problem stringlengths 10 5.15k | answer dict |
|---|---|
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ with an inclination angle $\alpha = 60^{\circ}$ is $\left\{\begin{array}{l}{x=2+t\cos\alpha}\\{y=t\sin\alpha}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole, and the non-negative half-axis of the $x$-axis as the polar axis. Establish a polar coordinate system with the same unit length as the rectangular coordinate system. The polar coordinate equation of the curve $C$ in the polar coordinate system is $\rho =\rho \cos ^{2}\theta +4\cos \theta$. <br/>$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$; <br/>$(2)$ Let point $P(2,0)$. The line $l$ intersects the curve $C$ at points $A$ and $B$, and the midpoint of chord $AB$ is $D$. Find the value of $\frac{|PD|}{|PA|}+\frac{|PD|}{|PB|}$. | {
"answer": "\\frac{\\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \(0 < x < \frac{\pi}{2}\), the value of the function \(y = \tan 3x \cdot \cot^3 x\) cannot take numbers within the open interval \((a, b)\). Find the value of \(a + b\). | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle PQR$ is similar to $\triangle XYZ$. What is the number of centimeters in the length of $\overline{YZ}$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(10,-2)--(8,6)--cycle);
label("10cm",(6,3),NW);
label("7cm",(10.2,2.5),NE);
draw((15,0)--(23,-1.8)--(22,4.5)--cycle);
label("$P$",(10,-2),E);
label("4cm",(21.2,1.3),NE);
label("$Q$",(8,6),N);
label("$R$",(0,0),SW);
label("$X$",(23,-1.8),E);
label("$Y$",(22,4.5),NW);
label("$Z$",(15,0),SW);
[/asy] | {
"answer": "5.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the drawing, there is a grid composed of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three equilateral triangles $ABC$, $BCD$, and $CDE$ are positioned such that $B$, $C$, and $D$ are collinear, and $C$ is the midpoint of $BD$. Triangle $CDE$ is positioned such that $E$ is on the same side of line $BD$ as $A$. What is the value of $AE \div BC$ when expressed in simplest radical form?
[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(12.5,4.35)--(15,0)--cycle);
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),S);
label("$D$",(15,0),SE);
label("$E$",(12.5,4.35),N);
[/asy] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be chosen randomly from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}5x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
A) $\frac{1}{9}$
B) $\frac{1}{10}$
C) $\frac{1}{8}$
D) $\frac{1}{7}$
E) $\frac{1}{6}$ | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A straight line $l$ passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to $l$ is one quarter of its minor axis length, calculate the eccentricity of the ellipse. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(XYZ,\) \(XY = 5,\) \(XZ = 7,\) \(YZ = 9,\) and \(W\) lies on \(\overline{YZ}\) such that \(\overline{XW}\) bisects \(\angle YXZ.\) Find \(\cos \angle YXW.\) | {
"answer": "\\frac{3\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack? | {
"answer": "\\frac{17}{442}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$. | {
"answer": "-\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the magnitude of vector $\overrightarrow {a}$ is 1, the magnitude of vector $\overrightarrow {b}$ is 2, and the magnitude of $\overrightarrow {a}+ \overrightarrow {b}$ is $\sqrt {7}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac {\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a region bounded by a larger quarter-circle with a radius of $5$ units, centered at the origin $(0,0)$ in the first quadrant, a smaller circle with radius $2$ units, centered at $(0,4)$ that lies entirely in the first quadrant, and the line segment from $(0,0)$ to $(5,0)$, calculate the area of the region. | {
"answer": "\\frac{9\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$. | {
"answer": "2001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Neznaika is drawing closed paths inside a $5 \times 8$ rectangle, traveling along the diagonals of $1 \times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$. | {
"answer": "\\sqrt{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the set of ordered triples $(x,y,z)$ of real numbers where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{5}{16}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}$ satisfies $a\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, then the minimum value of $\{\frac{S_{n}+4}{n}\}$ is ___. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arrived at $B$. On the way, the rabbit encounters 5 monkeys walking towards $B$, and continues to $A$ just as another monkey leaves $A$. If the rabbit's running speed is 3 km/h, find the distance between locations $A$ and $B$. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\left(a + \frac{1}{a}\right)^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
[/asy] | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?
A) 11:30
B) 11:45
C) 12:00
D) 12:30
E) 12:45 | {
"answer": "12:30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron $P-ABC$, $\Delta ABC$ is an equilateral triangle, and $PA=PB=PC=3$, $PA \perp PB$. The volume of the circumscribed sphere of the tetrahedron $P-ABC$ is __________. | {
"answer": "\\frac{27\\sqrt{3}\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $\overline{CE}$ bisects angle $C$ (no trisection this time), where $E$ is on $\overline{AB}$, $F$ is still on $\overline{AD}$, but now $BE=10$, and $AF=5$. Find the area of $ABCD$. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=15$, and it satisfies $\frac{a_{n+1}}{2n-3} = \frac{a_n}{2n-5}+1$, knowing $n$, $m\in\mathbb{N}$, and $n > m$, find the minimum value of $S_n - S_m$. | {
"answer": "-14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag, there are $4$ red balls, $m$ yellow balls, and $n$ green balls. Now, two balls are randomly selected from the bag. Let $\xi$ be the number of red balls selected. If the probability of selecting two red balls is $\frac{1}{6}$ and the probability of selecting one red and one yellow ball is $\frac{1}{3}$, then $m-n=$____, $E\left(\xi \right)=$____. | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$, and $A$, $B$, $C$ form an arithmetic sequence.
(I) If $b=7$ and $a+c=13$, find the area of $\triangle ABC$.
(II) Find the maximum value of $\sqrt{3}\sin A + \sin(C - \frac{\pi}{6})$ and the size of angle $A$ when the maximum value is reached. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receiving the task, the club members immediately started producing helium gas at a rate of $xL$ per day. It is known that the cost of raw materials for producing $1L$ of helium gas is $100$ yuan. If the daily production of helium gas is less than $4L$, the additional cost per day is $W_1=4x^2+16$ yuan. If the daily production of helium gas is greater than or equal to $4L$, the additional cost per day is $W_2=17x+\frac{9}{x}-3$ yuan. The production cost consists of raw material cost and additional cost.
$(1)$ Write the relationship between the total cost $W$ (in yuan) and the daily production $x$ (in $L$).
$(2)$ When the club produces how many liters of helium gas per day, the total cost is minimized? What is the minimum cost? | {
"answer": "4640",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original list price should the marked price be? | {
"answer": "107\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a village, a plot of land is shaped as a right triangle, where one of the legs measures 5 units and another measures 12 units. Farmer Euclid decides to leave a small unplanted square at the vertex of this right angle, and the shortest distance from this unplanted square to the hypotenuse is 3 units. Determine the fraction of the plot that is planted.
A) $\frac{412}{1000}$
B) $\frac{500}{1000}$
C) $\frac{290}{1000}$
D) $\frac{145}{1000}$
E) $\frac{873}{1000}$ | {
"answer": "\\frac{412}{1000}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Pascal's Triangle, we know each number is the combination of two numbers just above it. What is the sum of the middle three numbers in each of Rows 5, 6, and 7? | {
"answer": "157",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$ .
Find the greatest value of $M = abc$ | {
"answer": "6171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.
Determine the maximum possible number of positive integers among the 2018 integers. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line $\sqrt{3}x+y-1=0$ is ____. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron are equilateral triangles with a side length of 1 and form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that contains the given edge. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. | {
"answer": "\\frac{17\\pi}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \). Additionally, find its height dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Vertices:
- \( A_{1}(-1, 2, 4) \)
- \( A_{2}(-1, -2, -4) \)
- \( A_{3}(3, 0, -1) \)
- \( A_{4}(7, -3, 1) \) | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers between 200 and 400 are divisible by 8? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**p22.** Consider the series $\{A_n\}^{\infty}_{n=0}$ , where $A_0 = 1$ and for every $n > 0$ , $$ A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]}, $$ where $[x]$ denotes the largest integer value smaller than or equal to $x$ . Find the $(2023^{3^2}+20)$ -th element of the series.**p23.** The side lengths of triangle $\vartriangle ABC$ are $5$ , $7$ and $8$ . Construct equilateral triangles $\vartriangle A_1BC$ , $\vartriangle B_1CA$ , and $\vartriangle C_1AB$ such that $A_1$ , $B_1$ , $C_1$ lie outside of $\vartriangle ABC$ . Let $A_2$ , $B_2$ , and $C_2$ be the centers of $\vartriangle A_1BC$ , $\vartriangle B_1CA$ , and $\vartriangle C_1AB$ , respectively. What is the area of $\vartriangle A_2B_2C_2$ ?**p24.**There are $20$ people participating in a random tag game around an $20$ -gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$ -gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$ .
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | {
"answer": "653",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.
(I) Find the rectangular coordinate equation of curve $C\_2$ and the maximum value of the distance $|OP|$ between the moving point $P$ on curve $C\_1$ and the coordinate origin $O$;
(II) If curve $C\_2$ intersects with curve $C\_1$ at points $A$ and $B$, and intersects with the $x$-axis at point $E$, find the value of $|EA|+|EB|$. | {
"answer": "\\frac{6 \\sqrt{3}}{5}",
"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. It is given that $\sin C+\cos C=1-\sin \frac{C}{2}$.
$(1)$ Find the value of $\sin C$.
$(2)$ If $a^{2}+b^{2}=4(a+b)-8$, find the value of side $c$. | {
"answer": "1+ \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money. | {
"answer": "12.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of the function \( y = \tan x - \frac{2}{|\cos x|} \) is to be determined. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation when one piece of each product is sold. | {
"answer": "5.92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;
$(2)$ If $(2x-1)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{n}x^{n}$, find the value of $(a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}$. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the digits $1, 2, 3, 4$, 24 unique four-digit numbers can be formed without repeating any digit. If these 24 four-digit numbers are arranged in ascending order, find the sum of the two middle numbers. | {
"answer": "4844",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \). | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle $\triangle ABC$ with vertex angle $A = \frac{2\pi}{3}$ and base $BC = 2\sqrt{3}$, find the dot product $\vec{BA} \cdot \vec{AC}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as parallel bases, and $AB > CD$. A point $P$ inside the trapezoid connects to the vertices $A$, $B$, $C$, $D$, creating four triangles. The areas of these triangles, starting from the triangle with base $\overline{CD}$ moving clockwise, are $3$, $4$, $6$, and $7$. Determine the ratio $\frac{AB}{CD}$.
- **A**: $2$
- **B**: $\frac{5}{2}$
- **C**: $\frac{7}{3}$
- **D**: $3$
- **E**: $\sqrt{5}$ | {
"answer": "\\frac{7}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is | {
"answer": "$7 \\sqrt{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.) | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
What is the value of \( f(90) \)? | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Calculate the value of $b$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a sphere is $72\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bottom) of the cylinder. | {
"answer": "90\\pi \\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $2\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,
$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.
$(2)$ If three seeds are sown per hole, the probability of having seedlings in this hole is _______; the probability of having strong seedlings in this hole is _______. | {
"answer": "0.936",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈(0, \dfrac{π}{2})$ and $β∈(\dfrac{π}{2},π)$, with $\cos β=-\dfrac{1}{3}$ and $\sin (α+β)=\dfrac{7}{9}$.
(1) Find the value of $\sin α$;
(2) Find the value of $\sin (2α+β)$. | {
"answer": "\\dfrac{10\\sqrt{2}}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of x that satisfies the equation \( x^{x^x} = 2 \) is calculated. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functions: (1) $f(x) = \frac{1}{x}$; (2) $f(x) = x + 1$; (3) $f(x) = \begin{cases} -x^2 & \text{if}\ x \geq 0 \\ x^2 & \text{if}\ x < 0 \end{cases}$; identify which can be called an "ideal function" by their respective sequence numbers. | {
"answer": "(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$. | {
"answer": "\\frac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6}\left(\frac{x}{2}\right) \cos ^{2}\left(\frac{x}{2}\right) d x
$$ | {
"answer": "\\frac{5\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when she came back, she found Ivan and Foma in front of the door, with Foma having apparently arrived after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the shop to reopen. | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n = 2^{31} \times 3^{19} \times 5^7 \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | {
"answer": "13307",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p = 2027$ be the smallest prime greater than $2018$ , and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let $\mathrm{GF}(p)$ be the integers modulo $p$ , and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$ ). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$ , where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[
D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2}
\]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$ , and such that for any nonconstant polynomial $f$ , $D(f)$ is a polynomial with degree less than that of $f$ . If the number of possible values of $D(P(X))$ can be written as $a^b$ , where $a$ , $b$ are positive integers with $a$ minimized, compute $ab$ .
*Proposed by Brandon Wang* | {
"answer": "4114810",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a triangle is $80$ , and one side of the base angle is $60^\circ$ . The sum of the lengths of the other two sides is $90$ . The shortest side is: | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $M$ represents the number $9$ on the number line.<br/>$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.<br/>$(2)$ If point $M$ is moved $4$ units on the number line, then the number represented by point $M$ is ______. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$ . Lines $AM$ and $BC$ intersect in point $N$ . What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$ ? | {
"answer": "3/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two balls, one blue and one orange, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is $3^{-k}$ for $k = 1, 2, 3,...$. What is the probability that the blue ball is tossed into a higher-numbered bin than the orange ball?
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{1}{16}$
D) $\frac{7}{16}$
E) $\frac{3}{8}$ | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $x > 0$, find the maximum value of $f(x) = 1 - 2x - \frac{3}{x}$ and the value of $x$ at which it occurs. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$ . If $f(2004) = 2547$ , find $f(2547)$ . | {
"answer": "2547",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.298",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the fractions of the form $\frac{2}{n(n+2)}$, where $n$ takes on odd positive integers from 1 to 2011? Express your answer as a decimal to the nearest thousandth. | {
"answer": "0.999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock. | {
"answer": "12.925",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{a}=( \sqrt {2},m)(m > 0)$, $\overrightarrow{b}=(\sin x,\cos x)$, and the maximum value of the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$ is $2$.
1. Find $m$ and the smallest positive period of the function $f(x)$;
2. In $\triangle ABC$, $f(A- \frac {\pi}{4})+f(B- \frac {\pi}{4})=12 \sqrt {2}\sin A\sin B$, where $A$, $B$, $C$ are the angles opposite to sides $a$, $b$, $c$ respectively, and $C= \frac {\pi}{3}$, $c= \sqrt {6}$, find the area of $\triangle ABC$. | {
"answer": "\\frac { \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the sum \[ \sum_{n=0}^{332} (-1)^{n} {1008 \choose 3n} \] and find the remainder when the sum is divided by $500$. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=\log_{a}x$ and $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a\neq 1$, $t\in R$.
(1) If $0 < a < 1$, and $x\in[\frac{1}{4},2]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for the real number $t$;
(2) If $t=4$, and $x\in[\frac{1}{4},2]$ such that the minimum value of $F(x)=2g(x)-f(x)$ is $-2$, find the value of the real number $a$. | {
"answer": "a=\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | {
"answer": "\\frac{11+6\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$ . Find the least number $k$ such that $s(M,N)\le k$ , for all points $M,N$ .
*Dinu Șerbănescu* | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).
(I) Find the equation of the parabola C;
(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distances from points A and B to line l. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'? | {
"answer": "\\frac{11}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).
Find the maximum number \(n\) of such sets. | {
"answer": "871",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly bill for Package $B$ (in yuan), and $x$ represent the monthly call duration in minutes. <br/>$(1)$ Express the functions of $y_{1}$ with respect to $x$ and $y_{2}$ with respect to $x$. <br/>$(2)$ For how long should the monthly call duration be such that the charges for Packages $A$ and $B$ are the same? <br/>$(3)$ In what scenario is Package $A$ more cost-effective? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\sqrt{37}$. Find the cosine of the angle at the vertex. | {
"answer": "-\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equations of curve C as $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta \end{cases}(\theta\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\begin{cases} x'= \frac {1}{2}x \\ y'= \frac {1}{ \sqrt {3}}y \end{cases}$$ to obtain curve C'. With the origin as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established.
(Ⅰ) Find the polar equation of curve C';
(Ⅱ) If a line l passing through point $$A\left( \frac {3}{2},\pi\right)$$ (in polar coordinates) with a slope angle of $$\frac {\pi}{6}$$ intersects curve C' at points M and N, and the midpoint of chord MN is P, find the value of $$\frac {|AP|}{|AM|\cdot |AN|}$$. | {
"answer": "\\frac {3 \\sqrt {3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$ , $x^{2}\cdot x^{2}=x^{4}$ , $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating $x^{2006}$ . | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \(E\), and the point of intersection of the diagonals is labeled as \(F\). Similar right triangles \(BFC\) and \(DFA\) have corresponding sides \(x, y\) in the first triangle and \(4x, 4y\) in the second triangle.
The height of the trapezoid \(h\) is given by the sum of the heights of triangles \(BFC\) and \(AFD\):
\[ h = x y + \frac{4x \cdot 4y}{4} = 5xy \]
The area of the trapezoid is \(\frac{15}{16}\) of the area of triangle \(AED\):
\[ \frac{1}{2} AC \cdot BD = \frac{15}{16} \cdot \frac{1}{2} \cdot AE \cdot ED \cdot \sin 60^\circ = \frac{15 \sqrt{3}}{64} \cdot \frac{4}{3} AB \cdot \frac{4}{3} CD = \frac{15 \sqrt{3}}{36} AB \cdot CD \]
From this, we have:
\[ \frac{25}{2} xy = \frac{15 \sqrt{3}}{36} \sqrt{x^2 + 16y^2} \cdot \sqrt{y^2 + 16x^2} \]
Given that \(x^2 + y^2 = 1\), find:
\[ \frac{30}{\sqrt{3}} xy = \sqrt{1 + 15y^2} \cdot \sqrt{1 + 15x^2} \]
By simplifying:
\[ 300 x^2 y^2 = 1 + 15(x^2 + y^2) + 225 x^2 y^2 \]
\[ 75 x^2 y^2 = 16 \]
\[ 5xy = \frac{4}{\sqrt{3}} \] | {
"answer": "\\frac{4}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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