problem stringlengths 10 5.15k | answer dict |
|---|---|
Find the limit, when $n$ tends to the infinity, of $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$ | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^2 - 3x - 1$, find the derivative of $f(2)$ and $f'(1)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? | {
"answer": "1509",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \begin{cases} 2a-(x+ \frac {4}{x}),x < a\\x- \frac {4}{x},x\geqslant a\\end{cases}$.
(1) When $a=1$, if $f(x)=3$, then $x=$ \_\_\_\_\_\_;
(2) When $a\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=$ \_\_\_\_\_\_. | {
"answer": "- \\frac {11}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the ratio of the legs of a right triangle is $1:3$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $1:3$
B) $1:9$
C) $3:1$
D) $9:1$ | {
"answer": "9:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, forming a tray with taped edges. Determine the height of this tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges. | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube \(ABCDA_1B_1C_1D_1\) has edge length 1. Point \(M\) is taken on the side diagonal \(A_1D\), and point \(N\) is taken on \(CD_1\), such that the line segment \(MN\) is parallel to the diagonal plane \(A_1ACC_1\). Find the minimum length of \(MN\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle \( \triangle ABC \) where \(\angle ACB = 90^\circ\), \(CA = 3\), and \(CB = 4\), there is a point \(P\) inside \(\triangle ABC\) such that the sum of the distances from \(P\) to the three sides is \(\frac{13}{5}\). Find the length of the locus of point \(P\). | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\frac{(2200 - 2096)^2}{121}$? | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an acute-angled triangle $ABC$ , the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $ . Find the value of the smallest angle of triangle $ ABC $ .
(Black Maxim) | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle has altitudes of lengths 15, 21, and 35. Find its area. | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cells of a $20 \times 20$ table are colored in $n$ colors such that for any cell, in the union of its row and column, cells of all $n$ colors are present. Find the greatest possible number of blue cells if:
(a) $n=2$;
(b) $n=10$. | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$ | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares.
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
*Proposed by Arsenii Nikolaiev* | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, where the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, it is given that $2 \sqrt {3}ac\sin B = a^{2} + b^{2} - c^{2}$.
$(1)$ Determine the size of angle $C$;
$(2)$ If $b\sin (\pi - A) = a\cos B$ and $b= \sqrt {2}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt {3} + 1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored with two colors), then the coloring of the polygon is called "harmonious." Find the largest positive integer $N$ such that there exists a harmonious coloring of a regular $N$-gon. | {
"answer": "2017^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a shooting contest, 8 targets are arranged in two columns with 3 targets and one column with 2 targets. The rules are:
- The shooter can freely choose which column to shoot at.
- He must attempt the lowest target not yet hit.
a) If the shooter ignores the second rule, in how many ways can he choose only 3 positions out of the 8 distinct targets to shoot?
b) If the rules are followed, in how many ways can the 8 targets be hit? | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Jessie's 10th birthday, in 2010, her mother said, "My age is now five times your age." In what year will Jessie's mother be able to say, "My age is now 2.5 times your age," on Jessie's birthday? | {
"answer": "2027",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose point $P$ is on the curve represented by the equation $\sqrt{(x-5)^2+y^2} - \sqrt{(x+5)^2+y^2} = 6$, and $P$ is also on the line $y=4$. Determine the x-coordinate of point $P$. | {
"answer": "-3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$. | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function defined piecewise by
\[ f(x) = \left\{ \begin{aligned}
2x + 1 & \quad \text{if } x < 1 \\
x^2 & \quad \text{if } x \ge 1
\end{aligned} \right. \]
Determine the value of \( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \). | {
"answer": "1 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board we write a series of $n$ numbers, where $n \geq 40$ , and each one of them is equal to either $1$ or $-1$ , such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$ .
(ii) The sum of every $42$ consecutive numbers is not equal to $0$ .
We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$ . | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$? | {
"answer": "12.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $g(x)$ satisfies
\[ g(x + g(x)) = 5g(x) \]
for all $x$, and $g(1) = 5$. Find $g(26)$. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers $1, 2, \cdots, 10$, a number $a$ is randomly selected, and from the numbers $-1, -2, \cdots, -10$, a number $b$ is randomly selected. What is the probability that $a^{2} + b$ is divisible by 3? | {
"answer": "0.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | {
"answer": "\\dfrac{9}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere. | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $EFGH$ has area $2016$. An ellipse with area $2016\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? | {
"answer": "8\\sqrt{1008}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The moisture content of freshly cut grass is $60\%$, and the moisture content of hay is $15\%$. How much hay will be obtained from one ton of freshly cut grass? | {
"answer": "470.588",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Miki's father is saving money in a piggy bank for the family's vacation, adding to it once a week. Miki counts and notes how much money has accumulated every week and looks for patterns in the growth. Let $P_{n}$ denote the amount in the piggy bank in the $n$-th week (in forints). Here are a few observations:
(1) $P_{5} = 2P_{3}$,
(2) $P_{8} = P_{3} + 100$,
(3) $P_{9} = P_{4} + P_{7}$.
"The amount of forints has always been even, but it has never been divisible by 3."
"The number of forints today is a perfect square, and I also noticed that dad increases the deposit each week by the same amount that the third deposit exceeded the second deposit; thus the contents of our piggy bank will never be a perfect square again."
Which week does Miki's last observation refer to, and is Miki's prediction correct? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuckoo clock rings "cuckoo" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times did the clock ring "cuckoo" during this period? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ follows the rule:
\[a_{n+2} = a_{n+1} + a_n\]
for all $n \geq 1$. If $a_6 = 50$, find $a_7$. | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in the expansion of $(1-2x)^{n} (n \in \mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the product $(.331)^3$. | {
"answer": "0.037",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \_\_\_\_\_\_. | {
"answer": "5\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows.
\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]
Evaluate
\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\] | {
"answer": "e^{1/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?
Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$ . | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters. | {
"answer": "330",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is currently 3:15:15 PM on a 12-hour digital clock. After 196 hours, 58 minutes, and 16 seconds, what will the time be in the format $A:B:C$? What is the sum $A + B + C$? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest possible subset of {1, 2, ... , 15} such that the product of any three distinct elements of the subset is not a square. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1)$ Calculate: $\sqrt{12}-(-\frac{1}{2})^{-1}-|\sqrt{3}+3|+(2023-\pi)^0$<br/>$(2)$ Simplify the algebraic expression $\frac{3x-8}{x-1}-\frac{x+1}{x}÷\frac{{x}^{2}-1}{{x}^{2}-3x}$, and select appropriate integers from $0 \lt x\leqslant 3$ to substitute and find the value. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in $\triangle ABC$, $AB= \sqrt {3}$, $BC=1$, and $\sin C= \sqrt {3}\cos C$, the area of $\triangle ABC$ is ______. | {
"answer": "\\frac { \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the force with which water presses on a dam, whose cross-section has the shape of an isosceles trapezoid. The density of water is $\rho=1000 \, \text{kg} / \text{m}^{3}$, and the acceleration due to gravity $g$ is $10 \, \text{m} / \text{s}^{2}$.
Hint: The pressure at depth $x$ is $\rho g x$.
Given:
\[ a = 6.6 \, \text{m}, \quad b = 10.8 \, \text{m}, \quad h = 4.0 \, \text{m} \] | {
"answer": "640000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers between 200 and 400 are divisible by 8? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$ , $VB$ , $VC$ and $VD$ at $M$ , $N$ , $P$ , $Q$ respectively. Find $VQ : QD$ , if $VM : MA = 2 : 1$ , $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$ . | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A$ , $B$ , and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$ . Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$ . For all $i \ge 1$ , circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , and $\omega_{i - 1}$ . If
\[
S = \sum_{i = 1}^\infty r_i,
\]
then $S$ can be expressed as $a\sqrt{b} + c$ , where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$ .
*Proposed by Aaron Lin* | {
"answer": "233",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T = (1+i)^{19} - (1-i)^{19}$, where $i=\sqrt{-1}$. Determine $|T|$. | {
"answer": "512\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a chemistry quiz, there were $7y$ questions. Tim missed $2y$ questions. What percent of the questions did Tim answer correctly? | {
"answer": "71.43\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You flip a fair coin which results in heads ( $\text{H}$ ) or tails ( $\text{T}$ ) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$ ? | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of the circumcircle of the acute-angled triangle \(ABC\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \(A\), \(C\), and the orthocenter of triangle \(ABC\). Find \(AC\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest four-digit negative integer congruent to $1 \pmod{17}?$ | {
"answer": "-1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \(Z\) moves on \(|z| = 3\) in the complex plane, and \(w = \frac{1}{2}\left(z + \frac{1}{z}\right)\), where the trajectory of \(w\) is the curve \(\Gamma\). A line \(l\) passes through point \(P(1,0)\) and intersects the curve \(\Gamma\) at points \(A\) and \(B\), and intersects the imaginary axis at point \(M\). If \(\overrightarrow{M A} = t \overrightarrow{A P}\) and \(\overrightarrow{M B} = s \overrightarrow{B P}\), find the value of \(t + s\). | {
"answer": "-\\frac{25}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | {
"answer": "1365",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \perp l$, with the foot of the perpendicular being $Q$, and $\left( \overrightarrow{PC}+\frac{1}{2}\overrightarrow{PQ} \right)\cdot \left( \overrightarrow{PC}-\frac{1}{2}\overrightarrow{PQ} \right)=0$.
(1) Find the trajectory equation of the moving point $P$;
(2) If $EF$ is any diameter of circle $N: x^{2}+(y-1)^{2}=1$, find the maximum and minimum values of $\overrightarrow{PE}\cdot \overrightarrow{PF}$. | {
"answer": "12-4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new diagram, the grid is composed of squares. The grid is segmented into various levels that step upwards as you move to the right. Determine the area of the shaded region in the following configuration:
- The grid dimensions are 15 units wide and 5 units tall.
- The shaded region fills up from the bottom to a height of 2 units for the first 4 units of width, then rises to fill up to 3 units height until the 9th unit of width, then continues up to 4 units height until the 13th unit of width, and finally fills up to 5 units height until the 15th unit of width.
- An unshaded triangle is formed on the rightmost side, with a base of 15 units along the bottom and a height of 5 units. | {
"answer": "37.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{x|x^{2}+2x-3=0,x\in R\}$ and $B=\{x|x^{2}-\left(a+1\right)x+a=0,x\in R\}$.<br/>$(1)$ When $a=2$, find $A\cap C_{R}B$;<br/>$(2)$ If $A\cup B=A$, find the set of real numbers for $a$. | {
"answer": "\\{1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice's password consists of a two-digit number, followed by a symbol from the set {$!, @, #, $, %}, followed by another two-digit number. Calculate the probability that Alice's password consists of an even two-digit number followed by one of {$, %, @}, and another even two-digit number. | {
"answer": "\\frac{3}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola with its left and right foci being $F_1$ and $F_2$ respectively, and the length of chord $AB$ on the left branch passing through $F_1$ is 5. If $2a=8$, calculate the perimeter of $\triangle ABF_2$. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression \(\dfrac{\sqrt[4]{7}}{\sqrt[6]{7}}\). | {
"answer": "7^{\\frac{1}{12}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes. | {
"answer": "96\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_{n}\}$ such that $a_{1}+2a_{2}+\cdots +na_{n}=n$, and a sequence $\{b_{n}\}$ such that ${b_{m-1}}+{b_m}=\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ The sum of the first $20$ terms of $\{b_{n}\}$. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A taxi has a starting fare of 10 yuan. After exceeding 10 kilometers, for every additional kilometer, the fare increases by 1.50 yuan (if the increase is less than 1 kilometer, it is rounded up to 1 kilometer; if the increase is more than 1 kilometer but less than 2 kilometers, it is rounded up to 2 kilometers, etc.). Now, traveling from A to B costs 28 yuan. If one walks 600 meters from A before taking a taxi to B, the fare is still 28 yuan. If one takes a taxi from A, passes B, and goes to C, with the distance from A to B equal to the distance from B to C, how much is the taxi fare? | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Mathematics College Entrance Examination scores distribution $\xi$ closely follows the normal distribution $N(100, 5^2)$, and $P(\xi < 110) = 0.96$. Find the value of $P(90 < \xi < 100)$. | {
"answer": "0.46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The leadership team of a sports event needs to select 4 volunteers from 5 candidates named A, B, C, D, and E to undertake four different tasks: translation, tour guiding, protocol, and driving. If A and B can only undertake the first three tasks, while the other three candidates can undertake all four tasks, determine the number of different selection schemes. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 3012 positive numbers with both their sum and the sum of their reciprocals equal to 3013. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | {
"answer": "\\frac{12052}{3013}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\] | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the line $y=x-2$ intersects the hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a>0, b>0)$ at the intersection points $A$ and $B (not coincident)$. The perpendicular bisector of line segment $AB$ passes through the point $(4,0)$. The eccentricity of the hyperbola $C$ is ______. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adam and Sarah start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Sarah travels west at 5 mph. After how many hours are they 85 miles apart? | {
"answer": "7.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Harriet lives in a large family with 4 sisters and 6 brothers, and she has a cousin Jerry who lives with them. Determine the product of the number of sisters and brothers Jerry has in the house. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$ . | {
"answer": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the series:
\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \] | {
"answer": "305175780",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are six students with unique integer scores in a mathematics exam. The average score is 92.5, the highest score is 99, and the lowest score is 76. What is the minimum score of the student who ranks 3rd from the highest? | {
"answer": "95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | {
"answer": "\\frac{9}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( C \) and \( D \) have the same \( y \)-coordinate of 25, but different \( x \)-coordinates. What is the difference between the slope and the \( y \)-intercept of the line containing both points? | {
"answer": "-25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane, points $D(3, 15)$, $E(15, 0)$, and $F(0, t)$ form a triangle $\triangle DEF$. If the area of $\triangle DEF$ is 50, what is the value of $t$? | {
"answer": "\\frac{125}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $4$. Another die's faces are marked with $2$, $2$, $5$, $6$, $7$, and $8$. Find the probability that the sum of the top two numbers will be $7$, $9$, or $11$.
A) $\frac{1}{18}$
B) $\frac{1}{9}$
C) $\frac{1}{3}$
D) $\frac{4}{9}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angles of a triangle at points \( A, B \), and \( C \) are such that \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), calculate the value of \( x \). | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB=\sqrt{5}$, $BC=1$, and $AC=2$. $I$ is the incenter of $\triangle ABC$ and the circumcircle of $\triangle IBC$ intersects $AB$ at $P$. Find $BP$. | {
"answer": "\\sqrt{5} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand. | {
"answer": "2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the positive term geometric sequence $\\{a_n\\}$, $a\_$ and $a\_{48}$ are the two roots of the equation $2x^2 - 7x + 6 = 0$. Find the value of $a\_{1} \cdot a\_{2} \cdot a\_{25} \cdot a\_{48} \cdot a\_{49}$. | {
"answer": "9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, where $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3\cos Y$. What is the length of $XY$? | {
"answer": "\\frac{40\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected from the other $47$ points. | {
"answer": "\\frac{12}{47}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the xy-plane with a rectangular coordinate system, let vector $\overrightarrow {a}$ = (cosα, sinα) and vector $\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.
(1) If $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of α.
(2) If tan2α = -1/7, find the value of the dot product $\overrightarrow {a}$ • $\overrightarrow {b}$. | {
"answer": "\\frac{\\sqrt{6} - 7\\sqrt{2}}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)3-\left(-2\right)$;<br/>$(2)\left(-4\right)\times \left(-3\right)$;<br/>$(3)0\div \left(-3\right)$;<br/>$(4)|-12|+\left(-4\right)$;<br/>$(5)\left(+3\right)-14-\left(-5\right)+\left(-16\right)$;<br/>$(6)(-5)÷(-\frac{1}{5})×(-5)$;<br/>$(7)-24×(-\frac{5}{6}+\frac{3}{8}-\frac{1}{12})$;<br/>$(8)3\times \left(-4\right)+18\div \left(-6\right)-\left(-2\right)$;<br/>$(9)(-99\frac{15}{16})×4$. | {
"answer": "-399\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular $2018$ -gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$ -gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ( $2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers.
(Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle). | {
"answer": "2 \\times 1008!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle? | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative? | {
"answer": "-0.05",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the midpoints of $AD$ and $BC$, respectively. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find $p+q$, where the ratio is $p:q$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the linear function y=kx+b, where k and b are constants, and the table of function values, determine the incorrect function value. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. | {
"answer": "8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, it is known that the circle $C: x^{2} + y^{2} + 8x - m + 1 = 0$ intersects with the line $x + \sqrt{2}y + 1 = 0$ at points $A$ and $B$. If $\triangle ABC$ is an equilateral triangle, then the value of the real number $m$ is. | {
"answer": "-11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving point $A$ on the curve $y=x^{2}$, let $m$ be the tangent line to the curve at point $A$. Let $n$ be a line passing through point $A$, perpendicular to line $m$, and intersecting the curve at another point $B$. Determine the minimum length of the line segment $AB$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 99 positive integers, and their sum is 101101. The greatest possible value of the greatest common divisor of these 99 positive integers is: | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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