problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\triangle DEF$ to the area of quadrilateral $ABEF$ is: | {
"answer": "1: 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular swimming pool is 20 meters long, 15 meters wide, and 3 meters deep. Calculate its surface area. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular 14-gon, where each vertex is connected to every other vertex by a segment, three distinct segments are chosen at random. What is the probability that the lengths of these three segments refer to a triangle with positive area?
A) $\frac{73}{91}$
B) $\frac{74}{91}$
C) $\frac{75}{91}$
D) $\frac{76}{91}$
E) $\frac{77}{91}$ | {
"answer": "\\frac{77}{91}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ryan is learning number theory. He reads about the *Möbius function* $\mu : \mathbb N \to \mathbb Z$ , defined by $\mu(1)=1$ and
\[ \mu(n) = -\sum_{\substack{d\mid n d \neq n}} \mu(d) \]
for $n>1$ (here $\mathbb N$ is the set of positive integers).
However, Ryan doesn't like negative numbers, so he invents his own function: the *dubious function* $\delta : \mathbb N \to \mathbb N$ , defined by the relations $\delta(1)=1$ and
\[ \delta(n) = \sum_{\substack{d\mid n d \neq n}} \delta(d) \]
for $n > 1$ . Help Ryan determine the value of $1000p+q$ , where $p,q$ are relatively prime positive integers satisfying
\[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}. \]
*Proposed by Michael Kural* | {
"answer": "14013",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F$ , $D$ , and $E$ be points on the sides $[AB]$ , $[BC]$ , and $[CA]$ of $\triangle ABC$ , respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$ . The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$ , calculate the perimeter of $\triangle DEF$ . | {
"answer": "12\\sqrt{2} + 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the digits $1$, $2$, $3$, $4$, form a four-digit number with the first digit being $1$, and having exactly two identical digits in the number. How many such four-digit numbers are there? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\Delta APF$ is minimized, the area of this triangle is \_\_\_. | {
"answer": "\\dfrac{36}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | {
"answer": "1728",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Round $3.1415926$ to the nearest thousandth using the rounding rule, and determine the precision of the approximate number $3.0 \times 10^{6}$ up to which place. | {
"answer": "3.142",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jancsi usually waits for Juliska at the metro station in the afternoons. Once, he waited for 12 minutes, during which 5 trains arrived at the station. On another occasion, he waited for 20 minutes, and Juliska arrived on the seventh train. Yesterday, Jancsi waited for 30 minutes. How many trains could have arrived in the meantime, given that the time between the arrival of two trains is always the same? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the total area in square centimeters of the football-shaped area of regions II and III combined. Express your answer as a decimal to the nearest tenth. | {
"answer": "9.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD. | {
"answer": "\\frac{3}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain the answer. | {
"answer": "201320132013",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(A B C D\) is a rectangle, \(P\) is on \(B C\), \(Q\) is on \(C D\), and \(R\) is inside \(A B C D\). Also, \(\angle P R Q = 30^\circ\), \(\angle R Q D = w^\circ\), \(\angle P Q C = x^\circ\), \(\angle C P Q = y^\circ\), and \(\angle B P R = z^\circ\). What is the value of \(w + x + y + z\)? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct arrangements of the letters in the word "example" are there? | {
"answer": "5040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A_0 = (0,0) \). Points \( A_1, A_2, \dots \) lie on the \( x \)-axis, and distinct points \( B_1, B_2, \dots \) lie on the graph of \( y = x^2 \). For every positive integer \( n \), \( A_{n-1}B_nA_n \) is an equilateral triangle. What is the least \( n \) for which the length \( A_0A_n \geq 100 \)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of 11 people, including Ivanov and Petrov, are seated in a random order around a circular table. Find the probability that there will be exactly 3 people sitting between Ivanov and Petrov. | {
"answer": "1/10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If you write down twice in a row the grade I received in school for Latin, you will get my grandmother's age. What will you get if you divide this age by the number of my kittens? Imagine - you will get my morning grade, increased by fourteen-thirds.
How old is my grandmother? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $15\cdot\frac{16}{9}\cdot\frac{-45}{32}$. | {
"answer": "-\\frac{25}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the ancient Chinese mathematical masterpiece "The Mathematical Classic of Sunzi" Volume $26$, the $26$th question is: "There is an unknown quantity, when divided by $3$, the remainder is $2$; when divided by $5$, the remainder is $3; when divided by $7$, the remainder is $2$. What is the quantity?" The mathematical meaning of this question is: find a positive integer that leaves a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $2$ when divided by $7$. This is the famous "Chinese Remainder Theorem". If we arrange all positive integers that leave a remainder of $2$ when divided by $3$ and all positive integers that leave a remainder of $2$ when divided by $7$ in ascending order to form sequences $\{a_{n}\}$ and $\{b_{n}\}$ respectively, and take out the common terms from sequences $\{a_{n}\}$ and $\{b_{n}\}$ to form sequence $\{c_{n}\}$, then $c_{n}=$____; if sequence $\{d_{n}\}$ satisfies $d_{n}=c_{n}-20n+20$, and the sum of the first $n$ terms of the sequence $\left\{\frac{1}{d_{n}d_{n+1}}\right\}$ is $S_{n}$, then $S_{2023}=$____. | {
"answer": "\\frac{2023}{4050}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest prime factor of $15! + 18!$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a^2 = 16$, $|b| = 3$, $ab < 0$, find the value of $(a - b)^2 + ab^2$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 5$ passes through no lattice point with $0 < x \leq 150$ for all $m$ such that $0 < m < b$. What is the maximum possible value of $b$?
**A)** $\frac{1}{150}$
**B)** $\frac{1}{151}$
**C)** $\frac{1}{152}$
**D)** $\frac{1}{153}$
**E)** $\frac{1}{154}$ | {
"answer": "\\frac{1}{151}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which number appears most frequently in the second position when listing the winning numbers of a lottery draw in ascending order? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 4 are between 100 and 350? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis over the interval $0 \leq x \leq 2\pi$. | {
"answer": "\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the value \(\left(\frac{11}{12}\right)^{2}\), determine the interval in which this value lies. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $x+2y=a$ intersects the circle $x^2+y^2=4$ at points $A$ and $B$, and $|\vec{OA}+ \vec{OB}|=|\vec{OA}- \vec{OB}|$, where $O$ is the origin, determine the value of the real number $a$. | {
"answer": "-\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$). | {
"answer": "144\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of
\[
\frac{3x + 4y + 6}{\sqrt{x^2 + 4y^2 + 4}}
\]
over all real numbers \(x\) and \(y\). | {
"answer": "\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY = 540$ and $YZ = 360$. Points $N$ and $O$ are located on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XN = NY$, and $\overline{ZO}$ is the angle bisector of angle $Z$. Let $Q$ be the point of intersection of $\overline{YN}$ and $\overline{ZO}$, and let $R$ be the point on line $YN$ for which $N$ is the midpoint of $\overline{RQ}$. If $XR = 216$, find $OQ$. | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x = 3 \) and \( y = 7 \), then what is the value of \( \frac{x^5 + 3y^3}{9} \)? | {
"answer": "141",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student rolls two dice simultaneously, and the scores obtained are a and b respectively. The eccentricity e of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ (a > b > 0) is greater than $\frac{\sqrt{3}}{2}$. What is the probability of this happening? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.
$(1) q > 1$ $(2) T_{2013} > 1$ $(3) S_{2012}a_{2013} < S_{2013}a_{2012}$ $(4)$ The smallest natural number $n$ for which $T_n > 1$ is $4025$ $(5) \min \left( T_n \right) = T_{2012}$ | {
"answer": "(1)(3)(4)(5)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \[\left|\left(2 + 2i\right)^6 + 3\right|\] | {
"answer": "515",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $c_i$ denote the $i$ th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$ ) | {
"answer": "\\frac{12}{\\pi^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) be a non-zero real number such that
\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \].
Find the product of all possible values of \( x \). | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger originally used? | {
"answer": "285",
"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"
} |
A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$. | {
"answer": "\\sqrt{5 - \\sqrt{5}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the infinite geometric series: $$\frac{4}{3} - \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \dots$$ | {
"answer": "\\frac{64}{75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies? | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\). | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8$. Find the value of $r$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
 | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_n = n(2n+1)$ . Evaluate
\[
\biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |.
\] | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real axis of the hyperbola. Find the eccentricity of this hyperbola. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\cos C=-\frac{1}{4}$.
$(1)$ If $\sin A=2\sin B$, find $b$ and $c$;
$(2)$ If $\cos (A-\frac{π}{4})=\frac{4}{5}$, find $c$. | {
"answer": "\\frac{5\\sqrt{30}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right rectangular prism has edge lengths $\log_{5}x, \log_{8}x,$ and $\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.
A) $1,000,000$
B) $10,000,000$
C) $100,000,000$
D) $1,000,000,000$
E) $10,000,000,000$ | {
"answer": "100,000,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of diagonals of a regular $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$ -gon at right angles are still considered perpendicular.
*2018 CCA Math Bonanza Tiebreaker Round #1* | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 300 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 12 positive integer divisors? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the animal kingdom, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 of each of these three types of animals, divided into 100 groups. Each group has exactly 3 animals, with exactly 2 animals of one type and 1 animal of another type.
After the groups were formed, Kung Fu Panda asked each animal, "Is there a tiger in your group?" and 138 animals responded "yes." Kung Fu Panda then asked each animal, "Is there a fox in your group?" and 188 animals responded "yes."
How many monkeys told the truth both times? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three rays emanate from a single point and form pairs of angles of $60^{\circ}$. A sphere with a radius of one unit touches all three rays. Calculate the distance from the center of the sphere to the initial point of the rays. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangular prism $\mathrm{ABC}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$, the lengths of the base edges and the lateral edges are all 2. If $\mathrm{E}$ is the midpoint of $\mathrm{CC}_{1}$, what is the distance from $\mathrm{C}_{1}$ to the plane $\mathrm{AB} \mathrm{B}_{1} \mathrm{E}$? | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2, \cdots$ where the number of 2s between consecutive 1s increases by 1 each time, calculate the sum of the first 1234 terms. | {
"answer": "2419",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$ | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations. | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Sweet Tooth store, they are thinking about what promotion to announce before March 8. Manager Vasya suggests reducing the price of a box of candies by $20\%$ and hopes to sell twice as many goods as usual because of this. Meanwhile, Deputy Director Kolya says it would be more profitable to raise the price of the same box of candies by one third and announce a promotion: "the third box of candies as a gift," in which case sales will remain the same (excluding the gifts). In whose version of the promotion will the revenue be higher? In your answer, specify how much greater the revenue will be if the usual revenue from selling boxes of candies is 10,000 units. | {
"answer": "6000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse C: $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \quad (a>b>0)$$ which passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.
(I) Determine the standard equation of ellipse C;
(II) Let Q be a moving point on ellipse C that is not on the x-axis, with the origin O. Draw a parallel line to OQ through point $F_2$ intersecting ellipse C at two distinct points M and N.
(1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant value. If so, find this constant; if not, please explain why.
(2) Denote the area of $\triangle QF_2M$ as $S_1$ and the area of $\triangle OF_2N$ as $S_2$, and let $S = S_1 + S_2$. Find the maximum value of $S$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the XY-plane, mark all the lattice points $(x, y)$ where $0 \leq y \leq 10$. For an integer polynomial of degree 20, what is the maximum number of these marked lattice points that can lie on the polynomial? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.
a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the points $A_{1}, C_{1}$, and $D_{1}$.
b) Find the angle $B_{1} C_{1} A$.
c) It is additionally known that the radius of the sphere is 2. Find the volume of the prism. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $\triangle ABC$ and $\triangle DEC$ share side $BC$. Given that $AB = 7\ \text{cm}$, $AC = 15\ \text{cm}$, $EC = 9\ \text{cm}$, and $BD = 26\ \text{cm}$, what is the least possible integral number of centimeters in $BC$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Provide a negative integer solution that satisfies the inequality $3x + 13 \geq 0$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given ellipse $C_1$ and parabola $C_2$ whose foci are both on the x-axis, the center of $C_1$ and the vertex of $C_2$ are both at the origin $O$. Two points are taken from each curve, and their coordinates are recorded in the table. Calculate the distance between the left focus of $C_1$ and the directrix of $C_2$. | {
"answer": "\\sqrt {3}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $y$ if $y$ is positive and $y \cdot \lfloor y \rfloor = 132$. Express your answer as a decimal. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a shaded semicircle of diameter 4, from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded? | {
"answer": "$\\frac{1}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______. | {
"answer": "2618",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If acosB - bcosA = $$\frac {c}{3}$$, then the minimum value of $$\frac {acosA + bcosB}{acosB}$$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ . | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An entrepreneur invested \$12,000 in a three-month savings certificate that paid a simple annual interest rate of $8\%$. After three months, she invested the total value of her investment in another three-month certificate. After three more months, the investment was worth \$12,980. If the annual interest rate of the second certificate is $s\%$, what is $s?$ | {
"answer": "24\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express $0.7\overline{32}$ as a common fraction. | {
"answer": "\\frac{1013}{990}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$? | {
"answer": "12 \\sqrt[4]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{3}{5}$ and a minor axis length of $8$,
(1) Find the standard equation of the ellipse $C$;
(2) Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$, respectively. A line $l$ passing through $F_{2}$ intersects the ellipse $C$ at two distinct points $M$ and $N$. If the circumference of the inscribed circle of $\triangle F_{1}MN$ is $π$, and $M(x_{1},y_{1})$, $N(x_{2},y_{2})$, find the value of $|y_{1}-y_{2}|$. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $r$ be the speed in miles per hour at which a wheel, $15$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $4$ miles per hour. Determine the original speed $r$.
A) 9
B) 10
C) 11
D) 12
E) 13 | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mom decides to take Xiaohua on a road trip to 10 cities during the vacation. After checking the map, Xiaohua is surprised to find that for any three cities among these 10, either all three pairs of cities are connected by highways, or exactly two pairs of cities among the three are not connected by highways. What is the minimum number of highways that must be opened among these 10 cities? (Note: There can be at most one highway between any two cities.) | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle \(ABC\) with \(\angle A = 30^\circ\) and \(AB = AC\). Point \(D\) is the midpoint of \(BC\). Point \(P\) is chosen on segment \(AD\), and point \(Q\) is chosen on side \(AB\) such that \(PB = PQ\). What is the measure of angle \(PQC\)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $16$? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | {
"answer": "\\frac{29}{120}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the center of circle $C$ lies on the $x$-axis and circle $C$ is tangent to the line $x + \sqrt{3}y + n = 0$ at the point $(\frac{3}{2}, \frac{\sqrt{3}}{2})$, find:
1. The value of $n$ and the equation of circle $C$.
2. If circle $M: x^2 + (y - \sqrt{15})^2 = r^2 (r > 0)$ is tangent to circle $C$, find the length of the chord formed by intersecting circle $M$ with the line $\sqrt{3}x - \sqrt{2}y = 0$. | {
"answer": "2\\sqrt{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \le x \le 5$ and $1 \le y \le 5$? | {
"answer": "2160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Company $W$'s product $p$ produced by line $D$ follows a normal distribution $N(80, 0.25)$ in terms of size. $400$ randomly selected products were tested from the current production line, and the size distribution is summarized in the table below:
| Product Size ($mm$) | $[76,78.5]$ | $(78.5,79]$ | $(79,79.5]$ | $(79.5,80.5]$ | $(80.5,81]$ | $(81,81.5]$ | $(81.5,83]$ |
|---------------------|-------------|------------|-------------|--------------|------------|-------------|------------|
| Number of Items | $8$ | $54$ | $54$ | $160$ | $72$ | $40$ | $12$ |
According to the product quality standards and the actual situation of the production line, events outside the range $(\mu - 3\sigma, \mu + 3\sigma]$ are considered rare events. If a rare event occurs, it is considered an abnormality in the production line. Products within $(\mu - 3\sigma, \mu + 3\sigma]$ are considered as qualified products, while those outside are defective. $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$.
$(1)$ Determine if the production line is working normally and explain the reason.
$(2)$ Express the probability in terms of frequency. If $3$ more products are randomly selected from the production line for retesting, with a testing cost of $20$ dollars per qualified product and $30$ dollars per defective product, let the testing cost of these $3$ products be a random variable $X$. Find the expected value and variance of $X. | {
"answer": "\\frac{57}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rabbit and a hedgehog participated in a running race on a 550 m long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of 10 m/s and the hedgehog ran anticlockwise at a speed of 1 m/s. When they met, the rabbit continued as before, but the hedgehog turned around and ran clockwise. How many seconds after the rabbit did the hedgehog reach the finish? | {
"answer": "545",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the tangent line to the curve $f\left(x\right)=- \frac{ \sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\tan^{-1}\left( \frac{f'\left(1\right)}{\mid f'\left(1\right) \mid} \right)$, where $f'\left(x\right)$ is the derivative of $f\left(x\right)$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sin (\frac{\omega}{2}x+\varphi), 1)$ and $\overrightarrow{b}=(1, \cos (\frac{\omega}{2}x+\varphi))$, where $\omega > 0$ and $0 < \varphi < \frac{\pi}{4}$, define the function $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}-\overrightarrow{b})$. If the function $y=f(x)$ has a period of $4$ and passes through point $M(1, \frac{1}{2})$,
(1) Find the value of $\omega$;
(2) Find the maximum and minimum values of the function $f(x)$ when $-1 \leq x \leq 1$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit numbers exist that are 5 times the product of their digits? | {
"answer": "175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the roots of the polynomial $Q(x) = x^3 + px^2 + qx + r$ are $\sin \frac{3\pi}{7}, \sin \frac{5\pi}{7},$ and $\sin \frac{\pi}{7}$. Calculate the product $pqr$.
A) $0.725$
B) $1.45$
C) $1.0$
D) $1.725$ | {
"answer": "0.725",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many two digit numbers have exactly $4$ positive factors? $($ Here $1$ and the number $n$ are also considered as factors of $n. )$ | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral has vertices $P(a,b)$, $Q(2b,a)$, $R(-a, -b)$, and $S(-2b, -a)$, where $a$ and $b$ are integers and $a>b>0$. Find the area of quadrilateral $PQRS$.
**A)** $16$ square units
**B)** $20$ square units
**C)** $24$ square units
**D)** $28$ square units
**E)** $32$ square units | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x) = x^{3} + ax^{2} + bx + c\) has three real roots \(x_{1}, x_{2}, x_{3}\), and satisfy:
(1) \(x_{2} - x_{1} = \lambda\);
(2) \(x_{3} > \frac{1}{2}\left(x_{1} + x_{2}\right)\).
Find the maximum value of \(\frac{2a^{3} + 27c - 9ab}{\lambda^{3}}\). | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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