problem stringlengths 10 5.15k | answer dict |
|---|---|
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, and $c$ be three positive real numbers whose sum is 2. If no one of these numbers is more than three times any other, find the minimum value of the product $abc$. | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:
a) any three of them together have more than 1000 shares,
b) any three consecutive shareholders together have more than 1000 shares? | {
"answer": "6674",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$. | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly $1$ . What is the minimal number of liars? | {
"answer": "1012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point P moves on the parabola $y^2=4x$, and point Q moves on the line $x-y+5=0$. Find the minimum value of the sum of the distance $d$ from point P to the directrix of the parabola and the distance $|PQ|$ between points P and Q. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the fraction of the field that is planted. | {
"answer": "\\frac{431}{480}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the ratio of the area of a regular hexagon inscribed in an equilateral triangle with side length $s$ to the area of a regular hexagon inscribed in a circle with radius $r$? Assume the height of the equilateral triangle equals the diameter of the circle, thus $s = r \sqrt{3}$. | {
"answer": "\\dfrac{9}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit whole numbers have at least one 5 or consecutively have the digit 1 followed by the digit 2? | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\infty)$. Find the real number $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ and $AEFG$ be two faces of a cube with edge $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $AEFG$ at point $Q$, which is 3 units from $\overline{EF}$ and 6 units from $\overline{AG}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $p\sqrt{q}$, where $p$ and $q$ are integers and $q$ is not divisible by the square of any prime. Find $p+q$. | {
"answer": "155",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider three squares: $PQRS$, $TUVW$, and $WXYZ$, where each side of the squares has length $s=1$. $S$ is the midpoint of $WY$, and $R$ is the midpoint of $WU$. Calculate the ratio of the area of the shaded quadrilateral $PQSR$ to the sum of the areas of the three squares.
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac{1}{8}$
D) $\frac{1}{3}$
E) $\frac{1}{24}$ | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of:
\[\frac{abc(a + b + c)}{(a + b)^3 (b + c)^3}.\] | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a=2 \int_{-3}^{3} (x+|x|) \, dx$, determine the total number of terms in the expansion of $(\sqrt{x} - \frac{1}{\sqrt[3]{x}})^a$ where the power of $x$ is not an integer. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of the region in space defined by
\[|x - y + z| + |x - y - z| \le 10\]and $x, y, z \ge 0$. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ has an asymptote perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola. If $|F_1A|=2|F_2A|$, then $\cos \angle AF_2F_1=$ __________. | {
"answer": "\\dfrac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(\alpha) = \cos\alpha \sqrt{\frac{\cot\alpha - \cos\alpha}{\cot\alpha + \cos\alpha}} + \sin\alpha \sqrt{\frac{\tan\alpha - \sin\alpha}{\tan\alpha + \sin\alpha}}$, and $\alpha$ is an angle in the second quadrant.
(1) Simplify $f(\alpha)$.
(2) If $f(-\alpha) = \frac{1}{5}$, find the value of $\frac{1}{\tan\alpha} - \frac{1}{\cot\alpha}$. | {
"answer": "-\\frac{7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine if there are other composite numbers smaller than 20 that are also abundant besides number 12. If so, list them. If not, confirm 12 remains the smallest abundant number. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \geqslant \lambda(x-a)^{3}$ for all $x \geqslant 0$. Additionally, determine when equality holds in the given expression. | {
"answer": "-1/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum: \( S = 19 \cdot 20 \cdot 21 + 20 \cdot 21 \cdot 22 + \cdots + 1999 \cdot 2000 \cdot 2001 \). | {
"answer": "6 \\left( \\binom{2002}{4} - \\binom{21}{4} \\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A \(3\times 5\) rectangle and a \(4\times 6\) rectangle need to be contained within a square without any overlapping at their interior points, and the square's sides are parallel to the sides of the given rectangles. Determine the smallest possible area of this square. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | {
"answer": "152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate the total number of minutes he spends on the bus. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a three-digit number \( n = \overline{a b c} \). If the digits \( a, b, c \) can form an isosceles (including equilateral) triangle, calculate how many such three-digit numbers \( n \) are there. | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct products can you obtain by multiplying two or more distinct elements from the set $\{1, 2, 3, 5, 7, 11\}$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given acute angles $ \alpha $ and $ \beta $ satisfy $ \sin \alpha =\frac{\sqrt{5}}{5},\sin (\alpha -\beta )=-\frac{\sqrt{10}}{10} $, then $ \beta $ equals \_\_\_\_\_\_\_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Class A and Class B each send 2 students to participate in the grade math competition. The probability of each participating student passing the competition is 0.6, and the performance of the participating students does not affect each other. Find:
(1) The probability that there is exactly one student from each of Class A and Class B who passes the competition;
(2) The probability that at least one student from Class A and Class B passes the competition. | {
"answer": "0.9744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integer nearest to $500\sum_{n=4}^{10005}\frac{1}{n^2-9}$. | {
"answer": "174",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine $\sqrt[7]{218618940381251}$ without a calculator. | {
"answer": "102",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\frac{\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.
(1) Find the equation of the ellipse $E$;
(2) Find the maximum area of $\triangle OAB$. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse with its foci on the $y$-axis, an eccentricity of $\frac{2\sqrt{2}}{3}$, and one focus at $(0, 2\sqrt{2})$.
(1) Find the standard equation of the ellipse;
(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origin with a radius of $2$ at points $A$ and $B$. $C$ is a point on the ellipse such that $\overrightarrow{AB} \cdot \overrightarrow{CP} = 0$. Find the length of chord $AB$ when $|\overrightarrow{CP}|$ is at its maximum. | {
"answer": "\\frac{\\sqrt{62}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ , $b$ , $c$ be positive integers with $a \le 10$ . Suppose the parabola $y = ax^2 + bx + c$ meets the $x$ -axis at two distinct points $A$ and $B$ . Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$ . | {
"answer": "\\frac{\\sqrt{13}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right $ \triangle ABC$ has $ AB \equal{} 3$ , $ BC \equal{} 4$ , and $ AC \equal{} 5$ . Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$ , $ W$ on $ \overline{AB}$ , and $ Z$ on $ \overline{BC}$ . What is the side length of the square?
[asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt));
real s = (60/37);
pair A = (0,0);
pair C = (5,0);
pair B = dir(60)*3;
pair W = waypoint(B--A,(1/3));
pair X = foot(W,A,C);
pair Y = (X.x + s, X.y);
pair Z = (W.x + s, W.y);
label(" $A$ ",A,SW);
label(" $B$ ",B,NW);
label(" $C$ ",C,SE);
label(" $W$ ",W,NW);
label(" $X$ ",X,S);
label(" $Y$ ",Y,S);
label(" $Z$ ",Z,NE);
draw(A--B--C--cycle);
draw(X--W--Z--Y);[/asy] | {
"answer": "\\frac {60}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point P is taken on the circle x²+y²=4. A vertical line segment PD is drawn from point P to the x-axis, with D being the foot of the perpendicular. As point P moves along the circle, what is the trajectory of the midpoint M of line segment PD? Also, find the focus and eccentricity of this trajectory. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a} = (\cos \frac{3x}{2}, \sin \frac{3x}{2})$ and $\overrightarrow{b} = (\cos \frac{x}{2}, -\sin \frac{x}{2})$, with $x \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$,
(Ⅰ) Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $|\overrightarrow{a} + \overrightarrow{b}|$.
(Ⅱ) Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - |\overrightarrow{a} + \overrightarrow{b}|$, find the maximum and minimum values of $f(x)$. | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the rightmost three digits of $3^{1987}$? | {
"answer": "187",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\). | {
"answer": "433",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1)\sqrt{5}-27+|2-\sqrt{5}|-\sqrt{9}+(\frac{1}{2})^{2}$；<br/>$(2)2\sqrt{40}-5\sqrt{\frac{1}{10}}-\sqrt{10}$；<br/>$(3)(3\sqrt{12}-2\sqrt{\frac{1}{3}}-\sqrt{48})÷4\sqrt{3}-{(\sqrt{2}-1)^0}$；<br/>$(4)(-\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})+(-\sqrt{3}-1)^{2}$. | {
"answer": "2+2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A projectile is launched with an initial velocity of $u$ at an angle of $\alpha$ from the ground. The trajectory can be modeled by the parametric equations:
\[
x = ut \cos \alpha, \quad y = ut \sin \alpha - \frac{1}{2} kt^2,
\]
where $t$ denotes time and $k$ denotes a constant acceleration, forming a parabolic arch.
Suppose $u$ is constant, but $\alpha$ varies over $0^\circ \le \alpha \le 90^\circ$. The highest points of each parabolic arch are plotted. Determine the area enclosed by the curve traced by these highest points, and express it in the form:
\[
d \cdot \frac{u^4}{k^2}.
\] | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sample space be $\Omega =\{1,2,3,4,5,6,7,8\}$ with equally likely sample points, and events $A=\{1,2,3,4\}$, $B=\{1,2,3,5\}$, $C=\{1,m,n,8\}$, such that $p\left(ABC\right)=p\left(A\right)p\left(B\right)p\left(C\right)$, and satisfying that events $A$, $B$, and $C$ are not pairwise independent. Find $m+n$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integer.
[asy]
draw(arc((2,0), 1, 0,180));
draw((0,0)--(4,0));
draw((0,-2.5)--(4,-2.5));
draw(arc((3-sqrt(2)/2, -4+sqrt(2)/2+1.5), 1, -45, 135));
draw((3-sqrt(2), -4+sqrt(2)+1.5)--(3, -4+1.5));
draw(anglemark((3-sqrt(2), -4+sqrt(2)+1.5), (3, -4+1.5), (0, -4+1.5)));
label(" $\gamma$ ", (2.8, -3.9+1.5), WNW, fontsize(8));
[/asy]
*Problem proposed by Ahaan Rungta* | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given angles $\alpha$ and $\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\alpha, \beta \in (0, \pi)$. The terminal side of angle $\beta$ intersects the unit circle at a point whose x-coordinate is $-\frac{5}{13}$, and the terminal side of angle $\alpha + \beta$ intersects the unit circle at a point whose y-coordinate is $\frac{3}{5}$. Then, find the value of $\cos\alpha$. | {
"answer": "\\frac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and it satisfies $g(3i)=g(3+i)=0$. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $i = 2, 3, \ldots, 2020$ , let \[
a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}.
\]Let $x_2$ , $\ldots$ , $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$ . Let $S$ be the maximum possible value of \[
\sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{-2.25} x_i)
\] and let $m$ be the smallest positive integer such that $S^m$ is rational. When $S^m$ is written as a fraction in lowest terms, let its denominator be $p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ for prime numbers $p_1 < \cdots < p_k$ and positive integers $\alpha_i$ . Compute $p_1\alpha_1+p_2\alpha_2 + \cdots + p_k\alpha_k$ .
*Proposed by Edward Wan and Brandon Wang* | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2035\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\). | {
"answer": "1018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs? | {
"answer": "14.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange the sequence $\{2n+1\} (n\in\mathbb{N}^{*})$ sequentially in brackets with one number in the first bracket, two numbers in the second bracket, three numbers in the third bracket, four numbers in the fourth bracket, one number in the fifth bracket, and so on in a cycle, then calculate the sum of the numbers in the $120$th bracket. | {
"answer": "2392",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, the area of square \( QRST \) is 36. Also, the length of \( PQ \) is one-half of the length of \( QR \). What is the perimeter of rectangle \( PRSU \)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 20 positive divisors? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate to 3 decimal places the following expressions:
1.
$$
\frac{2 \sqrt{3}}{\sqrt{3}-\sqrt{2}}
$$
2.
$$
\frac{(3+\sqrt{3})(1+\sqrt{5})}{(5+\sqrt{5})(1+\sqrt{3})}
$$ | {
"answer": "0.775",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $i_{1}, i_{2}, \cdots, i_{n}$ be a permutation of the set $\{1, 2, \cdots, n\}$. If there exist $k < l$ and $i_{k} > i_{l}$, then the pair $\left(i_{k}, i_{l}\right)$ is called an inversion, and the total number of inversions in the permutation is called the inversion count of this permutation. For example, in the permutation 1432, the inversions are $43, 42,$ and $32$, so the inversion count of this permutation is 3. Given $n=6$ and $i_{3}=4$, find the sum of the inversion counts of all such permutations. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $m \gt 0$, $n \gt 0$, and $m+2n=1$, find the minimum value of $\frac{(m+1)(n+1)}{mn}$. | {
"answer": "8+4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \(\left\lfloor \frac{8(a+b)}{c} \right\rfloor + \left\lfloor \frac{8(a+c)}{b} \right\rfloor + \left\lfloor \frac{8(b+c)}{a} \right\rfloor\), where \(a\), \(b\), and \(c\) are arbitrary natural numbers. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x, y, z\) be nonzero real numbers such that \(x + y + z = 0\) and \(xy + xz + yz \neq 0\). Find all possible values of
\[
\frac{x^7 + y^7 + z^7}{xyz (xy + xz + yz)}.
\] | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, and ${a}_{n}+(-1)^{n}{a}_{n+1}=1-\frac{n}{2022}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{a_{n}\}$. Find $S_{2023}$. | {
"answer": "506",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, plane $ABDE$ is perpendicular to plane $ABC$. Triangle $ABC$ is an isosceles right triangle with $AC=BC=4$. Quadrilateral $ABDE$ is a right trapezoid with $BD \parallel AE$, $BD \perp AB$, $BD=2$, and $AE=4$. Points $O$ and $M$ are the midpoints of $CE$ and $AB$ respectively. Find the sine of the angle between line $CD$ and plane $ODM$. | {
"answer": "\\frac{\\sqrt{30}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
\[f(x) = \left\{
\begin{array}{cl}
x^2 + 3 & \text{if $x < 15$}, \\
3x - 2 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $f^{-1}(10) + f^{-1}(49).$ | {
"answer": "\\sqrt{7} + 17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( p(x) = x^4 - 4x^2 + 3x + 1 \), then find the coefficient of the \( x^3 \) term in the polynomial \( (p(x))^3 \). | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with magnitudes $|\overrightarrow {a}| = 6\sqrt {3}$ and $|\overrightarrow {b}| = \frac {1}{3}$, and their dot product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, determine the angle $\theta$ between $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the three-digit number \( n = abc \). If the digits \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle, how many such three-digit numbers exist? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sara baked 60 pies, of which one third contained chocolate, three-fifths contained berries, half contained cinnamon, and one-fifth contained poppy seeds. What is the largest possible number of pies that had none of these ingredients? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfying:
\[
\begin{array}{l}
\left|z_{1}\right| \leq 1, \left|z_{2}\right| \leq 2, \\
\left|2z_{3} - z_{1} - z_{2}\right| \leq \left|z_{1} - z_{2}\right|.
\end{array}
\]
What is the maximum value of \( \left|z_{3}\right| \)? | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The recurring decimal \(0 . \dot{x} y \dot{z}\), where \(x, y, z\) denote digits between 0 and 9 inclusive, is converted to a fraction in lowest term. How many different possible values may the numerator take? | {
"answer": "660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pete's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars. What is the maximum amount Pete can withdraw from the account if he has no other money? | {
"answer": "498",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An oreo shop now sells $5$ different flavors of oreos, $3$ different flavors of milk, and $2$ different flavors of cookies. Alpha and Gamma decide to purchase some items. Since Alpha is picky, he will order no more than two different items in total, avoiding replicas. To be equally strange, Gamma will only order oreos and cookies, and she will be willing to have repeats of these flavors. How many ways can they leave the store with exactly 4 products collectively? | {
"answer": "2100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, there are points $P_0$, $P_1$, $P_2$, $P_3$, ..., $P_{n-1}$, $P_n$. Let the coordinates of point $P_k$ be $(x_k,y_k)$ $(k\in\mathbb{N},k\leqslant n)$, where $x_k$, $y_k\in\mathbb{Z}$. Denote $\Delta x_k=x_k-x_{k-1}$, $\Delta y_k=y_k-y_{k-1}$, and it satisfies $|\Delta x_k|\cdot|\Delta y_k|=2$ $(k\in\mathbb{N}^*,k\leqslant n)$;
(1) Given point $P_0(0,1)$, and point $P_1$ satisfies $\Delta y_1 > \Delta x_1 > 0$, find the coordinates of $P_1$;
(2) Given point $P_0(0,1)$, $\Delta x_k=1$ $(k\in\mathbb{N}^*,k\leqslant n)$, and the sequence $\{y_k\}$ $(k\in\mathbb{N},k\leqslant n)$ is increasing, point $P_n$ is on the line $l$: $y=3x-8$, find $n$;
(3) If the coordinates of point $P_0$ are $(0,0)$, and $y_{2016}=100$, find the maximum value of $x_0+x_1+x_2+…+x_{2016}$. | {
"answer": "4066272",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the sum of all prime numbers $p$ for which there exists no integer solution $x$ to the congruence $5(8x+2)\equiv 3\pmod{p}$, and there exists no integer solution $y$ to the congruence $3(10y+3)\equiv 2\pmod{p}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student?
(The 7th American Junior High School Mathematics Examination, 1991) | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \). | {
"answer": "-13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An amoeba is placed in a puddle one day, and on that same day, it splits into two amoebas with a probability of 0.8. Each subsequent day, every amoeba in the puddle has a probability of 0.8 to split into two new amoebas. After one week, assuming no amoebas die, how many amoebas are there in the puddle on average? (Assume the puddle had no amoebas before the first one was placed in it.) | {
"answer": "26.8435456",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$, $y$, $z$, $u$, and $v$ be positive integers with $x+y+z+u+v=2505$. Let $N$ be the largest of the sums $x+y$, $y+z$, $z+u$, and $u+v$. Determine the smallest possible value of $N$. | {
"answer": "1253",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive odd integers greater than 1 and less than $200$ are square-free? | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of the expression $\frac{3\sqrt{12}}{\sqrt{3} + \sqrt{4} + \sqrt{6}}$.
A) $\sqrt{3} + 2\sqrt{2} - \sqrt{6}$
B) $4 - \sqrt{3} - 2\sqrt{2}$
C) $\sqrt{3} - 2\sqrt{2} + \sqrt{6}$
D) $\frac{1}{2}(\sqrt{3} + \sqrt{6} - 2\sqrt{2})$
E) $\frac{1}{3}(2\sqrt{2} + \sqrt{6} - \sqrt{3})$ | {
"answer": "\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three buckets, X, Y, and Z. The average weight of the watermelons in bucket X is 60 kg, the average weight of the watermelons in bucket Y is 70 kg. The average weight of the watermelons in the combined buckets X and Y is 64 kg, and the average weight of the watermelons in the combined buckets X and Z is 66 kg. Calculate the greatest possible integer value for the mean in kilograms of the watermelons in the combined buckets Y and Z. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The local cinema has two ticket windows opening simultaneously. In how many ways can eight people line up to buy a ticket if they can choose any of the two windows? | {
"answer": "10321920",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume the number of passengers traveling from location A to location B per day, $X$, follows a normal distribution $N(800, 50^2)$. Let $p_0$ denote the probability that the number of passengers traveling from A to B in a day does not exceed 900.
(1) Find the value of $p_0$. (Reference data: If $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X \leq \mu + \sigma) = 0.6826$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) = 0.9544$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) = 0.9974$)
(2) A passenger transport company uses two models of vehicles, A and B, for long-distance passenger transport services between locations A and B, with each vehicle making one round trip per day. The passenger capacities of models A and B are 36 and 60, respectively, and the operating costs from A to B are 1,600 yuan per vehicle for model A and 2,400 yuan per vehicle for model B. The company plans to form a passenger transport fleet of no more than 21 vehicles, with the number of model B vehicles not exceeding the number of model A vehicles by more than 7. If the company needs to transport all passengers from A to B each day with a probability of at least $p_0$ and aims to minimize the operating cost from A to B, how many vehicles of models A and B should be equipped? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | {
"answer": "\\frac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let set $M=\{x|-1\leq x\leq 5\}$, and set $N=\{x|x-k\leq 0\}$.
1. If $M\cap N$ has only one element, find the value of $k$.
2. If $k=2$, find $M\cap N$ and $M\cup N$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, where the common difference $d\neq 0$, and $S_{3}+S_{5}=50$, $a_{1}$, $a_{4}$, $a_{13}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $\{\frac{{b}_{n}}{{a}_{n}}\}$ be a geometric sequence with the first term being $1$ and the common ratio being $3$,<br/>① Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$;<br/>② If the inequality $λ{T}_{n}-{S}_{n}+2{n}^{2}≤0$ holds for all $n\in N^{*}$, find the maximum value of the real number $\lambda$. | {
"answer": "-\\frac{1}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sample data: $110$, $120$, $120$, $120$, $123$, $123$, $140$, $146$, $150$, $162$, $165$, $174$, $190$, $210$, $235$, $249$, $280$, $318$, $428$, $432$, find the $75$th percentile. | {
"answer": "242",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I planned to work 20 hours a week for 12 weeks this summer to earn $3000 to buy a used car. Unfortunately, due to unforeseen events, I wasn't able to work any hours during the first three weeks of the summer. How many hours per week do I need to work for the remaining summer to achieve my financial goal? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$, and it passes through point $A(2,1)$.
(Ⅰ) Find the equation of ellipse $C$;
(Ⅱ) If $P$, $Q$ are two points on ellipse $C$, and the angle bisector of $\angle PAQ$ always perpendicular to the x-axis, determine whether the slope of line $PQ$ is a constant value? If yes, find the value; if no, explain why. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ with eccentricity $e= \dfrac {2 \sqrt {3}}{3}$, calculate the angle between the two asymptotes. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}\cos x-x$.
(Ⅰ) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f (0))$;
(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | {
"answer": "-\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of a hyperbola $E$, and point $P$ is on $E$, with $\angle F_1 P F_2 = \frac{\pi}{6}$ and $(\overrightarrow{F_2 F_1} + \overrightarrow{F_2 P}) \cdot \overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyperbola $E$. | {
"answer": "\\frac{\\sqrt{3} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people, A and B, form a team called "Star Team" to participate in a guessing idiom activity. In each round, A and B each guess an idiom. It is known that the probability of A guessing correctly in each round is $\frac{2}{3}$, and the probability of B guessing correctly is $p$. In each round of the activity, the guesses of A and B do not affect each other, and the results of each round do not affect each other. It is known that the probability of "Star Team" guessing one idiom correctly in the first round of the activity is $\frac{1}{2}$.
$(1)$ Find the value of $p$;
$(2)$ Let $X$ denote the total number of idioms guessed correctly by "Star Team" in two rounds of activity. Find the probability distribution and expectation of $X$. | {
"answer": "\\frac{7}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | {
"answer": "152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \(ABC\), points \(K\), \(L\), and \(M\) are taken on sides \(AB\), \(BC\), and \(AD\) respectively. It is known that \(AK = 5\), \(KB = 3\), \(BL = 2\), \(LC = 7\), \(CM = 1\), and \(MA = 6\). Find the distance from point \(M\) to the midpoint of \(KL\). | {
"answer": "\\frac{1}{2} \\sqrt{\\frac{3529}{21}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the Spring Festival, a supermarket holds a promotional lottery event. When the amount spent by a customer reaches a certain threshold, they can participate in a lottery. The rules of the event are: from a box containing 3 black balls, 2 red balls, and 1 white ball (identical except for color), customers can draw balls.
(Ⅰ) If the customer spends more than 100 yuan but no more than 500 yuan, they can draw 2 balls at once. Each black ball drawn is rewarded with 1 yuan in cash, each red ball is rewarded with 2 yuan in cash, and each white ball is rewarded with 3 yuan in cash. Calculate the probability that the total reward money is at least 4 yuan.
(Ⅱ) If the purchase amount exceeds 500 yuan, they can draw twice from the box, drawing one ball each time and then returning it before drawing again. Each black ball and white ball drawn is rewarded with 5 yuan in cash, and each red ball is rewarded with 10 yuan in cash. Calculate the probability that the total reward money is less than 20 yuan. | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55. | {
"answer": "57.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Player A and player B are preparing for a badminton match. The rules state that the winner of a round will serve in the next round. If player A serves, the probability of player A winning the round is $\frac{3}{4}$; if player B serves, the probability of player A winning the round is $\frac{1}{4}$. The results of each round are independent. By drawing lots, it is determined that player A will serve in the first round.
$(1)$ Find the probability that player B will serve in the third round.
$(2)$ Find the probability that the number of rounds won by player A in the first three rounds is not less than the number won by player B. | {
"answer": "\\frac{21}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements. | {
"answer": "48",
"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.