problem stringlengths 10 5.15k | answer dict |
|---|---|
A rectangular prism has a volume of \(8 \mathrm{~cm}^{3}\), total surface area of \(32 \mathrm{~cm}^{2}\), and its length, width, and height are in geometric progression. Find the sum of all its edge lengths (in cm). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | {
"answer": "231",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after working a normal 8 hour day by that slow running clock, is | {
"answer": "$2.60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of two numbers is \( t \) and the positive difference between the squares of these two numbers is 208. What is the larger of the two numbers? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ and $\beta$ are acute angles, $\tan\alpha= \frac {1}{7}$, $\sin\beta= \frac { \sqrt {10}}{10}$, find $\alpha+2\beta$. | {
"answer": "\\frac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system \( xOy \), a moving line \( l \) is tangent to the parabola \( \Gamma: y^{2} = 4x \), and intersects the hyperbola \( \Omega: x^{2} - y^{2} = 1 \) at one point on each of its branches, left and right, labeled \( A \) and \( B \). Find the minimum area of \(\triangle AOB\). | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function \( g(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x-1} & \quad \text{ if } x \ge 5 \end{aligned} \right. \). Find the value of \( g^{-1}(-6) + g^{-1}(-5) + \dots + g^{-1}(4) + g^{-1}(5) \). | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate: $(12345679^2 \times 81 - 1) \div 11111111 \div 10 \times 9 - 8$ in billions. (Answer in billions) | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the side lengths are: $AB = 17, BC = 20$ and $CA = 21$. $M$ is the midpoint of side $AB$. The incircle of $\triangle ABC$ touches $BC$ at point $D$. Calculate the length of segment $MD$.
A) $7.5$
B) $8.5$
C) $\sqrt{8.75}$
D) $9.5$ | {
"answer": "\\sqrt{8.75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ unique integers $b_k$ ($1\le k\le s$) with each $b_k$ either $1$ or $- 1$ such that\[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 2012.\]Find $m_1 + m_2 + \cdots + m_s$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_n\}$ be an arithmetic sequence. If we select any 4 different numbers from $\{a_1, a_2, a_3, \ldots, a_{10}\}$ such that these 4 numbers still form an arithmetic sequence, then there are at most \_\_\_\_\_\_ such arithmetic sequences. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\sqrt[4]{256000000}$. | {
"answer": "40\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area? | {
"answer": "45 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p, q, r, s, t, u, v, w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[ (pt)^2 + (qu)^2 + (rv)^2 + (sw)^2. \] | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation using the completing the square method: $2x^{2}-4x-1=0$. | {
"answer": "\\frac{2-\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given eight cubes with volumes \(1, 8, 27, 64, 125, 216, 343,\) and \(512\) cubic units, and each cube having its volume decreasing upwards, determine the overall external surface area of this arrangement. | {
"answer": "1021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest possible sum of two perfect squares such that their difference is 175 and both squares are greater or equal to 36. | {
"answer": "625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a trapezoid $ABCD$ with bases $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \perp \overline{CD}$, suppose that $CD = 10$, $\tan C = 2$, and $\tan D = 1$. Calculate the length of $AB$ and determine the area of the trapezoid. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a triangle with $AB=85$ , $BC=125$ , $CA=140$ , and incircle $\omega$ . Let $D$ , $E$ , $F$ be the points of tangency of $\omega$ with $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ respectively, and furthermore denote by $X$ , $Y$ , and $Z$ the incenters of $\triangle AEF$ , $\triangle BFD$ , and $\triangle CDE$ , also respectively. Find the circumradius of $\triangle XYZ$ .
*Proposed by David Altizio* | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to the equation
\[\tan (7 \pi \cos \theta) = \cot (7 \pi \sin \theta)\]
where $\theta \in (0, 4 \pi).$ | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in her first two rolls? Express your answer as a common fraction. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distribute 6 volunteers into 4 groups, with each group having at least 1 and at most 2 people, and assign them to four different exhibition areas of the fifth Asia-Europe Expo. The number of different allocation schemes is ______ (answer with a number). | {
"answer": "1080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot? | {
"answer": "$\\frac{1}{8}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$. | {
"answer": "\\dfrac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis. | {
"answer": "\\frac{\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).
Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \). | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system Oxyz, given points A(2, 0, 0), B(2, 2, 0), C(0, 2, 0), and D(1, 1, $\sqrt{2}$), calculate the relationship between the areas of the orthogonal projections of the tetrahedron DABC onto the xOy, yOz, and zOx coordinate planes. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a-b+c=7$, $b-c+d=8$, $c-d+a=4$, and $d-a+b=1$. Determine the value of $2a+2b+2c+2d$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the side length of square $ABCD$ is 1, point $M$ is the midpoint of side $AD$, and with $M$ as the center and $AD$ as the diameter, a circle $\Gamma$ is drawn. Point $E$ is on segment $AB$, and line $CE$ is tangent to circle $\Gamma$. Find the area of $\triangle CBE$. | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? | {
"answer": "27.55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the data set $10$, $6$, $8$, $5$, $6$, calculate the variance $s^{2}=$ \_\_\_\_\_\_. | {
"answer": "\\frac{16}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? | {
"answer": "\\frac{5}{42}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square of a natural number \( a \) gives a remainder of 8 when divided by a natural number \( n \). The cube of the number \( a \) gives a remainder of 25 when divided by \( n \). Find \( n \). | {
"answer": "113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many two-digit numbers have digits whose sum is a prime number? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty-cent piece. What is the probability that at least 40 cents worth of coins land on heads? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation. | {
"answer": "\\frac{64\\pi}{105}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^4 b^6 = a^8 b^3 = 1.\] | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum at $x = 1$ with the value of 10, find the values of $a$ and $b$. | {
"answer": "-11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points respectively), with a total score of 120 points. The second round consists of 4 problem-solving questions covering plane geometry, algebra, number theory, and combinatorics. The first two questions are worth 40 points each, and the last two questions are worth 50 points each, with a total score of 180 points.
It is known that a certain math competition participant has a probability of $\frac{4}{5}$ of correctly answering each fill-in-the-blank question in the first round, and a probability of $\frac{3}{5}$ of correctly answering each problem-solving question in the first round. In the second round, the participant has a probability of $\frac{3}{5}$ of correctly answering each of the first two questions, and a probability of $\frac{2}{5}$ of correctly answering each of the last two questions. Assuming full marks for correct answers and 0 points for incorrect answers:
1. Let $X$ denote the participant's score in the second round. Find $P(X \geq 100)$.
2. Based on the historical competition results in the participant's province, if a participant scores 100 points or above in the first round, the probability of winning the provincial first prize is $\frac{9}{10}$, while if the score is below 100 points, the probability is $\frac{2}{5}$. Can the probability of the participant winning the provincial first prize reach $\frac{1}{2}$, and explain the reason.
(Reference data: $(\frac{4}{5})^8 \approx 0.168$, $(\frac{4}{5})^7 \approx 0.21$, $(\frac{4}{5})^6 \approx 0.262$.) | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers. | {
"answer": "525",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria wants to purchase a book which costs \$35.50. She checks her purse and discovers she has two \$20 bills, and twelve quarters, and a bunch of nickels. Determine the minimum number of nickels that Maria needs with her to buy the book. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, define \[g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\] Find the largest value of $n$, denoted as $M$, for which the domain of $g_n$ is nonempty. For this value of $M$, if the domain of $g_M$ consists of a single point $\{d\}$, compute $d$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area enclosed by the curve of $y = \arccos(\cos x)$ and the $x$-axis over the interval $\frac{\pi}{4} \le x \le \frac{9\pi}{4}.$ | {
"answer": "\\frac{3\\pi^2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\frac{(1 + 23) \left( 1 + \dfrac{23}{2} \right) \left( 1 + \dfrac{23}{3} \right) \dotsm \left( 1 + \dfrac{23}{25} \right)}{(1 + 27) \left( 1 + \dfrac{27}{2} \right) \left( 1 + \dfrac{27}{3} \right) \dotsm \left( 1 + \dfrac{27}{21} \right)}.
\] | {
"answer": "421200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are positive integers such that $xy - 8x + 9y = 632$, what is the minimal possible value of $|x - y|$? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) square centimeters. | {
"answer": "1176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the digits of the greatest prime number that is a divisor of $16,385$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( P \) is a moving point on the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\). The tangent line to the ellipse at point \( P \) intersects the circle \(\odot O\): \(x^{2}+y^{2}=12\) at points \( M \) and \( N \). The tangents to \(\odot O\) at \( M \) and \( N \) intersect at point \( Q \).
(1) Find the equation of the locus of point \( Q \);
(2) If \( P \) is in the first quadrant, find the maximum area of \(\triangle O P Q\). | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \le i \le 4$. The dimensions of the prism are given by lengths 2 along the x-axis, 3 along the y-axis, and 1 along the z-axis. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. Find the side length of the octahedron. | {
"answer": "\\frac{\\sqrt{14}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $ K$ , $ L$ , $ M$ , and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$ , $ BLC$ , $ CMD$ , and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$ , find the area of $ KLMN$. | {
"answer": "32 + 16\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Zuminglish-Advanced, all words still consist only of the letters $M, O,$ and $P$; however, there is a new rule that any occurrence of $M$ must be immediately followed by $P$ before any $O$ can occur again. Also, between any two $O's$, there must appear at least two consonants. Determine the number of $8$-letter words in Zuminglish-Advanced. Let $X$ denote this number and find $X \mod 100$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jake will roll two standard six-sided dice and make a two-digit number from the numbers he rolls. If he rolls a 4 and a 2, he can form either 42 or 24. What is the probability that he will be able to make an integer between 30 and 40, inclusive? Express your answer as a common fraction. | {
"answer": "\\frac{11}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-congruent triangles can be formed by selecting vertices from the ten points in the triangular array, where the bottom row has four points, the next row has three points directly above the gaps of the previous row, followed by two points, and finally one point at the top? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the equilateral triangle \(PQR\), \(S\) is the midpoint of \(PR\), and \(T\) is on \(PQ\) such that \(PT=1\) and \(TQ=3\). Many circles can be drawn inside the quadrilateral \(QRST\) such that no part extends outside of \(QRST\). The radius of the largest such circle is closest to: | {
"answer": "1.10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | {
"answer": "\\frac{7}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_{n}\}$ is an increasing sequence of integers, and $a_{1}\geqslant 3$, $a_{1}+a_{2}+a_{3}+\ldots +a_{n}=100$. Determine the maximum value of $n$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(P Q R S T\) is a pentagon with \(P Q=8\), \(Q R=2\), \(R S=13\), \(S T=13\), and \(T P=8\). Also, \(\angle T P Q=\angle P Q R=90^\circ\). What is the area of pentagon \(P Q R S T\) ? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | {
"answer": "950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill the nine numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid, placing one number in each cell, such that the numbers in each row increase from left to right and the numbers in each column decrease from top to bottom. How many different ways are there to achieve this arrangement? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). | {
"answer": "75600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the line $y = mx + 3$, find the maximum possible value of $a$ such that the line passes through no lattice point with $0 < x \leq 150$ for all $m$ satisfying $\frac{2}{3} < m < a$. | {
"answer": "\\frac{101}{151}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $1 - 2 - 3 + 4 + 5 + 6 + 7 + 8 - 9 - 10 - \dots + 9801$, where the signs change after each perfect square and repeat every two perfect squares. | {
"answer": "-9801",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd numbers? | {
"answer": "154",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively. | {
"answer": "\\frac{880}{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle is inscribed in a triangle if its vertices all lie on the boundary of the triangle. Given a triangle \( T \), let \( d \) be the shortest diagonal for any rectangle inscribed in \( T \). Find the maximum value of \( \frac{d^2}{\text{area } T} \) for all triangles \( T \). | {
"answer": "\\frac{4\\sqrt{3}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3* | {
"answer": "607",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive sequence $\{a_n\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \frac{5}{2}a_{n+1}a_n$, where $n \in \mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If $a_2 = \frac{3}{2}$, $a_3 = x$, and $a_4 = 4$, find the range of $x$.
2. Suppose the sequence $\{a_n\}$ is a geometric sequence with a common ratio of $q$. If $\frac{1}{2}S_n < S_{n+1} < 2S_n$ for $n \in \mathbb{N}^*$, find the range of $q$.
3. If $a_1, a_2, \ldots, a_k$ ($k \geq 3$) form an arithmetic sequence, and $a_1 + a_2 + \ldots + a_k = 120$, find the minimum value of the positive integer $k$, and the corresponding sequence $a_1, a_2, \ldots, a_k$ when $k$ takes the minimum value. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular decagon \( ABCDEFGHIJ \) has its center at \( K \). Each of the vertices and the center are to be associated with one of the digits \( 1 \) through \( 10 \), with each digit used exactly once, in such a way that the sums of the numbers on the lines \( AKF \), \( BKG \), \( CKH \), \( DKI \), and \( EKJ \) are all equal. Find the number of valid ways to associate the numbers. | {
"answer": "3840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose F_1 and F_2 are the two foci of a hyperbola C, and there exists a point P on the curve C that is symmetric to F_1 with respect to an asymptote of C. Calculate the eccentricity of the hyperbola C. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Timur and Alexander are counting the trees growing around the house. Both move in the same direction, but they start counting from different trees. What is the number of trees growing around the house if the tree that Timur called the 12th, Alexander counted as $33-m$, and the tree that Timur called the $105-m$, Alexander counted as the 8th? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | {
"answer": "63_{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) that satisfies the following properties: The distances from three of the points \(A, B, C, D\) to \(\alpha\) are equal, while the distance from the fourth point to \(\alpha\) is twice the distance of one of the three aforementioned points. Determine the number of such planes \(\alpha\). | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the length of side $y$ in the following diagram?
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(3))--cycle); // modified triangle lengths
draw((0,0)--(-2,0)--(0,2*sqrt(3))--cycle);
label("10",(-1,2*sqrt(3)/2),NW); // changed label
label("$y$",(2/2,2*sqrt(3)/2),NE);
draw("$30^{\circ}$",(2.5,0),NW); // modified angle
draw("$45^{\circ}$",(-1.9,0),NE);
draw(rightanglemark((0,2*sqrt(3)),(0,0),(2,0),4));
[/asy] | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A client of a brokerage company deposited 12,000 rubles into a brokerage account at a rate of 60 rubles per dollar with instructions to the broker to invest the amount in bonds of foreign banks, which have a guaranteed return of 12% per annum in dollars.
(a) Determine the amount in rubles that the client withdrew from their account after a year if the ruble exchange rate was 80 rubles per dollar, the currency conversion fee was 4%, and the broker's commission was 25% of the profit in the currency.
(b) Determine the effective (actual) annual rate of return on investments in rubles.
(c) Explain why the actual annual rate of return may differ from the one you found in point (b). In which direction will it differ from the above value? | {
"answer": "39.52\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest $5$ digit integer congruent to $17 \pmod{26}$? | {
"answer": "99997",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The FISS World Cup is a very popular football event among high school students worldwide. China successfully obtained the hosting rights for the International Middle School Sports Federation (FISS) World Cup in 2024, 2026, and 2028. After actively bidding by Dalian City and official recommendation by the Ministry of Education, Dalian ultimately became the host city for the 2024 FISS World Cup. During the preparation period, the organizing committee commissioned Factory A to produce a certain type of souvenir. The production of this souvenir requires an annual fixed cost of 30,000 yuan. For each x thousand pieces produced, an additional variable cost of P(x) yuan is required. When the annual production is less than 90,000 pieces, P(x) = 1/2x^2 + 2x (in thousand yuan). When the annual production is not less than 90,000 pieces, P(x) = 11x + 100/x - 53 (in thousand yuan). The selling price of each souvenir is 10 yuan. Through market analysis, it is determined that all souvenirs can be sold out in the same year.
$(1)$ Write the analytical expression of the function of annual profit $L(x)$ (in thousand yuan) with respect to the annual production $x$ (in thousand pieces). (Note: Annual profit = Annual sales revenue - Fixed cost - Variable cost)
$(2)$ For how many thousand pieces of annual production does the factory maximize its profit in the production of this souvenir? What is the maximum profit? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A magician writes the numbers 1 to 16 on 16 positions of a spinning wheel. Four audience members, A, B, C, and D, participate in the magic show. The magician closes his eyes, and then A selects a number from the wheel. B, C, and D, in that order, each choose the next number in a clockwise direction. Only A and D end up with even numbers on their hands. The magician then declares that he knows the numbers they picked. What is the product of the numbers chosen by A and D? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a 4-digit positive integer has four different digits, the leading digit is not zero, the integer is a multiple of 4, and 6 is the largest digit, determine the total count of such integers. | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the numbers 3, 0, 4, 8, and a decimal point to form decimals, the largest three-digit decimal is \_\_\_\_\_\_, the smallest decimal is \_\_\_\_\_\_, and their difference is \_\_\_\_\_\_. | {
"answer": "8.082",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles trapezoid \(ABCD\) is circumscribed around a circle. The lateral sides \(AB\) and \(CD\) are tangent to the circle at points \(M\) and \(N\), respectively, and \(K\) is the midpoint of \(AD\). In what ratio does the line \(BK\) divide the segment \(MN\)? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate how many numbers from 1 to 30030 are not divisible by any of the numbers between 2 and 16. | {
"answer": "5760",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any real number $x$, the symbol $[x]$ represents the integer part of $x$, i.e., $[x]$ is the largest integer not exceeding $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. This function $[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Given that the function $f(x) (x \in \mathbb{R})$ satisfies $f(x)=f(2-x)$, and when $x \geqslant 1$, $f(x)=\log _{2}x$, find the value of $[f(-16)]+[f(-15)]+\ldots+[f(15)]+[f(16)]$. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company needs 500 tons of raw materials to produce a batch of Product A, and each ton of raw material can generate a profit of 1.2 million yuan. Through equipment upgrades, the raw materials required to produce this batch of Product A were reduced by $x (x > 0)$ tons, and the profit generated per ton of raw material increased by $0.5x\%$. If the $x$ tons of raw materials saved are all used to produce the company's newly developed Product B, the profit generated per ton of raw material is $12(a-\frac{13}{1000}x)$ million yuan, where $a > 0$.
$(1)$ If the profit from producing this batch of Product A after the equipment upgrade is not less than the profit from producing this batch of Product A before the upgrade, find the range of values for $x$;
$(2)$ If the profit from producing this batch of Product B is always not higher than the profit from producing this batch of Product A after the equipment upgrade, find the maximum value of $a$. | {
"answer": "5.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ has mid-segments $EF$ and $GH$ such that $EF$ goes from midpoint of $AB$ to midpoint of $CD$, and $GH$ from midpoint of $BC$ to midpoint of $AD$. Given that $EF$ and $GH$ are perpendicular, and the lengths are $EF = 18$ and $GH = 24$, find the area of $ABCD$. | {
"answer": "864",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction. | {
"answer": "\\frac{39}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify completely: $$\sqrt[3]{80^3 + 100^3 + 120^3}.$$ | {
"answer": "20\\sqrt[3]{405}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six orange candies and four purple candies are available to create different flavors. A flavor is considered different if the percentage of orange candies is different. Combine some or all of these ten candies to determine how many unique flavors can be created based on their ratios. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers)
(1) The male students must stand together;
(2) The female students cannot stand next to each other;
(3) If the 4 female students have different heights, they must stand from left to right in order from tallest to shortest;
(4) The teachers cannot stand at the ends, and the male students must stand in the middle. | {
"answer": "1728",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six students stand in a row for a photo. Among them, student A and student B are next to each other, student C is not next to either student A or student B. The number of different ways the students can stand is ______ (express the result in numbers). | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.