problem stringlengths 10 5.15k | answer dict |
|---|---|
Let \(x,\) \(y,\) \(z\) be real numbers such that \(9x^2 + 4y^2 + 25z^2 = 1.\) Find the maximum value of
\[8x + 3y + 10z.\] | {
"answer": "\\sqrt{173}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an eight-sided die (numbered 1 to 8) is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
A) $\frac{1}{12}$
B) $\frac{1}{8}$
C) $\frac{1}{6}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$ | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x)=2\tan \frac{x}{4}\cdot \cos^2 \frac{x}{4}-2\cos^2\left(\frac{x}{4}+\frac{\pi }{12}\right)+1$.
(Ⅰ) Find the smallest positive period and the domain of $f(x)$;
(Ⅱ) Find the intervals of monotonicity and the extremum of $f(x)$ in the interval $[-\pi,0]$; | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$ . $X$ , $Y$ , and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$ ). Compute the square of the smallest possible area of $XYZ$ .
*Proposed by Michael Ren* | {
"answer": "7500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 5 people standing in a row, calculate the number of ways for person A and person B to stand such that there is exactly one person between them. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in terms of $\pi$. | {
"answer": "9\\pi\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$ . Find the maximum possible value of $n$ . | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a flood control emergency, a large oil tank drifting downstream from upstream needs to be exploded by shooting. It is known that there are only $5$ bullets. The first hit can only cause the oil to flow out, and the second hit can cause an explosion. Each shot is independent, and the probability of hitting each time is $\frac{2}{3}$.
$(1)$ Calculate the probability that the oil tank will explode;
$(2)$ If the oil tank explodes or the bullets run out, the shooting will stop. Let $X$ be the number of shots. Calculate the probability that $X$ is not less than $4$. | {
"answer": "\\frac{7}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability $1/2$ and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die $5$ times and gets a $1, 2, 3, 4$ and $5$ in some unspecified
order. Compute the probability that his next roll is a $6$ . | {
"answer": "3/26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 chicks were sitting with their beaks open and 11 were growing feathers.
a) How many chicks were sitting with their beaks open two weeks ago?
b) How many chicks will be growing feathers next week?
Record the product of these numbers as the answer. | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function be $$f(x)=\sin(2\omega x+ \frac {\pi}{3})+ \frac { \sqrt {3}}{2}+a(\omega＞0)$$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $$\frac {\pi}{6}$$.
(1) Find the value of $\omega$;
(2) If the minimum value of $f(x)$ in the interval $$[- \frac {\pi}{3}, \frac {5\pi}{6}]$$ is $$\sqrt {3}$$, find the value of $a$;
(3) If $g(x)=f(x)-a$, what transformations are applied to the graph of $y=\sin x$ ($x\in\mathbb{R}$) to obtain the graph of $g(x)$? Also, write down the axis of symmetry and the center of symmetry for $g(x)$. | {
"answer": "\\frac { \\sqrt {3}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 6 balls of each of the four colors: red, blue, yellow, and green. Each set of 6 balls of the same color is numbered from 1 to 6. If 3 balls with different numbers are randomly selected, and these 3 balls have different colors and their numbers are not consecutive, the number of ways to do this is ______. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\). | {
"answer": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a school's mentoring program, several first-grade students can befriend one sixth-grade student, while one sixth-grade student cannot befriend multiple first-grade students. It is known that $\frac{1}{3}$ of the sixth-grade students and $\frac{2}{5}$ of the first-grade students have become friends. What fraction of the total number of students in the first and sixth grades are these friends? | {
"answer": "$\\frac{4}{11}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence, and calculate the 2003rd term of this new sequence. | {
"answer": "2047",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:
$\frac{4}{3}\pi r^3 = 50,000$
Simplifying, we get $r^3 = \frac{50,000 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, we want to find the volume of the top portion of Logan’s model. It is given that this sphere should have a volume of 0.2 liters. Using the same formula, we can solve for the radius:
$\frac{4}{3}\pi r^3 = 0.2$
Simplifying, we get $r^3 = \frac{0.2 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, since the sphere is a model of the top portion of the water tower, the radius of the model is proportional to the radius of the real tower. Therefore, the ratio of the radius of the model to the radius of the real tower is equal to the ratio of the volume of the model to the volume of the real tower.
$\frac{r_{model}}{r_{real}} = \frac{0.2}{50000}$
We can now equate the two expressions for the radius and solve for the height of the model:
$\frac{r_{model}}{60} = \frac{0.2}{50000}$ | {
"answer": "0.95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the 20th term in the sequence: $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \ldots, \frac{1}{m+1}, \frac{2}{m+1}, \ldots, \frac{m}{m+1}, \ldots$ | {
"answer": "\\frac{6}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of square \(ABCD\) is 64. The midpoints of its sides are joined to form the square \(EFGH\). The midpoints of its sides are \(J, K, L,\) and \(M\). The area of the shaded region is: | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For environmental protection, Wuyang Mineral Water recycles empty bottles. Consumers can exchange 4 empty bottles for 1 bottle of mineral water (if there are fewer than 4 empty bottles, they cannot be exchanged). Huacheng Middle School bought 1999 bottles of Wuyang brand mineral water. If they exchange the empty bottles for new bottles of water as much as possible, then the teachers and students of Huacheng Middle School can drink a total of bottles of mineral water; conversely, if they can drink a total of 3126 bottles of mineral water, then they originally bought bottles of mineral water. | {
"answer": "2345",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One angle of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid. | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day before. If he is late for work, he changes his mode of transportation the next day. Given all this, how likely is it that Maurice will be late for work on his 467th trip? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(x + \frac{1}{y} = 3\) and \(y + \frac{1}{z} = 3\), what is the value of the product \(xyz\)? | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\sin A= \sqrt {3}a\cos C$.
(Ⅰ) Find angle $C$;
(Ⅱ) If $c= \sqrt {21}$, and $\sin C+\sin (B-A)=5\sin 2A$, find the area of $\triangle ABC$. | {
"answer": "\\frac {5 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four teams, including Quixajuba, are competing in a volleyball tournament where:
- Each team plays against every other team exactly once;
- Any match ends with one team winning;
- In any match, the teams have an equal probability of winning;
- At the end of the tournament, the teams are ranked by the number of victories.
a) Is it possible that, at the end of the tournament, all teams have the same number of victories? Why?
b) What is the probability that the tournament ends with Quixajuba alone in first place?
c) What is the probability that the tournament ends with three teams tied for first place? | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the quotient $2213_4 \div 13_4$ in base 4. | {
"answer": "53_4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$? | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \). | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $A=\{(x,y) \,|\, |x| \leq 1, |y| \leq 1, x, y \in \mathbb{R}\}$, and $B=\{(x,y) \,|\, (x-a)^2+(y-b)^2 \leq 1, x, y \in \mathbb{R}, (a,b) \in A\}$, then the area represented by set $B$ is \_\_\_\_\_\_. | {
"answer": "12 + \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of rectangle $ABCD$ have lengths $12$ and $14$. 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": "36\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that a car driving on this road does not stop at point A and the probability that it does not stop at point B and the probability that it does not stop at point C are $\left(1-\frac{25}{60}\right)$, $\left(1-\frac{35}{60}\right)$, and $\left(1-\frac{45}{60}\right)$, respectively. | {
"answer": "\\frac{35}{192}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right trapezoid \(ABCD\), it is known that \(AB \perp BC\), \(BC \parallel AD\), \(AB = 12\), \(BC = 10\), and \(AD = 6\). Point \(F\) is a movable point on a circle centered at point \(C\) with radius 8, and point \(E\) is a point on \(AB\). When the value of \(DE + EF\) is minimized, what is the length of \(AE\)? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $X \sim N(\mu, \sigma^2)$, $P(\mu-\sigma < X \leq \mu+\sigma) = 0.68$, $P(\mu-2\sigma < X \leq \mu+2\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scored above 120? | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the set of all three-digit numbers composed of the digits 0, 1, 2, 3, 4, 5, without any repeating digits, how many such numbers have a digit-sum of 9? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$. | {
"answer": "\\frac{3\\sqrt{6}}{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the line $x - 2y + 2k = 0$ encloses a triangle with an area of $1$ together with the two coordinate axes, find the value of the real number $k$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits. | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of the squares of which is equal to 1, and the segments carry the products of the numbers at their ends. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the maximum number that could be obtained? | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perpendicular to the side $AB$ of the trapezoid $ABCD$, passing through its midpoint $K$, intersects the side $CD$ at point $L$. It is known that the area of quadrilateral $AKLD$ is five times greater than the area of quadrilateral $BKLC$. Given $CL=3$, $DL=15$, and $KC=4$, find the length of segment $KD$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is ${-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}$? | {
"answer": "-199",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface),
if $x + y + z = s$, and $0 \leq x \leq y \leq z \leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\_\_\_\_\_\_\_\_\_\_$. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let ellipse M be defined by the equation $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defined by $x^{2}+y^{2}=4$.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and there is a point $P(1, \sqrt {2})$ on ellipse M, find the maximum area of triangle PAB. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The digits from 1 to 9 are each used exactly once to write three one-digit integers and three two-digit integers. The one-digit integers are equal to the length, width and height of a rectangular prism. The two-digit integers are equal to the areas of the faces of the same prism. What is the surface area of the rectangular prism? | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \begin{cases} \log_{10} x, & x > 0 \\ x^{-2}, & x < 0 \end{cases}$, if $f(x\_0) = 1$, find the value of $x\_0$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a factor tree, each value is the product of the two values below it, unless a value is a prime number or a preset integer product of primes. Using this structure, calculate the value of $X$ in the factor tree provided:
[asy]
draw((-2,-.3)--(0,0)--(2,-.3),linewidth(1));
draw((-3,-1.3)--(-2,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(2,-.8)--(3,-1.3),linewidth(1));
label("X",(0,0),N);
label("F",(-2,-.8),N);
label("7",(-3,-1.3),S);
label("G",(2,-.8),N);
label("4",(-1,-1.3),S);
label("11",(1,-1.3),S);
label("H",(3,-1.3),S);
draw((-2,-2.3)--(-1,-1.8)--(0,-2.3),linewidth(1));
draw((2,-2.3)--(3,-1.8)--(4,-2.3),linewidth(1));
label("7",(-2,-2.3),S);
label("2",(0,-2.3),S);
label("11",(2,-2.3),S);
label("2",(4,-2.3),S);
[/asy] | {
"answer": "6776",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$ , that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$ . | {
"answer": "9/14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in the population and the sample. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\). | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to the three internal angles are $a$, $b$, and $c$, respectively. Given that $\cos A= \frac{ \sqrt {10}}{10}$ and $a\sin A+b\sin B-c\sin C= \frac{ 2 \sqrt {5}}{5}a\sin B$.
1. Find the value of $B$;
2. If $b=10$, find the area $S$ of $\triangle ABC$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $ABCD$ and $EFGD$ are squares each with side lengths of 5 and 3 respectively, and $H$ is the midpoint of both $BC$ and $EF$. Calculate the total area of the polygon $ABHFGD$. | {
"answer": "25.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the system of equations for the positive numbers \(x, y, z\):
$$
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=108 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=124
\end{array}\right.
$$
Find the value of the expression \(xy + yz + xz\). | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \_\_\_\_\_ | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two particles move along the edges of a square $ABCD$ with \[A \Rightarrow B \Rightarrow C \Rightarrow D \Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at vertex $A$, and the other starts at the midpoint of side $CD$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of square $ABCD$?
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{9}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is half the height \(BH\) of triangle \(ABC\). Find the ratio \(AH:HC\). | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya wrote consecutive natural numbers \( N \), \( N+1 \), \( N+2 \), and \( N+3 \) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first two circles turned out to be 200, and the sum of the numbers in the third and fourth circles turned out to be 105. What is the sum of the numbers in the second and third circles? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron \( P-ABCD \), where each face is an equilateral triangle with side length 1, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( BC \), respectively. Find the distance between the skew lines \( MN \) and \( PC \). | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of
\[\frac{2}{a} + \frac{4}{b}.\] | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $AC$ and $CE$ are two diagonals of a regular hexagon $ABCDEF$, and points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$. | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(T\). | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\circ$? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there is a geometric sequence $\{a_n\}$ with a common ratio $q > 1$ and the sum of the first $n$ terms is $S_n$, $S_3 = 7$, the sequence $a_1+3$, $3a_2$, $a_3+4$ forms an arithmetic sequence. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and $6T_n = (3n+1)b_n + 2$ for $n \in \mathbb{N}^*$.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Find the general term formula for the sequence $\{b_n\}$.
(3) Let $A = \{a_1, a_2, \ldots, a_{10}\}$, $B = \{b_1, b_2, \ldots, b_{40}\}$, and $C = A \cup B$. Calculate the sum of all elements in the set $C$. | {
"answer": "3318",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations $x^3 + Cx + 20 = 0$ and $x^3 + Dx^2 + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an infinite increasing sequence of natural numbers, each number is divisible by at least one of the numbers 1005 and 1006, but none is divisible by 97. Additionally, any two consecutive numbers differ by no more than $k$. What is the smallest possible $k$ for this scenario? | {
"answer": "2011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.
(I) Find the standard equation of the ellipse;
(II) If a line passing through $F_1$ intersects the ellipse at points $B$ and $D$, and another line passing through $F_2$ intersects the ellipse at points $A$ and $C$, with $AC \perp BD$, find the minimum value of $|AC| + |BD|$. | {
"answer": "\\frac{16\\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | {
"answer": "227.052227052227",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are located on sides $YZ$, $XZ$, and $XY$, respectively. The cevians $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$. Given that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the value of $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$ , $Q\in\overline{AC}$ , and $N,P\in\overline{BC}$ .
Suppose that $ABC$ is an equilateral triangle of side length $2$ , and that $AMNPQ$ has a line of symmetry perpendicular to $BC$ . Then the area of $AMNPQ$ is $n-p\sqrt{q}$ , where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$ .
*Proposed by Michael Ren* | {
"answer": "5073",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what? | {
"answer": "4\\sqrt{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $EFGH$, has its sides parallel to the corresponding sides of the larger square, and each side of $EFGH$ is tangent to one of the semicircles. Provide the area of square $EFGH$.
A) $36$
B) $64$
C) $100$
D) $144$
E) $256$ | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals: | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A grid consists of multiple squares, as shown below. Count the different squares possible using the lines in the grid.
[asy]
unitsize(0.5 cm);
int i, j;
for(i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
for(i = 1; i <= 4; ++i) {
for (j = 1; j <= 4; ++j) {
if ((i > 1 && i < 4) || (j > 1 && j < 4)) continue;
draw((i,j)--(i,j+1)--(i+1,j+1)--(i+1,j)--cycle);
}
}
[/asy] | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | {
"answer": "2 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\cos \angle BNG$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each pair of adjacent red dots counts as one fault, and each pair of blue dots exactly two apart (separated by one dot) counts as another fault. What is the smallest number of faults the critic could find? | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a\cos C=c\sin A$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\triangle ABC$. | {
"answer": "9 + 6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number $b$ , let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$ ? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is to the left, right, up, or down, all four equally likely. Let $q$ be the probability that the object reaches $(3,3)$ in eight or fewer steps. Write $q$ in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b.$ | {
"answer": "4151",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the product \( S = \left(1+2^{-\frac{1}{32}}\right)\left(1+2^{-\frac{1}{16}}\right)\left(1+2^{-\frac{1}{8}}\right)\left(1+2^{-\frac{1}{4}}\right)\left(1+2^{-\frac{1}{2}}\right) \), calculate the value of \( S \). | {
"answer": "\\frac{1}{2}\\left(1 - 2^{-\\frac{1}{32}}\\right)^{-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A capacitor with a capacitance of $C_{1} = 20 \mu$F is charged to a voltage $U_{1} = 20$ V. A second capacitor with a capacitance of $C_{2} = 5 \mu$F is charged to a voltage $U_{2} = 5$ V. The capacitors are connected with opposite-charged plates. Determine the voltage that will be established across the plates. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 2\sin^2\left(\frac{\pi}{4} + x\right) - \sqrt{3}\cos{2x} - 1$, where $x \in \mathbb{R}$:
1. If the graph of function $h(x) = f(x + t)$ is symmetric about the point $\left(-\frac{\pi}{6}, 0\right)$, and $t \in \left(0, \frac{\pi}{2}\right)$, find the value of $t$.
2. In an acute triangle $ABC$, if angle $A$ satisfies $h(A) = 1$, find the range of $(\sqrt{3} - 1)\sin{B} + \sqrt{2}\sin{C}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B, answering correctly yields 90 points and answering incorrectly results in a loss of 90 points. If the total score of the four students is 0 points, how many different scoring situations are there? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cells of a $50 \times 50$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if
(a) $n=2$
(b) $n=25$. | {
"answer": "1300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\{1, 2, 3, 4, 5, 6\}$. Find $P(7)$. | {
"answer": "51840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube? | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all positive integers $m>10^{2022}$ , determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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