problem stringlengths 10 5.15k | answer dict |
|---|---|
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ . | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z \) be nonnegative real numbers. Define:
\[
A = \sqrt{x + 3} + \sqrt{y + 6} + \sqrt{z + 12},
\]
\[
B = \sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}.
\]
Find the minimum value of \( A^2 - B^2 \). | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest four-digit number that is divisible by $45$? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces, there are a total of ▲ different parking arrangements. | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A glass is filled to the brim with salty water. Fresh ice with mass \( m = 502 \) floats on the surface. What volume \( \Delta V \) of water will spill out of the glass by the time the ice melts? Neglect surface tension. The density of fresh ice is \( \rho_{n} = 0.92 \, \text{g/cm}^3 \), the density of salty ice is \( \rho_{c} = 0.952 \, \text{g/cm}^3 \), and the density of fresh water is \( \rho_{ns} = 12 \, \text{g/cm}^3 \). Neglect the change in total volume when mixing the two liquids. | {
"answer": "2.63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What integer \( n \) satisfies \( 0 \le n < 23 \) and
$$
54126 \equiv n \pmod{23}~?
$$ | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ms. Carr expands her reading list to 12 books and asks each student to choose any 6 books. Harold and Betty each randomly select 6 books from this list. Calculate the probability that there are exactly 3 books that they both select. | {
"answer": "\\frac{405}{2223}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(2\cos x,1)$, $\overrightarrow{b}=(\sqrt{3}\sin x+\cos x,-1)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
1. Find the maximum and minimum values of $f(x)$ in the interval $[0,\frac{\pi}{4}]$.
2. If $f(x_{0})=\frac{6}{5}$, $x_{0}\in[\frac{\pi}{4},\frac{\pi}{2}]$, find the value of $\cos 2x_{0}$.
3. If the function $y=f(\omega x)$ is monotonically increasing in the interval $(\frac{\pi}{3},\frac{2\pi}{3})$, find the range of positive values for $\omega$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a rectangle with dimensions 6 units by 7 units. A triangle is formed with its vertices on the sides of the rectangle. Vertex $A$ is on the left side, 3 units from the bottom. Vertex $B$ is on the bottom side, 5 units from the left. Vertex $C$ is on the top side, 2 units from the right. Calculate the area of triangle $ABC$. | {
"answer": "17.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two boards, one five inches wide and the other eight inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are separated, what is the area of the unpainted region on the five-inch board? (Neglect the holes caused by the nails.) | {
"answer": "40 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation about $x$, $2x^{2}-( \sqrt {3}+1)x+m=0$, its two roots are $\sin θ$ and $\cos θ$, where $θ∈(0,π)$. Find:
$(1)$ the value of $m$;
$(2)$ the value of $\frac {\tan θ\sin θ}{\tan θ-1}+ \frac {\cos θ}{1-\tan θ}$;
$(3)$ the two roots of the equation and the value of $θ$ at this time. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( C_1 \) and \( C_2 \) be circles defined by
\[
(x-12)^2 + y^2 = 25
\]
and
\[
(x+18)^2 + y^2 = 64,
\]
respectively. What is the length of the shortest line segment \( \overline{RS} \) that is tangent to \( C_1 \) at \( R \) and to \( C_2 \) at \( S \)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $\overrightarrow{a}+2\overrightarrow{b}$, where $\overrightarrow{a}=(2,0)$ and $|\overrightarrow{b}|=1$, and then calculate the magnitude of this vector. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\sin 9^\circ \sin 45^\circ \sin 69^\circ \sin 81^\circ.$ | {
"answer": "\\frac{0.6293 \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many six-digit numbers exist that have three even and three odd digits? | {
"answer": "281250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is referred to as a "turtle's knee." Given that in the turtle's knee $M-ABC$, $MA \perp$ plane $ABC$, and $MA=AB=BC=2$, the sum of the surface areas of the circumscribed sphere and the inscribed sphere of the turtle's knee is ______. | {
"answer": "24\\pi-8\\sqrt{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a straight segment of a one-way, single-lane highway, cars travel at the same speed and follow a safety rule where the distance from the back of one car to the front of the next is equal to the car’s speed divided by 10 kilometers per hour, rounded up to the nearest whole number (e.g., a car traveling at 52 kilometers per hour maintains a 6 car length distance to the car in front). Assume each car is 5 meters long, and the cars can travel at any speed. A sensor by the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Find $N$ divided by 20. | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 150$ such that $i^x+i^y$ is a real number. | {
"answer": "3515",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given that $\log_2{2} = a$, express $\log_8{20} - 2\log_2{20}$ in terms of $a$.
(2) Evaluate the expression: $(\ln{4})^0 + (\frac{9}{4})^{-0.5} + \sqrt{(1 - \sqrt{3})^2} - 2^{\log_4{3}}$. | {
"answer": "\\frac{9}{2} - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Student council has 24 members: 12 boys and 12 girls. A 5-person committee is selected at random. What is the probability that the committee includes at least one boy and at least one girl? | {
"answer": "\\frac{455}{472}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), point \(N\) lies on side \(AB\) such that \(AN = 3NB\); the median \(AM\) intersects \(CN\) at point \(O\). Find \(AB\) if \(AM = CN = 7\) cm and \(\angle NOM = 60^\circ\). | {
"answer": "4\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer that is both an integer power of 13 and is not a palindrome. | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of rectangle $ABCD$ have lengths $12$ (height) and $15$ (width). An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. | {
"answer": "369 \\sqrt{3} - 540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum:
\[
\sin^2 3^\circ + \sin^2 9^\circ + \sin^2 15^\circ + \dots + \sin^2 177^\circ.
\] | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g. | {
"answer": "2020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Can we find \( N \) such that all \( m \times n \) rectangles with \( m, n > N \) can be tiled with \( 4 \times 6 \) and \( 5 \times 7 \) rectangles? | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse has a major axis of length 12 and a minor axis of 10. Using one focus as a center, an external circle is tangent to the ellipse. Find the radius of the circle. | {
"answer": "\\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take? | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ be a nonreal root of $x^4 = 1.$ Compute
\[(1 - \alpha + \alpha^2 - \alpha^3)^4 + (1 + \alpha - \alpha^2 + \alpha^3)^4.\] | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$ | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
7.61 log₂ 3 + 2 log₄ x = x^(log₉ 16 / log₃ x). | {
"answer": "16/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=1-2x^{2}$ and $g(x)=x^{2}-2x$, let $F(x) = \begin{cases} f(x), & \text{if } f(x) \geq g(x) \\ g(x), & \text{if } f(x) < g(x) \end{cases}$. Determine the maximum value of $F(x)$. | {
"answer": "\\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c, d \) be positive integers such that \( \gcd(a, b) = 24 \), \( \gcd(b, c) = 36 \), \( \gcd(c, d) = 54 \), and \( 70 < \gcd(d, a) < 100 \). Which of the following numbers is a factor of \( a \)? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cubic polynomial $q(x)$ satisfies $q(1) = 5,$ $q(6) = 20,$ $q(14) = 12,$ and $q(19) = 30.$ Find
\[q(0) + q(1) + q(2) + \dots + q(20).\] | {
"answer": "357",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the first term of a geometric sequence $\{a\_n\}$ is $\frac{3}{2}$, and the sum of the first $n$ terms is $S\_n$, where $n \in \mathbb{N}^*$. Also, $-2S\_2$, $S\_3$, and $4S\_4$ form an arithmetic sequence.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. For a sequence $\{A\_n\}$, if there exists an interval $M$ such that $A\_i \in M$ for all $i = 1, 2, 3, ...$, then $M$ is called the "range interval" of sequence $\{A\_n\}$. Let $b\_n = S\_n + \frac{1}{S\_n}$, find the minimum length of the "range interval" of sequence $\{b\_n\}$. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( y = \cos \frac{2 \pi}{9} + i \sin \frac{2 \pi}{9} \). Compute the value of
\[
(3y + y^3)(3y^3 + y^9)(3y^6 + y^{18})(3y^2 + y^6)(3y^5 + y^{15})(3y^7 + y^{21}).
\] | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water, from which the excess liquid overflows. At what point in time will the amount of salt in both vessels be equal? | {
"answer": "333.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute \[\sum_{n=1}^{500} \frac{1}{n^2 + 2n}.\] | {
"answer": "\\frac{1499}{2008}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle \( \triangle ABC \) with legs \( AC=12 \), \( BC=5 \). Point \( D \) is a moving point on the hypotenuse \( AB \). The triangle is folded along \( CD \) to form a right dihedral angle \( A-CD-B \). When the length of \( AB \) is minimized, let the plane angle of the dihedral \( B-AC-D \) be \( \alpha \). Find \( \tan^{10} \alpha \). | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polynomial $p(x)$ leaves a remainder of $2$ when divided by $x - 3,$ a remainder of 1 when divided by $x - 4,$ and a remainder of 5 when divided by $x + 4.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 3)(x - 4)(x + 4).$ Find $r(5).$ | {
"answer": "\\frac{1}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct sequences of five letters can be made from the letters in COMPUTER if each letter can be used only once, each sequence must begin with M, end with R, and the third letter must be a vowel (A, E, I, O, U)? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( O \) be the origin, \( A_1, A_2, A_3, \ldots \) be points on the curve \( y = \sqrt{x} \) and \( B_1, B_2, B_3, \ldots \) be points on the positive \( x \)-axis such that the triangles \( O B_1 A_1, B_1 B_2 A_2, B_2 B_3 A_3, \ldots \) are all equilateral, with side lengths \( l_1, l_2, l_3, \ldots \) respectively. Find the value of \( l_1 + l_2 + l_3 + \cdots + l_{2005} \). | {
"answer": "4022030/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn^2,$ for all positive integers $m$ and $n.$ Find $b_{12}$. | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular. | {
"answer": "1 + \\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The "Hua Luogeng" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first "Hua Cup" is: \( A_1 = 1 + 9 + 8 + 6 = 24 \). The sum of the digits of the years of the first two "Hua Cup" contests is: \( A_2 = 1 + 9 + 8 + 6 + 1 + 9 + 8 + 8 = 50 \). Find the sum of the digits of the years of the first 50 "Hua Cup" contests, \( A_{50} \). | {
"answer": "629",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence defined by $O = \begin{cases} 3N + 2, & \text{if } N \text{ is odd} \\ \frac{N}{2}, & \text{if } N \text{ is even} \end{cases}$, for a given integer $N$, find the sum of all integers that, after being inputted repeatedly for 7 more times, ultimately result in 4. | {
"answer": "1016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $b-a=-6$ and $ab=7$, find the value of $a^2b-ab^2$. | {
"answer": "-42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the natural numbers from 1 to 700, inclusive, contain the digit 5 at least once? | {
"answer": "214",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B_{1}C_{1}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \] | {
"answer": "7.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers, all of which are nonprime? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a vertical vessel with straight walls closed by a piston, there is water. Its height is $h=2$ mm. There is no air in the vessel. To what height must the piston be raised for all the water to evaporate? The density of water is $\rho=1000$ kg / $\mathrm{m}^{3}$, the molar mass of water vapor is $M=0.018$ kg/mol, the pressure of saturated water vapor at a temperature of $T=50{ }^{\circ} \mathrm{C}$ is $p=12300$ Pa. The temperature of water and vapor is maintained constant. | {
"answer": "24.258",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, respectively, determine the population in $2003$. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $q(x) = 2x^6 - 3x^4 + Dx^2 + 6$ be a polynomial. When $q(x)$ is divided by $x - 2$, the remainder is 14. Find the remainder when $q(x)$ is divided by $x + 2$. | {
"answer": "158",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0} = \) _______. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny only if the weather changed exactly one year ago on that day. In 2015, January in Anchuria featured a variety of sunny and rainy days. In which year will the weather in January first change in exactly the same pattern as it did in January 2015? | {
"answer": "2047",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a $\spadesuit$? | {
"answer": "\\dfrac{1}{52}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amy rolls six fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly three of the dice show a prime number and at least one die shows an 8? | {
"answer": "\\frac{2899900}{16777216}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A room has a floor with dimensions \(7 \times 8\) square meters, and the ceiling height is 4 meters. A fly named Masha is sitting in one corner of the ceiling, while a spider named Petya is in the opposite corner of the ceiling. Masha decides to travel to visit Petya by the shortest route that includes touching the floor. Find the length of the path she travels. | {
"answer": "\\sqrt{265}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, where \( A B C \) is the base and \( A A_{1}, B B_{1}, C C_{1} \) are the lateral edges. The segment \( C D \) is the diameter of this sphere, and point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism if \( C K = 2 \sqrt{3} \) and \( D K = 2 \sqrt{2} \). | {
"answer": "9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the house numbers from his friend's house to the end of the street. It is also known that on the side of the street where his friend's house is located, there are more than 50 but fewer than 500 houses.
What is the house number where the storyteller's friend lives? | {
"answer": "204",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all positive integer divisors of $129,600$. Calculate the number of numbers that are the product of two distinct elements of $S$. | {
"answer": "488",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( m \) be an integer greater than 1, and let's define a sequence \( \{a_{n}\} \) as follows:
\[
\begin{array}{l}
a_{0}=m, \\
a_{1}=\varphi(m), \\
a_{2}=\varphi^{(2)}(m)=\varphi(\varphi(m)), \\
\vdots \\
a_{n}=\varphi^{(n)}(m)=\varphi\left(\varphi^{(n-1)}(m)\right),
\end{array}
\]
where \( \varphi(m) \) is the Euler's totient function.
If for any non-negative integer \( k \), \( a_{k+1} \) always divides \( a_{k} \), find the greatest positive integer \( m \) not exceeding 2016. | {
"answer": "1944",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest positive integer that is not the sum of a positive integral multiple of $36$ and a positive composite integer that is not a multiple of $4$? | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P(x) = (x-1)(x-4)(x-5)$. Determine how many polynomials $Q(x)$ there exist such that there exists a polynomial $R(x)$ of degree 3 with $P(Q(x)) = P(x) \cdot R(x)$, and the coefficient of $x$ in $Q(x)$ is 6. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\square + q = 74$ and $\square + 2q^2 = 180$, what is the value of $\square$? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given in usual (x, y) notation, where the leftmost, topmost corner is (0, 3) and the rightmost, bottommost corner is (6, 0).] | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in which they are arranged). What is the smallest $N$ for which the magician and the assistant can arrange the trick so that the magician can always correctly guess the covered digits? | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Container A holds 4 red balls and 6 green balls; container B holds 8 red balls and 6 green balls; container C holds 8 red balls and 6 green balls; container D holds 3 red balls and 7 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? | {
"answer": "\\frac{13}{28}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $a = (2, -1, 3)$, $b = (-1, 4, -2)$, and $c = (7, 5, \lambda)$, if vectors $a$, $b$, and $c$ are coplanar, the real number $\lambda$ equals ( ). | {
"answer": "\\frac{65}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4. At each step, she rolls the die and records which number is on the bottom face. She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.) What is the expected number (average number) of steps that she takes? | {
"answer": "625/256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle? | {
"answer": "1/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product $11 \cdot 30 \cdot N$ is an integer whose representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is the fourth power of an integer in decimal (base 10). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) - g(xy) = 2x + 2y\]for all real numbers $x$ and $y.$
Calculate the number $n$ of possible values of $g(2),$ and the sum $s$ of all possible values of $g(2),$ and find the product $n \times s.$ | {
"answer": "\\frac{28}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$ , where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$ , one will make this until remain two numbers $x, y$ with $x\geq y$ . Find the maximum value of $x$ . | {
"answer": "1320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer $n,$ let
\[G_n = 1^2 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2}.\]Compute
\[\sum_{n = 1}^\infty \frac{1}{(n + 1) G_n G_{n + 1}}.\] | {
"answer": "1 - \\frac{6}{\\pi^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with eight positive odd integer divisors and sixteen positive even integer divisors? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every integer $n \ge 1$ , the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$ . Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$ . (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$ .)
*Proposed by Lewis Chen* | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.) | {
"answer": "2003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A freight train was delayed on its route for 12 minutes. Then, over a distance of 60 km, it made up for the lost time by increasing its speed by 15 km/h. Find the original speed of the train. | {
"answer": "39.375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose Xiao Ming's family subscribes to a newspaper. The delivery person may deliver the newspaper to Xiao Ming's home between 6:30 and 7:30 in the morning. Xiao Ming's father leaves for work between 7:00 and 8:00 in the morning. What is the probability that Xiao Ming's father can get the newspaper before leaving home? | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(x\) if
\[2 + 7x + 12x^2 + 17x^3 + \dotsb = 100.\] | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets? | {
"answer": "87429",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $N$ such that among the four numbers $N$, $N+1$, $N+2$, and $N+3$, one is divisible by $3^2$, one by $5^2$, one by $7^2$, and one by $11^2$. | {
"answer": "363",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years in the ages of any two consecutive children. In years, how old is the oldest child? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set $H$ is defined by the points $(x, y)$ with integer coordinates, $-8 \leq x \leq 8$, $-8 \leq y \leq 8$. Calculate the number of squares of side length at least $9$ that have their four vertices in $H$. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the parts processed by a worker, on average, $4 \%$ are non-standard. Find the probability that among the 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the considered sample of 30 parts, and what is its probability? | {
"answer": "0.305",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How long should a covered lip pipe be to produce the fundamental pitch provided by a standard tuning fork? | {
"answer": "0.189",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $3015x + 3020y = 3025$ and $3018x + 3024y = 3030$, what is the value of $x - y$? | {
"answer": "11.1167",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\natural$ , $\sharp$ , $\flat$ such that for all $x, y\in \mathbb{R}$ , we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An *equatorial expression* over three real variables $x$ , $y$ , $z$ , along with the *complexity* of such expression, is defined recursively by the following:
- $x$ , $y$ , and $z$ are equatorial expressions of complexity 0;
- when $P$ and $Q$ are equatorial expressions with complexity $p$ and $q$ respectively, all of $P\mathbin\natural Q$ , $P\mathbin\sharp Q$ , $P\mathbin\flat Q$ are equatorial expressions with complexity $1+p+q$ .
Compute the number of distinct functions $f: \mathbb{R}^3\rightarrow \mathbb{R}$ that can be expressed as equatorial expressions of complexity at most 3.
*Proposed by Yannick Yao* | {
"answer": "419",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Automobile license plates for a state consist of three letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly two letters are each repeated once (yielding a total of four letters where two are the same), and the digits include exactly one repetition? | {
"answer": "877,500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \(ABCD\), \(BE = 5\), \(EC = 4\), \(CF = 4\), and \(FD = 1\), as shown in the diagram. What is the area of triangle \(\triangle AEF\)? | {
"answer": "42.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert $6532_8$ to base 5. | {
"answer": "102313_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a hypothetical math competition, contestants are given the problem to find three distinct positive integers $X$, $Y$, and $Z$ such that their product $X \cdot Y \cdot Z = 399$. What is the largest possible value of the sum $X+Y+Z$? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation \( x y z 1 = 4 \) can be rewritten as \( x y z = 4 \). | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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