problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation
\[ (3+i)^n = p_n + q_n i\] for all integers $n \geq 0$, where $i = \sqrt{-1}$. Determine the value of
\[ \sum_{n=0}^\infty \frac{p_n q_n}{9^n}.\]
A) $\frac{5}{8}$
B) $\frac{7}{8}$
C) $\frac{3}{4}$
D) $\frac{15}{16}$
E) $\frac{9}{20}$ | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution. | {
"answer": "1012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS. | {
"answer": "\\sqrt{89}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)
Height of the Players on the Basketball Team (inches)
$4|8$
$5|0\;1\;4\;6\;7\;7\;9$
$6|0\;3\;4\;5\;7\;9\;9$
$7|1\;2\;4$ | {
"answer": "61.44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ms. Garcia weighed the packages in three different pairings and obtained weights of 162, 164, and 168 pounds. Find the total weight of all four packages. | {
"answer": "247",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | {
"answer": "(3 + \\sqrt{15})^3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children could mention? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \le k \le 150$, how many $a_k$ are divisible by both 3 and 5? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library? | {
"answer": "95/159",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{12}(x+12)^{12}=(x^{2}-2x-2)^{6}$, where $a_{i}$ are constants. Find the value of $2a_{2}+6a_{3}+12a_{4}+20a_{5}+\ldots+132a_{12}$. | {
"answer": "492",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside this cylinder? | {
"answer": "2\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, 2)$
$A_{2}(-1, 1, 3)$
$A_{3}(2, -2, 4)$
$A_{4}(-1, 0, -2)$ | {
"answer": "\\sqrt{\\frac{35}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$ ?
<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details> | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Team A has a probability of $$\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match. | {
"answer": "\\frac{19}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Within a cube with edge length 6, there is a regular tetrahedron with edge length \( x \) that can rotate freely inside the cube. What is the maximum value of \( x \)? | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the expression: $\left( \pi - 1 \right)^{0} + \left( \frac{1}{2} \right)^{-1} + \left| 5 - \sqrt{27} \right| - 2 \sqrt{3}$. | {
"answer": "8 - 5 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(AB\) is the diameter of a circle; \(BC\) is a tangent; \(D\) is the point where line \(AC\) intersects the circle. It is known that \(AD = 32\) and \(DC = 18\). Find the radius of the circle. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ and $b \in R$, the function $f(x) = \ln(x + 1) - 2$ is tangent to the line $y = ax + b - \ln2$ at $x = -\frac{1}{2}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \leqslant g(x) \leqslant m^2 - 2$ holds true in the interval $[1, 2]$, determine the real number $m$. | {
"answer": "e + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, squares \( a \), \( b \), \( c \), \( d \), and \( e \) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \( e \). | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a \) is a positive real number and \( b \) is an integer between \( 2 \) and \( 500 \), inclusive, find the number of ordered pairs \( (a,b) \) that satisfy the equation \( (\log_b a)^{1001}=\log_b(a^{1001}) \). | {
"answer": "1497",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram $ABCD$ , the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$ . If the area of $XYZW$ is $10$ , find the area of $ABCD$ | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$ , compute $ABC$ .
*Proposed by Evan Chen* | {
"answer": "183",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice? | {
"answer": "5\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets $A=\{1, 3, 2m-1\}$ and $B=\{3, m^2\}$; if $B \subseteq A$, find the value of the real number $m$. | {
"answer": "-1",
"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"
} |
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$ . Suppose the area of $\vartriangle DOC$ is $2S/9$ . Find the value of $a/b$ . | {
"answer": "\\frac{2 + 3\\sqrt{2}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)4.7+\left(-2.5\right)-\left(-5.3\right)-7.5$;<br/>$(2)18+48\div \left(-2\right)^{2}-\left(-4\right)^{2}\times 5$;<br/>$(3)-1^{4}+\left(-2\right)^{2}\div 4\times [5-\left(-3\right)^{2}]$;<br/>$(4)(-19\frac{15}{16})×8$ (Solve using a simple method). | {
"answer": "-159\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A$ and $B$ are $46$ kilometers apart. Person A rides a bicycle from point $A$ to point $B$ at a speed of $15$ kilometers per hour. One hour later, person B rides a motorcycle along the same route from point $A$ to point $B$ at a speed of $40$ kilometers per hour.
$(1)$ After how many hours can person B catch up to person A?
$(2)$ If person B immediately returns to point $A after reaching point $B, how many kilometers away from point $B will they meet person A on the return journey? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one? | {
"answer": "124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can 7 people sit around a round table, considering that two seatings are the same if one is a rotation of the other, and additionally, one specific person must sit between two particular individuals? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(b_1,b_2,b_3,\ldots,b_{14})$ be a permutation of $(1,2,3,\ldots,14)$ for which
$b_1>b_2>b_3>b_4>b_5>b_6>b_7>b_8 \mathrm{\ and \ } b_8<b_9<b_{10}<b_{11}<b_{12}<b_{13}<b_{14}.$
Find the number of such permutations. | {
"answer": "1716",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex polygon, all its diagonals are drawn. These diagonals divide the polygon into several smaller polygons. What is the maximum number of sides that a polygon in the subdivision can have if the original polygon has:
a) 13 sides;
b) 1950 sides? | {
"answer": "1950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An employee arrives at the unit randomly between 7:50 and 8:30. Calculate the probability that he can clock in on time. | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1, x_2, x_3, \dots, x_{50}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{50}^2 = 1.$ Find the maximum value of
\[
\frac{x_1}{1 + x_1^2} + \frac{x_2}{1 + x_2^2} + \frac{x_3}{1 + x_3^2} + \dots + \frac{x_{50}}{1 + x_{50}^2}.
\] | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the eccentricities of a confocal ellipse and a hyperbola are \( e_1 \) and \( e_2 \), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \( \frac{1}{e_1} + \frac{1}{e_2} \). | {
"answer": "5/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a string of lights with a recurrent pattern of three blue lights followed by four yellow lights, spaced 7 inches apart. Determine the distance in feet between the 4th blue light and the 25th blue light, given that 1 foot equals 12 inches. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z \in [0, 1] \). The maximum value of \( M = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \) is ______ | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bees, in processing flower nectar into honey, remove a significant amount of water. Research has shown that nectar usually contains about $70\%$ water, while the honey produced from it contains only $17\%$ water. How many kilograms of nectar must bees process to obtain 1 kilogram of honey? | {
"answer": "2.77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two real numbers are selected independently at random from the interval $[-15, 15]$. The product of those numbers is considered only if both numbers are outside the interval $[-5, 5]$. What is the probability that the product of those numbers, when considered, is greater than zero? | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $(128)^{\frac{7}{3}}$ | {
"answer": "65536 \\cdot \\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the two diagonals of an $8 \times 10$ grid are drawn, calculate the number of the $1 \times 1$ squares that are not intersected by either diagonal. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a nonnegative integer \(n\), let \(r_{11}(7n)\) stand for the remainder left when \(n\) is divided by \(11.\) For example, \(r_{11}(7 \cdot 3) = 10.\)
What is the \(15^{\text{th}}\) entry in an ordered list of all nonnegative integers \(n\) that satisfy $$r_{11}(7n) \leq 5~?$$ (Note that the first entry in this list is \(0\).) | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{a_n\}$ is an arithmetic sequence with common difference $d$, and $S_n$ is the sum of the first $n$ terms, if only $S_4$ is the minimum term among $\{S_n\}$, then the correct conclusion(s) can be drawn is/are ______.
$(1) d > 0$ $(2) a_4 < 0$ $(3) a_5 > 0$ $(4) S_7 < 0$ $(5) S_8 > 0$. | {
"answer": "(1)(2)(3)(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the three internal angles $A$, $B$, and $C$ of $\triangle ABC$ have opposite sides $a$, $b$, and $c$ respectively, and it is given that $b(\sin B-\sin C)+(c-a)(\sin A+\sin C)=0$.
$(1)$ Find the size of angle $A$;
$(2)$ If $a=\sqrt{3}$ and $\sin C=\frac{1+\sqrt{3}}{2}\sin B$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3+\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \( x \) and \( y \), define the operation \( \star \) as follows: \( x \star y = xy + 4y - 3x \).
Compute the value of the expression
$$
((\ldots)(((2022 \star 2021) \star 2020) \star 2019) \star \ldots) \star 2) \star 1
$$ | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $8 \cdot 9\frac{2}{5}$. | {
"answer": "75\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial sequence is defined as follows: \( f_{0}(x)=1 \) and \( f_{n+1}(x)=\left(x^{2}-1\right) f_{n}(x)-2x \) for \( n=0,1,2, \ldots \). Find the sum of the absolute values of the coefficients of \( f_{6}(x) \). | {
"answer": "190",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\] | {
"answer": "1025",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a regular octagon and a square formed by drawing four diagonals of the octagon. The edges of the square have length 1. What is the area of the octagon?
A) \(\frac{\sqrt{6}}{2}\)
B) \(\frac{4}{3}\)
C) \(\frac{7}{5}\)
D) \(\sqrt{2}\)
E) \(\frac{3}{2}\) | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square ABCD, where AB=2, fold along the diagonal AC so that plane ABC is perpendicular to plane ACD, resulting in the pyramid B-ACD. Find the ratio of the volume of the circumscribed sphere of pyramid B-ACD to the volume of pyramid B-ACD. | {
"answer": "4\\pi:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = |\ln x|$, the solution set of the inequality $f(x) - f(x_0) \geq c(x - x_0)$ is $(0, +\infty)$, where $x_0 \in (0, +\infty)$, and $c$ is a constant. When $x_0 = 1$, the range of values for $c$ is \_\_\_\_\_\_; when $x_0 = \frac{1}{2}$, the value of $c$ is \_\_\_\_\_\_. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular hexagon \( A_6 \) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I have 7 books, three of which are identical copies of the same novel, and the others are distinct. If a particular book among these must always be placed at the start of the shelf, in how many ways can I arrange the rest of the books? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many lattice points lie on the hyperbola \(x^2 - y^2 = 999^2\)? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$. | {
"answer": "1641",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $BC = a$ , $CA = b$ , and $AB = c$ . The $A$ -excircle is tangent to $\overline{BC}$ at $A_1$ ; points $B_1$ and $C_1$ are similarly defined.
Determine the number of ways to select positive integers $a$ , $b$ , $c$ such that
- the numbers $-a+b+c$ , $a-b+c$ , and $a+b-c$ are even integers at most 100, and
- the circle through the midpoints of $\overline{AA_1}$ , $\overline{BB_1}$ , and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$ .
| {
"answer": "125000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .
*Author:Anderson Wang* | {
"answer": "1458",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Investigate the formula of \\(\cos nα\\) and draw the following conclusions:
\\(2\cos 2α=(2\cos α)^{2}-2\\),
\\(2\cos 3α=(2\cos α)^{3}-3(2\cos α)\\),
\\(2\cos 4α=(2\cos α)^{4}-4(2\cos α)^{2}+2\\),
\\(2\cos 5α=(2\cos α)^{5}-5(2\cos α)^{3}+5(2\cos α)\\),
\\(2\cos 6α=(2\cos α)^{6}-6(2\cos α)^{4}+9(2\cos α)^{2}-2\\),
\\(2\cos 7α=(2\cos α)^{7}-7(2\cos α)^{5}+14(2\cos α)^{3}-7(2\cos α)\\),
And so on. The next equation in the sequence would be:
\\(2\cos 8α=(2\cos α)^{m}+n(2\cos α)^{p}+q(2\cos α)^{4}-16(2\cos α)^{2}+r\\)
Determine the value of \\(m+n+p+q+r\\). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Olya, after covering one-fifth of the way from home to school, realized that she forgot her notebook. If she does not return for it, she will reach school 6 minutes before the bell rings, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the focus of the parabola $4y^{2}=x$, and points $A$ and $B$ are on the parabola and located on both sides of the $x$-axis. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = 15$ (where $O$ is the origin), determine the minimum value of the sum of the areas of $\triangle ABO$ and $\triangle AFO$. | {
"answer": "\\dfrac{ \\sqrt{65}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sqrt{3}\sin^2{x}-\sin\left(2x-\frac{\pi}{3}\right)$,
(Ⅰ) Find the smallest positive period of the function $f(x)$ and the intervals where $f(x)$ is monotonically increasing;
(Ⅱ) Suppose $\alpha\in(0,\pi)$ and $f\left(\frac{\alpha}{2}\right)=\frac{1}{2}+\sqrt{3}$, find the value of $\sin{\alpha}$;
(Ⅲ) If $x\in\left[-\frac{\pi}{2},0\right]$, find the maximum value of the function $f(x)$. | {
"answer": "\\sqrt{3}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ant starts at one vertex of an octahedron and moves along the edges according to a similar rule: at each vertex, the ant chooses one of the four available edges with equal probability, and all choices are independent. What is the probability that after six moves, the ant ends at the vertex exactly opposite to where it started?
A) $\frac{1}{64}$
B) $\frac{1}{128}$
C) $\frac{1}{256}$
D) $\frac{1}{512}$ | {
"answer": "\\frac{1}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \(C\) to the lower right-hand corner \(D\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagonally. Each domino covers exactly two of the unit squares on the grid. Determine how many distinct arrangements are possible for Jessica to achieve this, assuming the path only moves right or down. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The administrator accidentally mixed up the keys for 10 rooms. If each key can only open one room, what is the maximum number of attempts needed to match all keys to their corresponding rooms? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer $n$ , such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$ .
- There is a real number $\xi$ with $P(\xi)=0$ . | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six people form a circle to play a coin-tossing game (the coin is fair). Each person tosses a coin once. If the coin shows tails, the person has to perform; if it shows heads, they do not have to perform. What is the probability that no two performers (tails) are adjacent? | {
"answer": "9/32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x > 0$, $y > 0$, and ${\!\!}^{2x+2y}=2$, find the minimum value of $\frac{1}{x}+\frac{1}{y}$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the equation $p = 15q^2 - 5$. Determine the value of $q$ when $p = 40$.
A) $q = 1$
B) $q = 2$
C) $q = \sqrt{3}$
D) $q = \sqrt{6}$ | {
"answer": "q = \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the last two digits of $\tbinom{200}{100}$ . Express the answer as an integer between $0$ and $99$ . (e.g. if the last two digits are $05$ , just write $5$ .) | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \in \mathbf{R} \), the sum of the maximum and minimum values of the function \( f(x)=\max \left\{\sin x, \cos x, \frac{\sin x+\cos x}{\sqrt{2}}\right\} \) is equal to? | {
"answer": "1 - \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a mathematics competition, there are four problems carrying 1, 2, 3, and 4 marks respectively. For each question, full score is awarded if the answer is correct; otherwise, 0 mark will be given. The total score obtained by a contestant is multiplied by a time bonus of 4, 3, 2, or 1 according to the time taken to solve the problems. An additional bonus score of 20 will be added after multiplying by the time bonus if one gets all four problems correct. How many different final scores are possible? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to \(x\), and the lateral side is equal to 12. On the ray \(AC\), point \(D\) is marked such that \(AD = 24\). From point \(D\), a perpendicular \(DE\) is dropped to the line \(AB\). Find \(x\) given that \(BE = 6\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the interior numbers of the eighth row of Pascal's Triangle? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hypotenuse of a right triangle measures $8\sqrt{2}$ inches and one angle is $45^{\circ}$. Calculate both the area and the perimeter of the triangle. | {
"answer": "16 + 8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the fictional country of Novaguard, they use the same twelve-letter Rotokas alphabet for their license plates, which are also five letters long. However, for a special series of plates, the following rules apply:
- The plate must start with either P or T.
- The plate must end with R.
- The letter U cannot appear anywhere on the plate.
- No letters can be repeated.
How many such special license plates are possible? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an Olympic 100-meter final, there are 10 sprinters competing, among which 4 are Americans. The gold, silver, and bronze medals are awarded to first, second, and third place, respectively. Calculate the number of ways the medals can be awarded if at most two Americans are to receive medals. | {
"answer": "588",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, find the sum of the values of $f(k)$ from $k=1$ to $k=22$. | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not below the $x$-axis? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Luis wrote the sequence of natural numbers, that is,
$$
1,2,3,4,5,6,7,8,9,10,11,12, \ldots
$$
When did he write the digit 3 for the 25th time? | {
"answer": "134",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Tuanjie Village, a cement road of $\frac {1}{2}$ kilometer long is being constructed. On the first day, $\frac {1}{10}$ of the total length was completed, and on the second day, $\frac {1}{5}$ of the total length was completed. What fraction of the total length is still unfinished? | {
"answer": "\\frac {7}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jacqueline has 200 liters of a chemical solution. Liliane has 30% more of this chemical solution than Jacqueline, and Alice has 15% more than Jacqueline. Determine the percentage difference in the amount of chemical solution between Liliane and Alice. | {
"answer": "13.04\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $\angle Q=90^\circ$, $PQ=9$ and $QR=12$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=6$, then what is the length of $PS$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks.
How many penalty kicks are needed to ensure that every player has had a chance to shoot against each of the goalies? | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(a+ \frac {1}{a})\ln x-x+ \frac {1}{x}$, where $a > 0$.
(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of values for $a$;
(II) Let $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$, does $M(a)$ have a maximum value? If it exists, find its maximum value; if not, explain why. | {
"answer": "\\frac {4}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jack and Jill run a 12 km circuit. First, they run 7 km to a certain point and then the remaining 5 km back to the starting point by different, uneven routes. Jack has a 12-minute head start and runs at the rate of 12 km/hr uphill and 15 km/hr downhill. Jill runs 14 km/hr uphill and 18 km/hr downhill. How far from the turning point are they when they pass each other, assuming their downhill paths are the same but differ in uphill routes (in km)?
A) $\frac{226}{145}$
B) $\frac{371}{145}$
C) $\frac{772}{145}$
D) $\frac{249}{145}$
E) $\frac{524}{145}$ | {
"answer": "\\frac{772}{145}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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