problem stringlengths 10 5.15k | answer dict |
|---|---|
Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $x-y-1=0$ and $x-y-5=0$ both intersect circle $C$ creating chords of length 10, find the area of circle $C$. | {
"answer": "27\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $c=2$, $b=\sqrt{2}a$. The maximum area of $\triangle ABC$ is ______. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$. | {
"answer": "\\frac {175}{264}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(1000,0),(1000,1005),$ and $(0,1005)$. What is the probability that $x > 5y$? Express your answer as a common fraction. | {
"answer": "\\frac{20}{201}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a given point $P$ on the curve $x^2 - y - \ln x = 0$, what is the minimum distance from point $P$ to the line $y = x - 2$? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram. | {
"answer": "35 + 2\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can a President, a Vice-President, and a Secretary be chosen from a group of 6 people, given the following constraints:
- The President must be one of the first three members of the group.
- The Vice-President must be one of the last four members of the group.
- No person can hold more than one office. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a person A is taller or heavier than another person B, then we note that A is *not worse than* B. In 100 persons, if someone is *not worse than* other 99 people, we call him *excellent boy*. What's the maximum value of the number of *excellent boys*? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, angle $R$ is a right angle, $PR=15$ units, and $Q$ is on a circle centered at $R$. Squares $PQRS$ and $PRUT$ are formed on sides $PQ$ and $PR$, respectively. What is the number of square units in the sum of the areas of the two squares $PQRS$ and $PRUT$? | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A malfunctioning thermometer shows a temperature of $+1^{\circ}$ in freezing water and $+105^{\circ}$ in the steam of boiling water. Currently, this thermometer shows $+17^{\circ}$; what is the true temperature? | {
"answer": "15.38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \gt 0$, $b\in R$, if the inequality $\left(ax-2\right)(-x^{2}-bx+4)\leqslant 0$ holds for all $x \gt 0$, then the minimum value of $b+\frac{3}{a}$ is ______. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\cos B+b\sin A$.
(1) Find $A$;
(2) If $a=2$ and $b=c$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $\angle BAC=40^\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\parallel OG$ , find $\angle ACB$ . | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. Both calculated the sum of the numbers they wrote. By how much is the sum that Mena calculated greater than the sum that Emily calculated? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $a$ , $b$ , $c$ which are differ from $1$ satisfies the following conditions;
(1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$ . | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=\ln (1+ \frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$. | {
"answer": "\\frac {20}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One night, 21 people exchanged phone calls $n$ times. It is known that among these people, there are $m$ people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ called $a_{i+1}$ (for $i=1,2, \cdots, m$ and $a_{m+1}=a_{1}$), and $m$ is an odd number. If no three people among these 21 people have all exchanged calls with each other, determine the maximum value of $n$. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $7 \cdot 9\frac{2}{5}$. | {
"answer": "65 \\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains 4 tan, 3 pink, 5 violet, and 2 green chips. If all 14 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the 4 tan chips, the 3 pink chips, and the 5 violet chips are each drawn consecutively, and there is at least one green chip placed between any two groups of these chips of other colors? Express your answer as a common fraction. | {
"answer": "\\frac{1440}{14!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angles A, B, C of triangle ABC correspond to the sides a, b, c respectively, and vectors $\overrightarrow {m}$ = (a, $- \sqrt {3}b$) and $\overrightarrow {n}$ = (cosA, sinB), and $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$.
(1) Find angle A.
(2) If $a = \sqrt{39}$ and $c = 5$, find the area of triangle ABC. | {
"answer": "\\frac{5\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. Find the x-coordinate of the point on the $x$-axis where a line, tangent to both circles, intersects. The line should intersect the x-axis to the right of the origin. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\triangle DEO$, $\triangle EFO$, $\triangle FGO$, and half of $\triangle DCO$. | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points A(1, 2, 1), B(-2, $$\frac{7}{2}$$, 4), and D(1, 1, 1), if $$\vec{AP} = 2\vec{PB}$$, then the value of |$$\vec{PD}$$| is ______. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathcal{S}\) be a set of 16 points in the plane, no three collinear. Let \(\chi(\mathcal{S})\) denote the number of ways to draw 8 line segments with endpoints in \(\mathcal{S}\), such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of \(\chi(\mathcal{S})\) across all such \(\mathcal{S}\). | {
"answer": "1430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______. | {
"answer": "9+4 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}. | {
"answer": "94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radian measure of -150° is equal to what fraction of π. | {
"answer": "-\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x) = 2\sin x\cos x + 2\cos\left(x + \frac{\pi}{4}\right)\cos\left(x - \frac{\pi}{4}\right)$,
1. Find the interval(s) where $f(x)$ is monotonically decreasing.
2. If $\alpha \in (0, \pi)$ and $f\left(\frac{\alpha}{2}\right) = \frac{\sqrt{2}}{2}$, find the value of $\sin \alpha$. | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side surface of a cylinder unfolds into a rectangle with side lengths of $6\pi$ and $4\pi$. The surface area of the cylinder is ______. | {
"answer": "24\\pi^2 + 8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the triangle with vertices at $(1,4,5)$, $(3,4,1)$, and $(1,1,1)$. | {
"answer": "\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $(a>b>0)$, the “companion point” of a point M$(x_0, y_0)$ on the ellipse C is defined as $$N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$$.
(1) Find the equation of the trajectory of the “companion point” N of point M on the ellipse C;
(2) If the “companion point” of the point $(1, \frac{3}{2})$ on the ellipse C is $\left(\frac{1}{2}, \frac{3}{2b}\right)$, find the range of values for $\overrightarrow{OM} \cdot \overrightarrow{ON}$ for any point M on ellipse C and its “companion point” N;
(3) When $a=2$, $b= \sqrt{3}$, a line l intersects the ellipse C at points A and B. If the “companion points” of A and B are P and Q respectively, and the circle with diameter PQ passes through the origin O, find the area of $\triangle OAB$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $ | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can we arrange 4 math books, 6 English books, and 2 Science books on a shelf if:
1. All books of the same subject must stay together.
2. The Science books can be placed in any order, but cannot be placed next to each other.
(The math, English, and Science books are all different.) | {
"answer": "207360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively.
$(1)$ If the event "$x+y=8$" is denoted as event $A$, find the probability of event $A$ occurring;
$(2)$ If the event "$x^{2}+y^{2} \leqslant 12$" is denoted as event $B$, find the probability of event $B$ occurring. | {
"answer": "\\dfrac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the lengths of sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively. Given that $(2b - \sqrt{3}c)\cos A = \sqrt{3}a\cos C$.
1. Find the measure of angle $A$.
2. If angle $B = \frac{\pi}{6}$ and the length of the median $AM$ on side $BC$ is $\sqrt{7}$, find the area of triangle $ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\) and \(b\) be angles such that
\[\cos (a - b) = \cos a - \cos b.\]
Find the maximum value of \(\cos a\). | {
"answer": "\\sqrt{\\frac{3+\\sqrt{5}}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\frac{cos2α}{sin(α-\frac{π}{4})}=-\frac{\sqrt{6}}{2}$, express $cos(α-\frac{π}{4})$ in terms of radicals. | {
"answer": "\\frac{\\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance between the foci of the ellipse
\[\frac{x^2}{45} + \frac{y^2}{5} = 9.\] | {
"answer": "12\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\left(\frac{x}{x-1}-1\right) \div \frac{{x}^{2}-1}{{x}^{2}-2x+1}$, where $x=\sqrt{5}-1$. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction. | {
"answer": "\\frac{130}{1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sarah's six assignment scores are 87, 90, 86, 93, 89, and 92. What is the arithmetic mean of these six scores? | {
"answer": "89.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\frac {x^{ \frac {1}{2}}-y^{ \frac {1}{2}}}{x^{ \frac {1}{2}}+y^{ \frac {1}{2}}}=$ ___. | {
"answer": "- \\frac { \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=4x$, and the line $l$: $y=- \frac {1}{2}x+b$ intersects the parabola at points $A$ and $B$.
(I) If the $x$-axis is tangent to the circle with $AB$ as its diameter, find the equation of the circle;
(II) If the line $l$ intersects the negative semi-axis of $y$, find the maximum area of $\triangle AOB$. | {
"answer": "\\frac {32 \\sqrt {3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that an ellipse has the equation $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $e = \frac {\sqrt {6}}{3}$. The distance from the origin to the line that passes through points $A(0,-b)$ and $B(a,0)$ is $\frac {\sqrt {3}}{2}$.
$(1)$ Find the equation of the ellipse.
$(2)$ Given the fixed point $E(-1,0)$, if the line $y = kx + 2 \ (k \neq 0)$ intersects the ellipse at points $C$ and $D$, is there a value of $k$ such that the circle with diameter $CD$ passes through point $E$? Please provide an explanation. | {
"answer": "\\frac {7}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A merchant buys $n$ radios for $d$ dollars, where $d$ is a positive integer. The merchant sells two radios at half the cost price to a charity sale, and the remaining radios at a profit of 8 dollars each. If the total profit is 72 dollars, what is the smallest possible value of $n$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | {
"answer": "1936",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $θ$ is an angle in the second quadrant, and $\sin θ$ and $\cos θ$ are the two roots of the equation $2x^2 + (\sqrt{3} - 1)x + m = 0$ (where $m \in \mathbb{R}$), find the value of $\sin θ - \cos θ$. | {
"answer": "\\frac{1 + \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of 5739204.742 and -176817.835, and round the result to the nearest integer. | {
"answer": "5562387",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;
(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \frac{1}{ \sqrt{x}})^{n}$ is $16$, find the coefficient of the $x$ term in the expansion. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that circle $\odot M$ passes through the point $(1,0)$ and is tangent to the line $x=-1$, $S$ is a moving point on the trajectory of the center $M$ of the circle, and $T$ is a moving point on the line $x+y+4=0$. Find the minimum value of $|ST|$. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive value of $x$ that satisfies the equation:
\[\log_2 (x + 2) + \log_{4} (x^2 - 2) + \log_{\frac{1}{2}} (x + 2) = 5.\] | {
"answer": "\\sqrt{1026}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all real numbers \(x\) for which \(|x^2 - 14x + 45| = 3?\)
A) 12
B) 14
C) 16
D) 18 | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of the jacket was increased and then decreased by a certain percent, and then a 10% discount was applied, resulting in a final price that is 75% of the original price. Determine the percent by which the price was increased and then decreased. | {
"answer": "40.82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag, there are 2 black balls labeled $1$ and $2$, and 3 white balls labeled $3$, $4$, and $5$. These 5 balls are identical except for their labels and colors.
$(1)$ If two balls are randomly drawn from the bag with replacement, one at a time, what is the probability of drawing a black ball first and then a white ball?
$(2)$ If two balls are randomly drawn from the bag without replacement, denoting the label of a black ball as $x$ and the label of a white ball as $y$, what is the probability that $y-x \gt 2$? | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system \( xO y \), the circle \( \Omega \) and the parabola \( \Gamma: y^{2} = 4x \) share exactly one common point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | {
"answer": "\\frac{4 \\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy] | {
"answer": "13.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=\sin^{2}x+2 \sqrt {3}\sin x\cos x- \frac {1}{2}\cos 2x$$, where $x\in\mathbb{R}$.
(I) Find the smallest positive period and the range of $f(x)$.
(II) If $$x_{0}(0\leq x_{0}\leq \frac {\pi}{2})$$ is a zero of $f(x)$, find the value of $\sin 2x_{0}$. | {
"answer": "\\frac { \\sqrt {15}- \\sqrt {3}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n \) such that for any given \( n \) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \( R_{1}, R_{2}, R_{3} \) such that \( R_{1} \) can be nested inside \( R_{2} \) and \( R_{2} \) can be nested inside \( R_{3} \). | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with center $O$, the measure of $\angle SIP$ is $48^\circ$ and $OS=12$ cm. Find the number of centimeters in the length of arc $SP$ and also determine the length of arc $SXP$, where $X$ is a point on the arc $SP$ such that $\angle SXP = 24^\circ$. Express your answer in terms of $\pi$. | {
"answer": "3.2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $y=-x^{2}+bx+c$ where $b$ and $c$ are constants.
$(1)$ If $y=0$ and the corresponding values of $x$ are $-1$ and $3$, find the maximum value of the quadratic function.
$(2)$ If $c=-5$, and the quadratic function $y=-x^{2}+bx+c$ intersects the line $y=1$ at a unique point, find the expression of the quadratic function in this case.
$(3)$ If $c=b^{2}$, and the maximum value of the function $y=-x^{2}+bx+c$ is $20$ when $b\leqslant x\leqslant b+3$, find the value of $b$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is isosceles with $AB = AC = 2$ and $BC = 1.5$. Points $E$ and $G$ are on segment $\overline{AC}$, and points $D$ and $F$ are on segment $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$, trapezoid $DFGE$, and trapezoid $FBCG$ all have the same perimeter. Find the sum $DE+FG$.
A) $\frac{17}{6}$
B) $\frac{19}{6}$
C) $\frac{21}{6}$
D) $\frac{11}{3}$ | {
"answer": "\\frac{19}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\overrightarrow {AD}=3 \overrightarrow {DC}$, $\overrightarrow {BP}=2 \overrightarrow {PD}$, if $\overrightarrow {AP}=λ \overrightarrow {BA}+μ \overrightarrow {BC}$, then $λ+μ=\_\_\_\_\_\_$. | {
"answer": "- \\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and $P$ is a point on hyperbola $C$ in the second quadrant. If the line $y=\dfrac{b}{a}x$ is exactly the perpendicular bisector of the segment $PF_2$, then find the eccentricity of the hyperbola $C$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\] | {
"answer": "\\frac{1}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Out of 8 shots, 3 hit the target, and we are interested in the total number of ways in which exactly 2 hits are consecutive. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} |\ln x|, & (0 < x\leqslant e^{3}) \\ e^{3}+3-x, & (x > e^{3})\end{cases}$, there exist $x\_1 < x\_2 < x\_3$ such that $f(x\_1)=f(x\_2)=f(x\_3)$. Find the maximum value of $\frac{f(x\_3)}{x\_2}$. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$, where $a_3$ and $a_7$ are the two roots of the quadratic equation $x^2+7x+9=0$, calculate the value of $a_5$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse and a hyperbola have the same foci $F\_1(-c,0)$, $F\_2(c,0)$. One endpoint of the ellipse's minor axis is $B$, and the line $F\_1B$ is parallel to one of the hyperbola's asymptotes. If the eccentricities of the ellipse and hyperbola are $e\_1$ and $e\_2$, respectively, find the minimum value of $3e\_1^2+e\_2^2$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{16} w_j.\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A=\{0, |x|\}$ and $B=\{1, 0, -1\}$. If $A \subseteq B$, then $x$ equals \_\_\_\_\_\_; The union of sets $A$ and $B$, denoted $A \cup B$, equals \_\_\_\_\_\_; The complement of $A$ in $B$, denoted $\complement_B A$, equals \_\_\_\_\_\_. | {
"answer": "\\{-1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(x \geqslant 1\), the minimum value of the function \(y=f(x)= \frac {4x^{2}-2x+16}{2x-1}\) is \_\_\_\_\_\_, and the corresponding value of \(x\) is \_\_\_\_\_\_. | {
"answer": "\\frac {5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a board, the 2014 positive integers from 1 to 2014 are written. The allowed operation is to choose two numbers \( a \) and \( b \), erase them, and write in their place the numbers \( \text{gcd}(a, b) \) (greatest common divisor) and \( \text{lcm}(a, b) \) (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1's that we can leave on the board. | {
"answer": "1007",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line y=b intersects with the function f(x)=2x+3 and the function g(x)=ax+ln x (where a is a real constant in the interval [0, 3/2]), find the minimum value of |AB|. | {
"answer": "2 - \\frac{\\ln 2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number has 6 on both its tens and hundredths places, and 0 on both its ones and tenths places. This number is written as \_\_\_\_\_\_. | {
"answer": "60.06",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$ , write the square and the cube of a positive integer. Determine what that number can be. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the inequality $x^{2}+ax+1\geqslant 0$, if this inequality holds for all $x\in(0, \frac {1}{2}]$, find the minimum value of the real number $a$. | {
"answer": "-\\frac {5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\dbinom{60}{3}$. | {
"answer": "57020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we call this number $M$ a "heart's desire number"; if this two-digit number can also be divided by the sum of the thousands digit and units digit of $M$, then we call this number $M$ not only a "heart's desire" but also a "desire fulfilled". ["Heart's desire, desire fulfilled" comes from "Analects of Confucius. On Governance", meaning that what is desired in the heart becomes wishes, and all wishes can be fulfilled.] For example, $M=3456$, since $45 \gt 3+6$, and $45\div \left(3+6\right)=5$, $3456$ is not only a "heart's desire" but also a "desire fulfilled". Now there is a four-digit natural number $M=1000a+100b+10c+d$, where $1\leqslant a\leqslant 9$, $1\leqslant b\leqslant 9$, $1\leqslant c\leqslant 9$, $1\leqslant d\leqslant 9$, $a$, $b$, $c$, $d$ are all integers, and $c \gt d$. If $M$ is not only a "heart's desire" but also a "desire fulfilled", where $\frac{{10b+c}}{{a+d}}=11$, let $F\left(M\right)=10\left(a+b\right)+3c$. If $F\left(M\right)$ can be divided by $7$, then the maximum value of the natural number $M$ that meets the conditions is ____. | {
"answer": "5883",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x) = 2\cos x \sin (x - A) (x \in \mathbb{R})$ reaches its minimum value at $x = \frac{11\pi}{12}$.
1. Find the measure of angle $A$.
2. If $a = 7$ and $\sin B + \sin C = \frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$. | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least possible value of
\[(x+2)(x+3)(x+4)(x+5) + 2024\] where \( x \) is a real number?
A) 2022
B) 2023
C) 2024
D) 2025
E) 2026 | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest four-digit negative integer congruent to $2 \pmod{17}$? | {
"answer": "-1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos \alpha =-\dfrac{3}{4}, \sin \beta =\dfrac{2}{3}$, with $\alpha$ in the third quadrant and $\beta \in (\dfrac{\pi }{2}, \pi )$.
(I) Find the value of $\sin 2\alpha$;
(II) Find the value of $\cos (2\alpha + \beta )$. | {
"answer": "-\\dfrac{\\sqrt{5} + 6\\sqrt{7}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is \(1\tfrac{1}{2}\) divided by \(\tfrac{5}{6}\)? | {
"answer": "\\tfrac{9}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \] | {
"answer": "1022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, with \\(a=4\\), \\(b=5\\), and the area of \\(\triangle ABC\\) is \\(5\sqrt{3}\\). Find the value of side \\(c=\\) ______. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the power function $f(x)=x^{\alpha}$ ($\alpha$ is a constant) always passes through point $A$, and the line ${kx}{-}y{+}2k{+}1{+}\sqrt{3}{=}0$ always passes through point $B$, then the angle of inclination of line $AB$ is _____. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C_{1}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ and the hyperbola $C_{2}: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with asymptote equations $x \pm \sqrt{3}y = 0$, determine the product of the eccentricities of $C_{1}$ and $C_{2}$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of \((x + y + z)^8\), determine the sum of the coefficients of all terms of the form \(x^2 y^a z^b\) (\(a, b \in \mathbf{N}\)). | {
"answer": "1792",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lisa drew graphs of all functions of the form \( y = ax + b \), where \( a \) and \( b \) take all natural values from 1 to 100. How many of these graphs pass through the point \((3, 333)\)? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed to get 40 parts? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.