) closer to \( B \). If \( \angle O_1AO_2 = 120^\circ \), calculate the measure of \( \angle CAD \). | [
"60^\\circ"
] | [
"Two circumferences with centers \\( O_1 \\) and \\( O_2 \\) intersect at points \\( A \\) and \\( B \\). A line is tangent to the circumferences at points \\( C \\) and \\( D \\), with \\( CD \\) closer to \\( B \\). If \\( \\angle O_1AO_2 = 120^\\circ \\), calculate the measure of \\( \\angle CAD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-5dd50f6e4b1cac159ced728f500ed5147e8462a60a2f4e0c367a4c14d1dc2c75 | Two cities are $150$ miles apart using a superhighway and $231$ miles apart using a scenic route. The scenic route requires an average speed that is $18$ mph slower than the average speed on the superhighway and takes three hours longer to travel. How long does the direct route (superhighway) take? | [
"\\( \\frac{5}{2} \\) hours or 2.5 hours."
] | [
"Two cities are $150$ miles apart using a superhighway and $231$ miles apart using a scenic route. The scenic route requires an average speed that is $18$ mph slower than the average speed on the superhighway and takes three hours longer to travel. How long does the direct route (superhighway) take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-cdbffb0093bacdbb221b5f5c5e89532526d169670cfbe5a535c44508a73b3fd0 | Two cities, Harmony and Dissension, are 455 miles apart. Paul Haymaker leaves Harmony at 12:00 noon traveling towards Dissension at 60 miles per hour. Simultaneously, Nick Plougham leaves Dissension traveling towards Harmony at 45 miles per hour. At what time will they meet? | [
"4:20 PM"
] | [
"Two cities, Harmony and Dissension, are 455 miles apart. Paul Haymaker leaves Harmony at 12:00 noon traveling towards Dissension at 60 miles per hour. Simultaneously, Nick Plougham leaves Dissension traveling towards Harmony at 45 miles per hour. At what time will they meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c29d1b17563ee5dacbe2abcc803af60faeee54e3bad1a8633449311d178fe919 | Two clerks, A and B, go out for lunch for one hour each. Clerk A goes out between 1 pm and 2 pm, and Clerk B goes out between 2 pm and 3 pm. Find the probability that they will not be out for lunch at the same time. | [
"1"
] | [
"Two clerks, A and B, go out for lunch for one hour each. Clerk A goes out between 1 pm and 2 pm, and Clerk B goes out between 2 pm and 3 pm. Find the probability that they will not be out for lunch at the same time."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-537db5f86afc3b59452a0e0337edbb70cfdbb80c873f9bfdb615e516397f57c2 | Two clocks agree at a certain time. The first clock loses 3.4 seconds every hour, and the second clock gains 2.6 seconds every hour. How many hours will it take until the clocks show the same time again? | [
"7200"
] | [
"Two clocks agree at a certain time. The first clock loses 3.4 seconds every hour, and the second clock gains 2.6 seconds every hour. How many hours will it take until the clocks show the same time again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-291ee353f275b26679ce68fd29c4812582737b50ed0c7e3a01ab41a01fe897bb | Two clocks start at 12 noon. The first clock gains 2 minutes every 12 hours, and the second clock loses 2 minutes every 36 hours. What is the correct time the next time both clocks show the same time? | [
"12:00 noon, 135 days from the starting time."
] | [
"Two clocks start at 12 noon. The first clock gains 2 minutes every 12 hours, and the second clock loses 2 minutes every 36 hours. What is the correct time the next time both clocks show the same time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 19
} | 41 | 0 | 0 |
omr-ws-36db6f68fb58afd25a7143fbe2cffd1f938688b1a0821f4022d3089a77528bb6 | Two cockroaches start running in the same direction from a marked point on a circle with a 1-meter circumference. Whenever the faster one catches up with the slower one, the slower one turns around and runs in the opposite direction. Whenever they meet face-to-face, the faster one turns around and runs in the opposite ... | [
"0"
] | [
"Two cockroaches start running in the same direction from a marked point on a circle with a 1-meter circumference. Whenever the faster one catches up with the slower one, the slower one turns around and runs in the opposite direction. Whenever they meet face-to-face, the faster one turns around and runs in the oppo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-bd95b3b90fb20ba89a195974a13f413d39bb14a9fb86055f5d8fa43e776438a7 | Two coins are biased such that when both are tossed once: - The probability of getting two heads is the same as the probability of getting two tails. - The probability of getting one head and one tail is $\frac{5}{8}$. For each coin, determine the probability of getting a head. | [
"\\(\\frac{1}{4}\\) and \\(\\frac{3}{4}\\)"
] | [
"Two coins are biased such that when both are tossed once:\n- The probability of getting two heads is the same as the probability of getting two tails.\n- The probability of getting one head and one tail is $\\frac{5}{8}$.\n\nFor each coin, determine the probability of getting a head."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-7fefa9609b47b90140e4d6b81d339d2543c11e12b2e6648cec16c1d635ac41a1 | Two coins are removed randomly and without replacement from a box containing 3 nickels, 2 dimes, and 1 coin of value $0$. What is the probability that one of the removed coins is worth five cents more than the other? | [
"\\dfrac{3}{5}"
] | [
"Two coins are removed randomly and without replacement from a box containing 3 nickels, 2 dimes, and 1 coin of value $0$. What is the probability that one of the removed coins is worth five cents more than the other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a33c920f8f86d0bb3b7d52f2380e98c4e6818c7f7b240429cf773b673ff57209 | Two coins are tossed independently with probabilities of heads being \( p_1 \) and \( p_2 \) respectively. Given that at least one head has occurred, what is the probability that the coins produced different outcomes? | [
"\\(\\frac{p_1(1-p_2) + (1-p_1)p_2}{1 - (1-p_1)(1-p_2)}\\)"
] | [
"Two coins are tossed independently with probabilities of heads being \\( p_1 \\) and \\( p_2 \\) respectively. Given that at least one head has occurred, what is the probability that the coins produced different outcomes?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-e5658144c69ea2f578af1e1cf0de6a7dde079d5c4efccf07a7b736ba0e7e382e | Two coins are tossed simultaneously 300 times, resulting in the following outcomes: Two tails: 83 times, One tail: 140 times, No tails: 77 times. Find the probability of each of these events occurring. | [
"\\( \\frac{83}{300}, \\frac{140}{300}, \\frac{77}{300} \\)"
] | [
"Two coins are tossed simultaneously 300 times, resulting in the following outcomes: Two tails: 83 times, One tail: 140 times, No tails: 77 times. Find the probability of each of these events occurring."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 19
} | 25 | 0 | 0 |
omr-ws-3ebd8aac8815aed9bf6ab4e3eae0ec6ef9b460917de220ab6404b0448dabef18 | Two collective farmers together brought 100 eggs to market. Having sold their eggs at different prices, both farmers made equal sums of money. If the first farmer had sold as many eggs as the second, she would have received 72 roubles; if the second farmer had sold as many eggs as the first, she would have received 32 ... | [
"The first had 40 eggs; the second, 60 eggs"
] | [
"Two collective farmers together brought 100 eggs to market. Having sold their eggs at different prices, both farmers made equal sums of money. If the first farmer had sold as many eggs as the second, she would have received 72 roubles; if the second farmer had sold as many eggs as the first, she would have receive... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-775624440b02965f98f2a09380d1cc55a47b575c1dd9e6008f1b2a9c15e225a5 | Two collective farmers with a total of a lit res of milk, though selling the milk at different prices, made equal sums of money. If the first farmer had sold as much milk as the second, she would have received $m$ roubles, and if the second farmer had sold as much milk as the first, she would have received $n$ roubles ... | [
"The first had $\\frac{a\\sqrt n}{\\sqrt m+\\sqrt n}$ liters; the second, $\\frac{a\\sqrt m}{\\sqrt m+\\sqrt n}$ liters"
] | [
"Two collective farmers with a total of a lit res of milk, though selling the milk at different prices, made equal sums of money. If the first farmer had sold as much milk as the second, she would have received $m$ roubles, and if the second farmer had sold as much milk as the first, she would have received $n$ rou... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01963240e847297f06531bcd3aecdc81f6d4a3413721cb2d38b3f2d82e3e85ce | Two combines stand in opposite corners of a $2004 \times 2004$ field. Two players move their combines alternately to one of the neighboring cells (two cells are neighbors if they share a common side). The combine that reaches a cell first collects the wheat in it. The player who collects wheat from more than half of th... | [
"The second player has a winning strategy."
] | [
"Two combines stand in opposite corners of a $2004 \\times 2004$ field. Two players move their combines alternately to one of the neighboring cells (two cells are neighbors if they share a common side). The combine that reaches a cell first collects the wheat in it. The player who collects wheat from more than half... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-399595fcab0afd916a4a773fcbf938852237da284629973e9900abd00f55461d | Two commercial airlines are flying at the same altitude along straight line courses that intersect at right angles. Plane A is approaching the intersection point at 400 mph and plane B is approaching the intersection point at 500 mph. At what rate is the distance between the planes changing when plane A is 6 miles from... | [
"-440 mph"
] | [
"Two commercial airlines are flying at the same altitude along straight line courses that intersect at right angles. Plane A is approaching the intersection point at 400 mph and plane B is approaching the intersection point at 500 mph. At what rate is the distance between the planes changing when plane A is 6 miles... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-8e07088e041a8820f13b06e56143c3198d08547fc152cee2994d48ceff311a29 | Two common tangents of two intersecting circles meet at a point $A$. Let $B$ be a point of intersection of the two circles, and $C$ and $D$ be the points in which one of the tangents touches the circles. If the radius of the circle passing through $B$, $C$, and $D$ is $r$ and the distance from $A$ to $B$ is $d$, find t... | [
"1"
] | [
"Two common tangents of two intersecting circles meet at a point $A$. Let $B$ be a point of intersection of the two circles, and $C$ and $D$ be the points in which one of the tangents touches the circles. If the radius of the circle passing through $B$, $C$, and $D$ is $r$ and the distance from $A$ to $B$ is $d$, f... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8f3e2ae84fee6fa8dca80372321523659fd09bc6a988e667bc5dffdbf66ee494 | Two common tangents of two intersecting circles meet at a point $A$. Let $B$ be a point of intersection of the two circles, and $C$ and $D$ be the points where one of the tangents touches the circles. If the radius of the circle passing through $B, C$, and $D$ is $r$, and the distance from $A$ to $B$ is $d$, find the v... | [
"2"
] | [
"Two common tangents of two intersecting circles meet at a point $A$. Let $B$ be a point of intersection of the two circles, and $C$ and $D$ be the points where one of the tangents touches the circles. If the radius of the circle passing through $B, C$, and $D$ is $r$, and the distance from $A$ to $B$ is $d$, find ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5a864ed5b88c2e91eb1336d631789b628dae34e8d6b302cd76461006ff83fda1 | Two common tangents touch circles $(O_1)$ and $(O_2)$ at points $X, Y$ and $Q, P$ respectively. The circle $(O_3)$ passes through $X$ and $Y$, and the circle $(O_4)$ passes through $P$ and $Q$. Circles $(O_3)$ and $(O_4)$ intersect at points $B$ and $C$. Circle $(O_5)$ passing through $B$ and $C$ intersects $(O_1)$ and... | [
"180"
] | [
"Two common tangents touch circles $(O_1)$ and $(O_2)$ at points $X, Y$ and $Q, P$ respectively. The circle $(O_3)$ passes through $X$ and $Y$, and the circle $(O_4)$ passes through $P$ and $Q$. Circles $(O_3)$ and $(O_4)$ intersect at points $B$ and $C$. Circle $(O_5)$ passing through $B$ and $C$ intersects $(O_1)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-cec026bf6ee63e3d551a3a78b99b1b8561a348753bcdc76c8fccf2d949cf5bfa | Two competitive ducks decide to have a race in the first quadrant of the $xy$ plane. They both start at the origin, and the race ends when one of the ducks reaches the line $y = \frac{1}{2}$. The first duck follows the graph of $y = \frac{x}{3}$ and the second duck follows the graph of $y = \frac{x}{5}$. If the two duc... | [
"\\(\\frac{5\\sqrt{65}}{39}\\)"
] | [
"Two competitive ducks decide to have a race in the first quadrant of the $xy$ plane. They both start at the origin, and the race ends when one of the ducks reaches the line $y = \\frac{1}{2}$. The first duck follows the graph of $y = \\frac{x}{3}$ and the second duck follows the graph of $y = \\frac{x}{5}$. If the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5d4e6c0caf1b8eb13efdcba0cb4bd6a378ef381f8f75268d7a76b93f96b22ec7 | Two competitors, $A$ and $B$, contest a duel. A large fixed integer $n$ is set. Each dueler must choose an integer between $1$ and $n$ without knowing the other's choice. The dueler who picks the lower integer gets to shoot first; if both pick the same number, a coin toss determines who shoots first. A dueler picking n... | [
"\\frac{1}{i H_n}"
] | [
"Two competitors, $A$ and $B$, contest a duel. A large fixed integer $n$ is set. Each dueler must choose an integer between $1$ and $n$ without knowing the other's choice. The dueler who picks the lower integer gets to shoot first; if both pick the same number, a coin toss determines who shoots first. A dueler pick... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-bed7dc8aafd55e40f661578d3c5756a7585a21a139e789a5078702f317905a93 | Two complementary angles are in a ratio of $3:2$. What is the measure, in degrees, of the smaller angle? | [
"36"
] | [
"Two complementary angles are in a ratio of $3:2$. What is the measure, in degrees, of the smaller angle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1f458292e8edee230cbe9045fc3846a0f32d430482cf24352e0520b28ece755 | Two complex numbers $z$ and $w$ are chosen at random such that $|z|=|w|=1$. What is the probability that $\text{Re}(zw)>0$ and $\text{Re}\left(\frac{z}{w}\right)>0$? | [
"\\(\\frac{1}{4}\\)"
] | [
"Two complex numbers $z$ and $w$ are chosen at random such that $|z|=|w|=1$. What is the probability that $\\text{Re}(zw)>0$ and $\\text{Re}\\left(\\frac{z}{w}\\right)>0$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 17
} | 23 | 0 | 0 |
omr-ws-14478d15f87c8071ca355bfdbe173c658c912af412fd357188912c5068e57c9c | Two complex numbers $z_1$ and $z_2$ are chosen uniformly and at random from the disk $0 \le |z| \le 1$. Let $z_1 = a + bi$ and $z_2 = c + di$, where $a, b, c, d$ are real numbers. The probability that $$\frac{a^2 + b^2}{2abcd + a^2c^2 + b^2d^2} \le \frac{a^2 + b^2}{-2abcd + a^2d^2 + b^2c^2} \le 4$$ can be written as $\... | [
"162"
] | [
"Two complex numbers $z_1$ and $z_2$ are chosen uniformly and at random from the disk $0 \\le |z| \\le 1$. Let $z_1 = a + bi$ and $z_2 = c + di$, where $a, b, c, d$ are real numbers. The probability that\n$$\\frac{a^2 + b^2}{2abcd + a^2c^2 + b^2d^2} \\le \\frac{a^2 + b^2}{-2abcd + a^2d^2 + b^2c^2} \\le 4$$\ncan be ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-be30e31d8dc2519b361360473164ffac2f44ec3165764166dd2cc01e7f9e993c | Two components with failure times \(X\) and \(Y\) (in years) have the joint density function given by \[ f_{XY}(x,y) = 0.2 \exp(-(0.4x + 0.5y)) \quad \text{for} \quad x, y \geq 0 \] \[ f_{XY}(x,y) = 0 \quad \text{otherwise} \] Find the probability that the circuit will fail in less than 3 years if: a) The two component... | [
"1 - e^{-2.7}"
] | [
"Two components with failure times \\(X\\) and \\(Y\\) (in years) have the joint density function given by\n\n\\[ f_{XY}(x,y) = 0.2 \\exp(-(0.4x + 0.5y)) \\quad \\text{for} \\quad x, y \\geq 0 \\]\n\\[ f_{XY}(x,y) = 0 \\quad \\text{otherwise} \\]\n\nFind the probability that the circuit will fail in less than 3 yea... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b8fb96466286fbb89c6a9f7b59da2d806cfc213fe1450022711f11d9b7e48c4e | Two concentric circles $O_1$ and $O_2$ have radii $7$ and $15$, respectively. A point $P$ is chosen at random inside $O_2$. Let $Q$ be the closest point to $P$ on the circumference of $O_1$. What is the probability that $PQ > 5$? | [
"\\(\\frac{17}{45}\\)"
] | [
"Two concentric circles $O_1$ and $O_2$ have radii $7$ and $15$, respectively. A point $P$ is chosen at random inside $O_2$. Let $Q$ be the closest point to $P$ on the circumference of $O_1$. What is the probability that $PQ > 5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b45d7a8b6aaeeca21c80f450e863fc9bd6bd87f0029dc1bfee3eaeeb260e232b | Two concentric circles $\omega$ and $\Omega$ have radii 8 and 13, respectively. $AB$ is a diameter of $\Omega$, and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$? | [
"19"
] | [
"Two concentric circles $\\omega$ and $\\Omega$ have radii 8 and 13, respectively. $AB$ is a diameter of $\\Omega$, and the tangent from $B$ to $\\omega$ touches $\\omega$ at $D$. What is the length of $AD$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 22
} | 69 | 0 | 0 |
omr-ws-509d65170e2855d52d976d4fbb3bc18712de8d21f3d0e4e5569c00db92af613d | Two concentric circles $k$ and $k_0$ are located such that there are exactly six circles congruent to $k_0$, which touch $k_0$ from the outside, $k$ from the inside, and each other pairwise. Calculate the area of the hatched region depending on the radius $r$ of the circle $k_0$. | [
"\\( r^2 \\left( \\frac{5\\pi}{6} - \\sqrt{3} \\right) \\)"
] | [
"Two concentric circles $k$ and $k_0$ are located such that there are exactly six circles congruent to $k_0$, which touch $k_0$ from the outside, $k$ from the inside, and each other pairwise. Calculate the area of the hatched region depending on the radius $r$ of the circle $k_0$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1178eac191a89bf31ca945deeb4dd2d7188ab4d3e52f2ec67bff3b9e3da24d3 | Two concentric circles \( C_1 \) and \( C_2 \) are given, with \( C_1 \) larger than \( C_2 \). Point \( A \) is on \( C_1 \), point \( B \) is on \( C_2 \), and \( AB \) is tangent to \( C_2 \). The line \( AB \) is extended to meet \( C_1 \) again at point \( C \neq A \). Point \( D \) is the midpoint of \( AB \). Po... | [
"1"
] | [
"Two concentric circles \\( C_1 \\) and \\( C_2 \\) are given, with \\( C_1 \\) larger than \\( C_2 \\). Point \\( A \\) is on \\( C_1 \\), point \\( B \\) is on \\( C_2 \\), and \\( AB \\) is tangent to \\( C_2 \\). The line \\( AB \\) is extended to meet \\( C_1 \\) again at point \\( C \\neq A \\). Point \\( D \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-8baf8ca1bf6b6944fac057ae756c8819666cfd989310bc2a561f842a84a1060b | Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your a... | [
"\\frac{16}{25}"
] | [
"Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ec31f9ab4c8fa76321536aa9adb775a4c476afbf031d2245cf784bfc236d947a | Two concentric circles are divided by \( n \) radii into \( 2n \) parts. Two parts are called neighbors if they share either a common side or a common arc. Initially, there are \( 4n + 1 \) frogs inside the parts. At each second, if there are three or more frogs in one part, then three of the frogs in that part will ju... | [
"9"
] | [
"Two concentric circles are divided by \\( n \\) radii into \\( 2n \\) parts. Two parts are called neighbors if they share either a common side or a common arc. Initially, there are \\( 4n + 1 \\) frogs inside the parts. At each second, if there are three or more frogs in one part, then three of the frogs in that p... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d06c196cc03c3318e4c8ba77bf7990cadf8a25571c6e8a62ca499130f4a6cbee | Two concentric circles are drawn so that the tangent segment to the smaller circle is a chord of the larger circle. If the area of the annulus (region outside the smaller circle and inside the larger one) is 100 square units, find the length of the chord shown. | [
"\\dfrac{20}{\\sqrt{\\pi}}"
] | [
"Two concentric circles are drawn so that the tangent segment to the smaller circle is a chord of the larger circle. If the area of the annulus (region outside the smaller circle and inside the larger one) is 100 square units, find the length of the chord shown."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-172f177aa333fcfc91a1fc4c1f32dc35a02e9924116b97a6e6e53330c78d020a | Two concentric circles are given. A fixed point \( P \) is chosen on the larger circle. A non-fixed point \( S \) is chosen on the smaller circle, and points \( Q \) and \( R \) are chosen on segment \( PS \) such that \( PQ = QR = RS \) (they trisect \( PS \)). As \( S \) is moved about its circle, \( Q \) and \( R \)... | [
"3"
] | [
"Two concentric circles are given. A fixed point \\( P \\) is chosen on the larger circle. A non-fixed point \\( S \\) is chosen on the smaller circle, and points \\( Q \\) and \\( R \\) are chosen on segment \\( PS \\) such that \\( PQ = QR = RS \\) (they trisect \\( PS \\)). As \\( S \\) is moved about its circle... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-6124eb1c85716824586d179e61e8efc44e6167fe7a83cf641982c2089b095094 | Two concentric circles are given. In the larger circle, a chord is drawn that is tangent to the smaller circle. Find the area of the ring between the circles, knowing that the length of this chord is 2009. | [
"\\( 1009020.25\\pi \\)"
] | [
"Two concentric circles are given. In the larger circle, a chord is drawn that is tangent to the smaller circle. Find the area of the ring between the circles, knowing that the length of this chord is 2009."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-cf681bf625f7c95114ad800fc7879924ef9e54647c2a96ae35bcb3df5b15e927 | Two concentric circles are given. The tangent to the smaller circle divides the larger circle in a ratio of \(1:5\). Find the ratio of the areas of the circles bounded by these circles. | [
"\\dfrac{3}{4}"
] | [
"Two concentric circles are given. The tangent to the smaller circle divides the larger circle in a ratio of \\(1:5\\). Find the ratio of the areas of the circles bounded by these circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-7689fd0ee03e3562f819e4d65197c540c8790952640b14db7e3a5b2e72ba746a | Two concentric circles are shown. The radius of the inner circle is 3, and the distance between the circles is 3. A line segment of length 4 has its endpoints on both circles. Compute the distance from point A to point B. | [
"\\(\\sqrt{14}\\)"
] | [
"Two concentric circles are shown. The radius of the inner circle is 3, and the distance between the circles is 3. A line segment of length 4 has its endpoints on both circles. Compute the distance from point A to point B."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7737f24c191a28dbad34548cf52becd8f4a788b20db34b7dc3233bd3b62f6139 | Two concentric circles are such that a chord of the larger circle, 36 inches long, is trisected by the smaller circle. The sum of the radii of the two circles is 36 inches. Find the radius of the larger circle. | [
"22"
] | [
"Two concentric circles are such that a chord of the larger circle, 36 inches long, is trisected by the smaller circle. The sum of the radii of the two circles is 36 inches. Find the radius of the larger circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-2544072a33ff514fa8b1f226daf93c78267bc088e08a0a5f12119f2720fd49d7 | Two concentric circles have center \( C \). Points \( A \) and \( B \) are on the smaller circle with radius 10 such that \( \angle ABC = 52^\circ \). Point \( M \) is on the larger circle such that \( \angle CAM = 150^\circ \) and \( \angle CBM = 135^\circ \). Find the radius of the larger circle. | [
"10\\sqrt{2}"
] | [
"Two concentric circles have center \\( C \\). Points \\( A \\) and \\( B \\) are on the smaller circle with radius 10 such that \\( \\angle ABC = 52^\\circ \\). Point \\( M \\) is on the larger circle such that \\( \\angle CAM = 150^\\circ \\) and \\( \\angle CBM = 135^\\circ \\). Find the radius of the larger cir... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b829d3653e00e1a8a8c2c0232655a44bb23558c6eeec33e9cf2b8484d74669e3 | Two concentric circles have radii $1$ and $2$ with center $O$. The vertex $A$ of an equilateral triangle $ABC$ lies on the larger circle, while the midpoint of side $BC$ lies on the smaller circle. If $B$, $O$, and $C$ are not collinear, what measure can the angle $\angle BOC$ have? | [
"\\(120^\\circ\\) or \\(60^\\circ\\)"
] | [
"Two concentric circles have radii $1$ and $2$ with center $O$. The vertex $A$ of an equilateral triangle $ABC$ lies on the larger circle, while the midpoint of side $BC$ lies on the smaller circle. If $B$, $O$, and $C$ are not collinear, what measure can the angle $\\angle BOC$ have?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-3f90fc8b7ee55cf7d2c7eef67df0ffb289f7ed80814424249c1d9aa46e9c008c | Two concentric circles have radii \(r\) and \(R > r\). Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find \(\frac{R}{r}\). | [
"7 + 4\\sqrt{3}"
] | [
"Two concentric circles have radii \\(r\\) and \\(R > r\\). Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find \\(\\frac{R}{r}\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b1e3445b73c7c7ec5cd94d34664008cf7c6376a4529a0d357b62d2b9af616a3c | Two concentric rectangular hyperbolas, whose axes meet at an angle of $45^\circ$, cut an angle $2\theta$. Determine the minimum value of $sec^2\theta$. | [
"2"
] | [
"Two concentric rectangular hyperbolas, whose axes meet at an angle of $45^\\circ$, cut an angle $2\\theta$. Determine the minimum value of $sec^2\\theta$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9efd7b604693ca751747db2c34c60fa52d4bf227bd291d50f3d42037710d7eb0 | Two concentric spheres have radii \( r \) and \( R \) with \( r < R \). Let \( A, B, \) and \( C \) be points on the surface of the larger sphere such that all sides of the triangle \( ABC \) are tangent to the surface of the smaller sphere. If \( R = 2r \), find the area of triangle \( ABC \) in terms of \( r \). | [
"3\\sqrt{3}r^2"
] | [
"Two concentric spheres have radii \\( r \\) and \\( R \\) with \\( r < R \\). Let \\( A, B, \\) and \\( C \\) be points on the surface of the larger sphere such that all sides of the triangle \\( ABC \\) are tangent to the surface of the smaller sphere. If \\( R = 2r \\), find the area of triangle \\( ABC \\) in t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ca2b5526a3f2e4569438c7e6e1eb59ad9abbf95f11f26cf3446a807625b855ae | Two concentric squares have side lengths of 8 and 6, respectively, with corresponding edges parallel. A third square is inscribed inside the outer square and circumscribed around the inner square. What is the area of this third square? | [
"48"
] | [
"Two concentric squares have side lengths of 8 and 6, respectively, with corresponding edges parallel. A third square is inscribed inside the outer square and circumscribed around the inner square. What is the area of this third square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-9ad93215f3455cd9b09ccd6800c1111f2d0f97f2230ab9633c6e9e2dedf7d80c | Two cone trunks $ T_1 $ and $ T_2 $ have a common base of radius 8 cm, with the other bases being concentric circles. The radius of the larger base of $ T_1 $ is 15 cm, and the volume of $ T_1 $ is three times the volume of $ T_2 $. Determine the ratio between the base area of the trunk $ T_2 $ and the base area of the... | [
"\\(\\frac{(-12 + \\sqrt{795})^2}{2025}\\)"
] | [
"Two cone trunks $ T_1 $ and $ T_2 $ have a common base of radius 8 cm, with the other bases being concentric circles. The radius of the larger base of $ T_1 $ is 15 cm, and the volume of $ T_1 $ is three times the volume of $ T_2 $. Determine the ratio between the base area of the trunk $ T_2 $ and the base area o... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-86fe5da1b6189ffe03e4e8ea83322ff6a5d55074bef809ed8b638acd643a1698 | Two cones have the altitudes $h_1$ and $h_2$ and a common base of radius $R$, and their vertices lie on different sides of the plane of the base. A ball is inscribed into the surface formed by the lateral surfaces of those cones. Find the radius of the other ball which touches both the lateral surface of the first cone... | [
"$\\frac{R(h_1+h_2)}{\\sqrt {R^2+h_1^2}+\\sqrt {R^2+h_2^2}}\\cdot \\frac{\\sqrt {R^2+h_1^2}-R}{\\sqrt {R^2+h_1^2}+R}$"
] | [
"Two cones have the altitudes $h_1$ and $h_2$ and a common base of radius $R$, and their vertices lie on different sides of the plane of the base. A ball is inscribed into the surface formed by the lateral surfaces of those cones. Find the radius of the other ball which touches both the lateral surface of the first... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4acfa95d519109bd98e245c7f02cc0f1c3fdbbb604727a93ee81d12abb6b4b81 | Two cones with a common base have a height that is half the generatrix. Three balls are placed in the area between the cones, contacting each other externally. Two balls are the same and touch both cones, and the third ball touches the boundary of the region. What is the maximum ratio of the radius of the third ball to... | [
"\\(\\frac{7-\\sqrt{22}}{3}\\)"
] | [
"Two cones with a common base have a height that is half the generatrix. Three balls are placed in the area between the cones, contacting each other externally. Two balls are the same and touch both cones, and the third ball touches the boundary of the region. What is the maximum ratio of the radius of the third ba... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f04120e276d4dc0cc646801f1975d9935d7498d70ed3a71045b6da25ac0047d | Two congruent $30^\circ - 60^\circ - 90^\circ$ triangles are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, what is the area common to both triangles? | [
"\\(12\\sqrt{3}\\)"
] | [
"Two congruent $30^\\circ - 60^\\circ - 90^\\circ$ triangles are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, what is the area common to both triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-aef39ebbae7e7e8bd3e3ca7ed44563314bdc1c68192856450d2298df647adea5 | Two congruent $30^\circ$-$60^\circ$-$90^\circ$ triangles are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, what is the area common to both triangles? | [
"\\( 12 \\sqrt{3} \\)"
] | [
"Two congruent $30^\\circ$-$60^\\circ$-$90^\\circ$ triangles are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, what is the area common to both triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 29
} | 66 | 0 | 0 |
omr-ws-9b2d1e13dd3e9dab0cb633e2c97f9e741ff6b3cbfedae883c7bcb441ca567b2e | Two congruent 10-by-10 squares overlap such that a vertex of one square is the center of the other square. What is the largest possible overlapping area? | [
"25"
] | [
"Two congruent 10-by-10 squares overlap such that a vertex of one square is the center of the other square. What is the largest possible overlapping area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-18496be9194c0dda59fa9e1bb05b2ff9354a2263c3ce36df15fb0e18891d0c15 | Two congruent circles $\omega_1$ and $\omega_2$ are both tangent to line $l$ from the same side. The common internal tangent $k$ of $\omega_1$ and $\omega_2$ intersects $l$ at $B$. Points $D$ and $E$ are chosen on $k$ (and on the same side of $l$ as the circles). Draw a tangent from $D$ to $\omega_1$ and from $E$ to $\... | [
"90^\\circ"
] | [
"Two congruent circles $\\omega_1$ and $\\omega_2$ are both tangent to line $l$ from the same side. The common internal tangent $k$ of $\\omega_1$ and $\\omega_2$ intersects $l$ at $B$. Points $D$ and $E$ are chosen on $k$ (and on the same side of $l$ as the circles). Draw a tangent from $D$ to $\\omega_1$ and from... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8153bebcd6278a6932920ebbee95787d0a5fe0c24a296bad1d1c48778ba1da38 | Two congruent circles $\omega_1$ and $\omega_2$ intersect at points $B$ and $C$. Select a point $A$ on $\omega_1$. Let $AB$ and $AC$ intersect $\omega_2$ again at points $A_1$ and $A_2$. Let $X$ be the midpoint of $BC$. Let $A_1X$ and $A_2X$ intersect $\omega_1$ at points $P_1$ and $P_2$ such that $P_1$ and $P_2$ are o... | [
"140"
] | [
"Two congruent circles $\\omega_1$ and $\\omega_2$ intersect at points $B$ and $C$. Select a point $A$ on $\\omega_1$. Let $AB$ and $AC$ intersect $\\omega_2$ again at points $A_1$ and $A_2$. Let $X$ be the midpoint of $BC$. Let $A_1X$ and $A_2X$ intersect $\\omega_1$ at points $P_1$ and $P_2$ such that $P_1$ and $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-93473e6a852aafbb0de6dabaa0f72ee8f26edd0bac61536683833dcb9f60759c | Two congruent circles \( w1 \) and \( w2 \), each with radius 2, are centered at points \( A \) and \( B \), respectively. Point \( O \) is the midpoint of segment \( AB \), and \( OA = 2\sqrt{2} \). Segments \( OC \) and \( OD \) are tangent to \( w1 \) and \( w2 \) at points \( C \) and \( D \), respectively, such th... | [
"\\(8\\sqrt{2} - 4 - \\pi\\)"
] | [
"Two congruent circles \\( w1 \\) and \\( w2 \\), each with radius 2, are centered at points \\( A \\) and \\( B \\), respectively. Point \\( O \\) is the midpoint of segment \\( AB \\), and \\( OA = 2\\sqrt{2} \\). Segments \\( OC \\) and \\( OD \\) are tangent to \\( w1 \\) and \\( w2 \\) at points \\( C \\) and ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f61146bd047a1e07c9d3cf46aba531c29ff042d891df422ac3256b256fe48b5b | Two congruent circles \(O\) and \(Q\) intersect at points \(A\) and \(B\). \(C\) is the midpoint of chord \(AB\). Lines \(DA\) and \(DB\) intersect circle \(Q\) at points \(E\) and \(F\), respectively. Lines \(EC\) and \(FC\) intersect circle \(O\) at points \(G\) and \(H\), respectively. If the radius of each circle i... | [
"16"
] | [
"Two congruent circles \\(O\\) and \\(Q\\) intersect at points \\(A\\) and \\(B\\). \\(C\\) is the midpoint of chord \\(AB\\). Lines \\(DA\\) and \\(DB\\) intersect circle \\(Q\\) at points \\(E\\) and \\(F\\), respectively. Lines \\(EC\\) and \\(FC\\) intersect circle \\(O\\) at points \\(G\\) and \\(H\\), respect... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c923d3b666e7d9a1a6f45bb28fe358bb0244b722d5823dd5c07e2f94f379018 | Two congruent circles are inscribed in a 3-4-5 triangle so that each touches the hypotenuse, a leg, and the other circle. What is their radius? | [
"\\(\\frac{5}{7}\\)"
] | [
"Two congruent circles are inscribed in a 3-4-5 triangle so that each touches the hypotenuse, a leg, and the other circle. What is their radius?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-eb8ed9448d0156447300e703c4a4c7ad9127588ac96ad995dc4019d38fb5ad2d | Two congruent circles intersect at points $A$ and $B$. Two more circles of the same radius are drawn: one through $A$, the other through $B$. Let $P$, $Q$, $R$, and $S$ be the four points of intersection of all four circles other than $A$ and $B$. If the distance between $A$ and $B$ is $10$ units and the radius of each... | [
"624"
] | [
"Two congruent circles intersect at points $A$ and $B$. Two more circles of the same radius are drawn: one through $A$, the other through $B$. Let $P$, $Q$, $R$, and $S$ be the four points of intersection of all four circles other than $A$ and $B$. If the distance between $A$ and $B$ is $10$ units and the radius of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-05f26de356fd1688d3b131f66b1257a23b10671b0ce6ee2c5b9891e1633d33ca | Two congruent circles of radius \( r \) have their centers on each other. Consider a circle that is tangent to one of the circles internally, tangent to the other circle externally, and also tangent to the line containing the centers of the two circles. Find the radius of this circle and describe a method to construct ... | [
"\\dfrac{\\sqrt{3}}{4}r"
] | [
"Two congruent circles of radius \\( r \\) have their centers on each other. Consider a circle that is tangent to one of the circles internally, tangent to the other circle externally, and also tangent to the line containing the centers of the two circles. Find the radius of this circle and describe a method to con... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e81eddbe02e3778a2803977e9191d29774b66852d4e073fbe94bc131121eac7 | Two congruent circles with radius \( r \) and a larger circle are tangent to each other and to the sides of a rectangle. Find the length of the rectangle's longer side. | [
"\\((3 + 2\\sqrt{2})r\\)"
] | [
"Two congruent circles with radius \\( r \\) and a larger circle are tangent to each other and to the sides of a rectangle. Find the length of the rectangle's longer side."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-941470993df4b35b5d4c09eb39dac4eaf5315aac8496db99ad8c2d88ba65fae4 | Two congruent circles with radius \( r \) and a larger circle are tangent to each other and to the sides of a rectangle. Find the length of the rectangle’s longer side. | [
"\\( r(3 + 2\\sqrt{2}) \\)"
] | [
"Two congruent circles with radius \\( r \\) and a larger circle are tangent to each other and to the sides of a rectangle. Find the length of the rectangle’s longer side."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 6,
"tir": 18
} | 66 | 0 | 0 |
omr-ws-2ac41d46483c4fee911be415d652aea6012eff1854322c4c61ee5f2b666fa070 | Two congruent circles, \( \omega_1 \) and \( \omega_2 \), are tangent to each other and both tangent to line \( AB \) with \( \omega_1 \) tangent at \( C \) and \( \omega_2 \) tangent at \( D \). Point \( E \) lies on line \( AB \) such that \( C \) is between \( E \) and \( D \). Point \( F \), distinct from \( C \), ... | [
"144"
] | [
"Two congruent circles, \\( \\omega_1 \\) and \\( \\omega_2 \\), are tangent to each other and both tangent to line \\( AB \\) with \\( \\omega_1 \\) tangent at \\( C \\) and \\( \\omega_2 \\) tangent at \\( D \\). Point \\( E \\) lies on line \\( AB \\) such that \\( C \\) is between \\( E \\) and \\( D \\). Point... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-e9de0d764c6a981974d3ed294e385c18cd70af7f9dda18746668b41434dcd125 | Two congruent circular coins are touching at point P. Coin A is held stationary while coin Z is rolled around it one time in such a way that the two coins remain tangent at all times. How many times will coin Z revolve around its center? | [
"2"
] | [
"Two congruent circular coins are touching at point P. Coin A is held stationary while coin Z is rolled around it one time in such a way that the two coins remain tangent at all times. How many times will coin Z revolve around its center?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-f2bf4f16740d15299cfff299326721220727f50f17ea8306117b667b1d2bfac2 | Two congruent circular fields are tangent to a trail, a river, and each other. A third smaller field is externally tangent to the two fields and the river. If the distance between the river and the trail is $200$ feet, what is the radius of the smaller field? | [
"25"
] | [
"Two congruent circular fields are tangent to a trail, a river, and each other. A third smaller field is externally tangent to the two fields and the river. If the distance between the river and the trail is $200$ feet, what is the radius of the smaller field?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6fe4a44d1e72a5bdbec81fe94c1723ae0d8b78e372ef2feb220679fd4eb5017 | Two congruent circular gates at the bottom of a 24000-gallon pool let out a total of 360 gpm (gallons per minute) of water. The flow rate is directly proportional to the area of a gate. Find the time it takes to empty the pool if the diameter of one gate is tripled while the other's is divided by 3. | [
"7 minutes"
] | [
"Two congruent circular gates at the bottom of a 24000-gallon pool let out a total of 360 gpm (gallons per minute) of water. The flow rate is directly proportional to the area of a gate. Find the time it takes to empty the pool if the diameter of one gate is tripled while the other's is divided by 3."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3227cd1db51e681564fa8067f0b6b45fee769b68034829a897237b9539af0519 | Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Express your answer in terms of $\pi$. | [
"2304\\pi"
] | [
"Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Express your answer in terms of $\\pi$."
... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78521677f909f00b797ab7691672571ae99dd1d953b688b642d1184f131749a4 | Two congruent coplanar parabolas have the same line as their axis and open in the same direction. Let the outer parabola be given by $y = x^2 + 4$ and the inner parabola by $y = x^2$. Tangents are drawn to the inner parabola from a point $(a, a^2 + 4)$ on the outer parabola. Calculate the area bounded by the tangents a... | [
"\\dfrac{16}{3}"
] | [
"Two congruent coplanar parabolas have the same line as their axis and open in the same direction. Let the outer parabola be given by $y = x^2 + 4$ and the inner parabola by $y = x^2$. Tangents are drawn to the inner parabola from a point $(a, a^2 + 4)$ on the outer parabola. Calculate the area bounded by the tange... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-7449ee7b5a3610b46b40b30b66120771959b72570b67713ce4fa16f248076db3 | Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. | [
"\\frac{16}{3}"
] | [
"Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-501066cc0a48858fed5c220a76e8955268c0448fc65434e1ac1edfedc0f73cc2 | Two congruent equilateral triangles \( ABC \) and \( A'B'C' \) in the plane are given. Let \( M, N, P \) be the midpoints of segments \( AA', BB', CC' \) respectively. If \( M, N, P \) form an equilateral triangle, find the ratio of the area of triangle \( MNP \) to the area of triangle \( ABC \). | [
"\\dfrac{1}{4}"
] | [
"Two congruent equilateral triangles \\( ABC \\) and \\( A'B'C' \\) in the plane are given. Let \\( M, N, P \\) be the midpoints of segments \\( AA', BB', CC' \\) respectively. If \\( M, N, P \\) form an equilateral triangle, find the ratio of the area of triangle \\( MNP \\) to the area of triangle \\( ABC \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-4956271cb3ca9a153de2886c7e7e59eecd6360469d02203fd16647c225a3c96c | Two congruent equilateral triangles intersect at the points \(A, B, C, D, E, F\). If the side length of each equilateral triangle is \(s\), calculate the value of \(\frac{AB^2 + CD^2 + EF^2}{BC^2 + DE^2 + FA^2}\). | [
"1"
] | [
"Two congruent equilateral triangles intersect at the points \\(A, B, C, D, E, F\\). If the side length of each equilateral triangle is \\(s\\), calculate the value of \\(\\frac{AB^2 + CD^2 + EF^2}{BC^2 + DE^2 + FA^2}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-00c51449611d1de379efe103fe858dc6500e46b26dde28fbeddc8e79eadd7a21 | Two congruent equilateral triangles, one with red sides and one with blue sides, overlap so that their sides intersect at six points, forming a hexagon. If \( r_1, r_2, r_3, b_1, b_2, b_3 \) are the red and blue sides of the hexagon respectively, calculate the value of \( \frac{r_1^2 + r_2^2 + r_3^2}{b_1^2 + b_2^2 + b_... | [
"2"
] | [
"Two congruent equilateral triangles, one with red sides and one with blue sides, overlap so that their sides intersect at six points, forming a hexagon. If \\( r_1, r_2, r_3, b_1, b_2, b_3 \\) are the red and blue sides of the hexagon respectively, calculate the value of \\( \\frac{r_1^2 + r_2^2 + r_3^2}{b_1^2 + b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e25baf4da76ce2aa3612225b6200def3fdc55d3e6166601b1df3c51ad71f22da | Two congruent non-overlapping equilateral triangles are placed wholly within a square of side length $1$. What is the maximum area they could cover? | [
"\\(\\frac{2\\sqrt{3} + 3}{12}\\)"
] | [
"Two congruent non-overlapping equilateral triangles are placed wholly within a square of side length $1$. What is the maximum area they could cover?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-d66fef539a85ece029e3e1d0d0c4e2a6226b04a8b33fe438e8ae4ce23f41d73a | Two congruent rectangles are positioned in a right triangle with legs \( a \) and \( b \). Both rectangles have a vertex on the hypotenuse and a long side on a leg. If the short side of each rectangle is \( kb \) for some real number \( k \), compute the largest possible value of \( k \). | [
"\\(\\frac{1}{5}\\)"
] | [
"Two congruent rectangles are positioned in a right triangle with legs \\( a \\) and \\( b \\). Both rectangles have a vertex on the hypotenuse and a long side on a leg. If the short side of each rectangle is \\( kb \\) for some real number \\( k \\), compute the largest possible value of \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04a39ad8acf7b8ca97514aa16023e93bbf6d4b4048c0abea8b1b961c83a8abfb | Two congruent rectangles intersect at the points \(A, B, C, D, E, F, P, K\). Calculate the value of \(AB + CD + EF + PK - (BC + DE + FP + KA)\). | [
"0"
] | [
"Two congruent rectangles intersect at the points \\(A, B, C, D, E, F, P, K\\). Calculate the value of \\(AB + CD + EF + PK - (BC + DE + FP + KA)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe15f59519e7aeccaeffc72e0cfd305b302ba7fbac9c2dfcae795fcb4327374b | Two congruent regular 15-sided polygons are placed side by side such that vertex \( A_2 \) of the first polygon coincides with vertex \( B_2 \) of the second polygon, and vertex \( A_3 \) coincides with vertex \( B_3 \). The vertices of the first polygon are labeled counterclockwise \( A_1, A_2, A_3, \ldots, A_{15} \),... | [
"168"
] | [
"Two congruent regular 15-sided polygons are placed side by side such that vertex \\( A_2 \\) of the first polygon coincides with vertex \\( B_2 \\) of the second polygon, and vertex \\( A_3 \\) coincides with vertex \\( B_3 \\). The vertices of the first polygon are labeled counterclockwise \\( A_1, A_2, A_3, \\ld... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-372114967a4fa7c0078ce1f744c9d71df18a2c6c400dbb84f2a8a7109d455a5e | Two congruent regular hexagons are cut into 6 parts each. These 12 parts are rearranged to form an equilateral triangle without gaps or overlaps. If the side length of each hexagon is 2 units, find the area of the equilateral triangle formed. | [
"12\\sqrt{3}"
] | [
"Two congruent regular hexagons are cut into 6 parts each. These 12 parts are rearranged to form an equilateral triangle without gaps or overlaps. If the side length of each hexagon is 2 units, find the area of the equilateral triangle formed."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69f9c3d329c850822c2c890d85410d95e70baf3ff1da5126cc1f090b0ffbff70 | Two congruent regular pentagons are positioned such that the shortest distance between any vertex of one pentagon and any vertex of the other pentagon is 1. A polyhedron is formed with 10 congruent equilateral triangular faces, where the vertices of the pentagons are part of this polyhedron. Let the length of the edge ... | [
"\\dfrac{5 + \\sqrt{5}}{4}"
] | [
"Two congruent regular pentagons are positioned such that the shortest distance between any vertex of one pentagon and any vertex of the other pentagon is 1. A polyhedron is formed with 10 congruent equilateral triangular faces, where the vertices of the pentagons are part of this polyhedron. Let the length of the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-fbb890cf138ff07696b3ea6d6cfa9c928875a264e9c424bee4e91ab385c4f153 | Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac{m}{n}$, wher... | [
"298"
] | [
"Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\\frac{m}{n}$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 8,
"tir": 3
} | 45 | 0 | 0 |
omr-ws-4ec893543197755a01f3e2cf6e855ef73ce10fb6ff8d763174ac93290ebdfa28 | Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where... | [
"298"
] | [
"Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\\frac{m}{n}$,... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-934eef61e14ff6cf8ed9ef6250f346cabbaf9a9234594647e8ffa0ba8061ce4a | Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, w... | [
"298"
] | [
"Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\\frac{m}{... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 7,
"tir": 6
} | 35 | 0 | 0 |
omr-ws-ee614eec58581563917666a8d60eb6b8653323cecccdd470ec800381ac4a7b4c | Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, ... | [
"298"
] | [
"Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\\frac{m}... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3b4ab3c9553b93fa4b371c3493e2225897aeb04158deb5a72eafd90b0d19de54 | Two congruent right-angled isosceles triangles, each with a base length of 1, slide on a line. What is the maximal area of overlap between the two triangles? | [
"\\(\\frac{1}{3}\\)"
] | [
"Two congruent right-angled isosceles triangles, each with a base length of 1, slide on a line. What is the maximal area of overlap between the two triangles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7957d4a3a14393d57b381277b0042afbdb4176d2d04b282fb01c638747063181 | Two congruent semicircles are placed inside a 1 by 2 rectangle without overlapping. Denote by \( r \) the maximum radius of the semicircles. Calculate the value of \( \lfloor 100r \rfloor \). | [
"62"
] | [
"Two congruent semicircles are placed inside a 1 by 2 rectangle without overlapping. Denote by \\( r \\) the maximum radius of the semicircles. Calculate the value of \\( \\lfloor 100r \\rfloor \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-95c986cc5363303800360dcd2c6bd47f6834e0a35cf755f30e25728727bbfb8b | Two congruent shaded semicircles are placed inside a larger semicircle. What fraction of the area of the larger semicircle do the two smaller semicircles cover in total? | [
"\\(\\frac{2}{3}\\)"
] | [
"Two congruent shaded semicircles are placed inside a larger semicircle. What fraction of the area of the larger semicircle do the two smaller semicircles cover in total?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad10fb88f2014bbe9f48acf74a5de88e4e534863e67b5d96c573a068760f24ce | Two congruent spheres are to be inserted into a cube with edge length $a$ such that they touch each other and three cube faces starting from a corner of the cube. Calculate the radius of these spheres in terms of $a$. | [
"\\dfrac{a(3 - \\sqrt{3})}{4}"
] | [
"Two congruent spheres are to be inserted into a cube with edge length $a$ such that they touch each other and three cube faces starting from a corner of the cube. Calculate the radius of these spheres in terms of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9ad4252f625a3d57fa42a0a70bd3ca6dba427cdfb5f5cc57505f75dcd0253c4f | Two congruent, equilateral triangles \( ABC \) and \( PQR \) with parallel sides are positioned such that one triangle has a vertex pointing up and the other has a vertex pointing down. The triangles are placed above each other so that the area of intersection is a hexagon \( A_1A_2A_3A_4A_5A_6 \) (labelled countercloc... | [
"2"
] | [
"Two congruent, equilateral triangles \\( ABC \\) and \\( PQR \\) with parallel sides are positioned such that one triangle has a vertex pointing up and the other has a vertex pointing down. The triangles are placed above each other so that the area of intersection is a hexagon \\( A_1A_2A_3A_4A_5A_6 \\) (labelled ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-187b5756de40fe5bb4df0e15117f47007fef65b2166866f2813106e10788cb0e | Two consecutive numbers $a$ and $b$ from the first $n$ natural numbers are removed. The arithmetic mean of the remaining numbers is $\frac{105}{4}$. Find the value of $a + b + n$. | [
"65"
] | [
"Two consecutive numbers $a$ and $b$ from the first $n$ natural numbers are removed. The arithmetic mean of the remaining numbers is $\\frac{105}{4}$. Find the value of $a + b + n$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-00d7922350b65fd79d8482d9ac995d83ee7ca3326e8715fb5e50ace034c59fb2 | Two consecutive numbers from the natural numbers \(1, 2, 3, \ldots, n\) are removed. The arithmetic mean of the remaining numbers is \(\frac{105}{4}\). Find the value of \(n\). | [
"50"
] | [
"Two consecutive numbers from the natural numbers \\(1, 2, 3, \\ldots, n\\) are removed. The arithmetic mean of the remaining numbers is \\(\\frac{105}{4}\\). Find the value of \\(n\\)."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-c56b69c469a155865273a54f2da4b7f507b0f0e637e55b8beeb11ca9a5b8fe6e | Two consecutive positive integers \(a\) and \(b\) are given to two mathematicians, A and B, respectively. Mathematician A knows \(a\) and Mathematician B knows \(b\). They take turns asking each other if they know the other's number. If Mathematician A starts the game, after how many turns will Mathematician B be able ... | [
"2023"
] | [
"Two consecutive positive integers \\(a\\) and \\(b\\) are given to two mathematicians, A and B, respectively. Mathematician A knows \\(a\\) and Mathematician B knows \\(b\\). They take turns asking each other if they know the other's number. If Mathematician A starts the game, after how many turns will Mathematici... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-08fc25be13148c621ef7e98892eefa9dc5120951bb435acaeee3e77cc12ddc1f | Two containers initially contain 10 litres of wine and 10 litres of water, respectively. A quantity \(0 < q_1 < 10\) litres of wine is transferred from the wine container to the water container, then the same quantity \(q_1\) of the blended mixture is transferred back to the wine container. Determine whether there is m... | [
"the same."
] | [
"Two containers initially contain 10 litres of wine and 10 litres of water, respectively. A quantity \\(0 < q_1 < 10\\) litres of wine is transferred from the wine container to the water container, then the same quantity \\(q_1\\) of the blended mixture is transferred back to the wine container. Determine whether t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9872aeb00fcf65251c4de9a3da2f075f8669c2da41785be5c6ef047fdc0496eb | Two contestants run a $3$ kilometer race along a circular course of length $300$ meters. If their speeds are in the ratio of $4:3$, how often and where would the winner pass the other? Note: The initial start-off is not counted as passing. | [
"The winner passes the other contestant 2 times at the starting point."
] | [
"Two contestants run a $3$ kilometer race along a circular course of length $300$ meters. If their speeds are in the ratio of $4:3$, how often and where would the winner pass the other? Note: The initial start-off is not counted as passing."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.