problem stringlengths 10 5.15k | answer dict |
|---|---|
Let \( n = 2^{40}5^{15} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | {
"answer": "599",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-2,0)$ and $P(1, \frac{3}{2})$ on the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.
(I) Find the equation of ellipse $M$ and its eccentricity.
(II) If quadrilateral $PABC$ is a parallelogram, find the value of $k$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = 1$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$ . | {
"answer": "4037",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-4 & \mbox{if }n\ge 1010 \\ g(g(n+7)) & \mbox{if }n<1010. \end{cases}\] Find $g(77)$. | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a WeChat group, there are 5 individuals: A, B, C, D, and E, playing a game involving grabbing red envelopes. There are 4 red envelopes, each person may grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many situations are there where both A and B grab a red envelope? (Answer with a numeral). | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 782^\circ$. | {
"answer": "-62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A manager schedules an informal review at a café with two of his team leads. He forgets to communicate a specific time, resulting in all parties arriving randomly between 2:00 and 4:30 p.m. The manager will wait for both team leads, but only if at least one has arrived before him or arrives within 30 minutes after him. Each team lead will wait for up to one hour if the other isn’t present, but not past 5:00 p.m. What is the probability that the review meeting successfully occurs? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of triangle $DEF$ are in the ratio $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. If the length of side $DE$ (the base) is 12 inches, what is the length, in inches, of the longer segment of side $EF$ once the bisector is drawn from $E$ to $EF$? | {
"answer": "\\frac{80}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of two consecutive even negative integers is 2496. What is the sum of these two integers? | {
"answer": "-102",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. | {
"answer": "6000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle and the radius of the circle? | {
"answer": "\\frac{10}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real number \( a \) makes the equation \( 4^{x} - 4^{-x} = 2 \cos(ax) \) have exactly 2015 solutions. For this \( a \), how many solutions does the equation \( 4^{x} + 4^{-x} = 2 \cos(ax) + 4 \) have? | {
"answer": "4030",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and a minor axis length of $2$.
1. Find the equation of the ellipse $E$;
2. A line $l$ is tangent to a circle $C$: $x^{2}+y^{2}=r^{2}(0 < r < b)$ at any point and intersects the ellipse $E$ at points $A$ and $B$, with $OA \perp OB$ ($O$ is the origin of the coordinate system), find the value of $r$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \). | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The tangent value of the angle between the slant height and the base is when the lateral area of the cone with volume $\frac{\pi}{6}$ is minimum. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = \_\_\_\_\_\_$. | {
"answer": "-\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After folding the long rope in half 6 times, calculate the number of segments the rope will be cut into. | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let \( T \) be the sum of this series. Find \( T \). | {
"answer": "\\frac{15}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alan, Beth, Carla, and Dave weigh themselves in pairs. Together, Alan and Beth weigh 280 pounds, Beth and Carla weigh 230 pounds, Carla and Dave weigh 250 pounds, and Alan and Dave weigh 300 pounds. How many pounds do Alan and Carla weigh together? | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: taking one pill daily. These pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect - absence of absent-mindedness, if it existed.
Another doctor prescribed the Scientist pills for absent-mindedness - also one per day for 10 consecutive days. These pills cure absent-mindedness in $80\%$ of cases, but in $5\%$ of cases, there is a side effect - knees stop hurting.
The bottles with pills look similar, and when the Scientist left for a ten-day business trip, he took one bottle with him, but paid no attention to which one. He took one pill a day for ten days and returned completely healthy. Both absent-mindedness and knee pain were gone. Find the probability that the Scientist took the pills for absent-mindedness. | {
"answer": "0.69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's line passes through (0,0) and (6,72), and Carlos’ line passes through (0,0) and (6,60). Determine how many more miles Alberto has traveled compared to Bjorn and Carlos individually after six hours. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the integers between 1 and 1500, inclusive, can be expressed as the difference of the squares of two positive integers? | {
"answer": "1124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
*Proposed by Jacob Weiner* | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vertex of angle $\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\frac{3}{5}, -\frac{4}{5})$.
(1) Find the value of $\sin(\alpha + \pi)$;
(2) If angle $\beta$ satisfies $\sin(\alpha + \beta) = \frac{5}{13}$, find the value of $\cos(\beta)$. | {
"answer": "\\frac{16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rooks do not attack across the cut-out cells? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form. | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=1}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers \( z \) and \( \omega \) satisfying the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( |z|, 2, |\omega| \) form an arithmetic sequence.
Is there a maximum value for \( \cos(\arg z - \arg \omega) \)? If so, find it. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \in \mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.
1. Find the value of $a$.
2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is given that $3a\cos A = \sqrt{6}(b\cos C + c\cos B)$.
(1) Calculate the value of $\tan 2A$.
(2) If $\sin\left(\frac{\pi}{2} + B\right) = \frac{1}{3}$ and $c = 2\sqrt{2}$, find the area of triangle $ABC$. | {
"answer": "\\frac{8}{5}\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For two lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel, determine the value of $a$ that satisfies this condition. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex pentagon \( P Q R S T \), the angle \( P R T \) is half of the angle \( Q R S \), and all sides are equal. Find the angle \( P R T \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY = 12$, $YZ = 16$, and $XZ = 20$, with $ZD$ as the angle bisector. Find the length of $ZD$. | {
"answer": "\\frac{16\\sqrt{10}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse C: $$\frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$$, F is the right focus, and l is a line passing through point F (not parallel to the y-axis), intersecting the ellipse at points A and B. l′ is the perpendicular bisector of AB, intersecting the major axis of the ellipse at point D. Then the value of $$\frac {DF}{AB}$$ is __________. | {
"answer": "\\frac {2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the coefficients of all terms in the expanded form of $(C_4^1x + C_4^2x^2 + C_4^3x^3 + C_4^4x^4)^2$ is 256. | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the following expression and find angle $\theta$ if the number can be expressed as $r e^{i \theta}$, where $0 \le \theta < 2\pi$:
\[ e^{11\pi i/60} + e^{21\pi i/60} + e^{31 \pi i/60} + e^{41\pi i /60} + e^{51 \pi i /60} \] | {
"answer": "\\frac{31\\pi}{60}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $1-2+3-4+\dots+100-101$. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
(1) $2x^2-3x-2=0$;
(2) $2x^2-3x-1=0$ (using the method of completing the square). | {
"answer": "\\frac{3-\\sqrt{17}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence $B=(b_1,b_2,\dots,b_{50})$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + \cdots + T_{50}}{50},\] where $T_k = b_1 + \cdots + b_k$ and $1 \leq k \leq 50$.
If the Cesaro sum of the 50-term sequence $(b_1,\dots,b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1,\dots,b_{50})$? | {
"answer": "492",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six equilateral triangles, each with side $4$, are arranged in a line such that the midpoint of the base of one triangle is the vertex of the next triangle. Calculate the area of the region of the plane that is covered by the union of the six triangular regions.
A) $16\sqrt{3}$
B) $19\sqrt{3}$
C) $24\sqrt{3}$
D) $18\sqrt{3}$
E) $20\sqrt{3}$ | {
"answer": "19\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer $n$ not exceeding $100$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $q$, and if $n > 60$, then the probability of choosing $n$ is $2q$. Find the probability that a perfect square is chosen.
- **A)** $\frac{1}{35}$
- **B)** $\frac{2}{35}$
- **C)** $\frac{3}{35}$
- **D)** $\frac{4}{35}$
- **E)** $\frac{6}{35}$ | {
"answer": "\\frac{3}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, triangles $ABC$ and $CBD$ are isosceles with $\angle ABC = \angle BAC$ and $\angle CBD = \angle CDB$. The perimeter of $\triangle CBD$ is $18,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $BD$ is $8.$ If $\angle ABC = \angle CBD$, find the length of $AB.$ | {
"answer": "14",
"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 left and right foci $F_1$ and $F_2$, respectively. Point A is the upper vertex of the ellipse, $|F_{1}A|= \sqrt {2}$, and the area of △$F_{1}AF_{2}$ is 1.
(1) Find the standard equation of the ellipse.
(2) Let M and N be two moving points on the ellipse such that $|AM|^2+|AN|^2=|MN|^2$. Find the equation of line MN when the area of △AMN reaches its maximum value. | {
"answer": "y=- \\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$ , and returns the sum of the positive integer divisors of $n$ . For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$ . What is the largest number n such that $\sigma_1(n) = 1854$ ?
| {
"answer": "1234",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\frac{\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\frac{\sqrt{3}}{2}$. Further, the line $y=kx+m$ ($k \neq 0$, $m \neq 0$) intersects the ellipse at two distinct points C and D, and points C and D both lie on the same circle centered at A.
(1) Find the equation of the ellipse;
(2) When $k = \frac{\sqrt{6}}{3}$, find the value of $m$ and the area of triangle $\triangle ACD$. | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the time forward by 7 hours. How many times did the cuckoo make a sound during this time? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $PQR$ where $\angle R = 90^\circ$, the lengths of sides $PQ = 15$ and $PR = 9$. Find $\sin Q$ and $\cos Q$. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $({{a}\_{n+1}}-{{a}\_{n}}={{(-1)}^{n+1}}\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\{(-1)^{n}a_{n}\}$. | {
"answer": "\\frac{20}{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many diagonals within a regular nine-sided polygon span an odd number of vertices between their endpoints? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola \( C_1: 2x^2 - y^2 = 1 \) and the ellipse \( C_2: 4x^2 + y^2 = 1 \). If \( M \) and \( N \) are moving points on the hyperbola \( C_1 \) and ellipse \( C_2 \) respectively, such that \( OM \perp ON \) and \( O \) is the origin, find the distance from the origin \( O \) to the line \( MN \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$? | {
"answer": "409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fixed point A (3, 4), and point P is a moving point on the parabola $y^2=4x$, the distance from point P to the line $x=-1$ is denoted as $d$. Find the minimum value of $|PA|+d$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the product of the numbers at the endpoints is allocated to each segment. Then, the sum of the numbers on the red segments is subtracted by the sum of the numbers on the blue segments. What is the maximum possible result? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the number $99,\!999,\!999,\!999$ squared. Following a pattern observed in previous problems, determine how many zeros are in the decimal expansion of this number squared. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $\angle Q=90^\circ$, $PQ=15$ and $QR=20$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=12$, then what is the length of $PS$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jason rolls four fair standard six-sided dice. He looks at the rolls and decides to either reroll all four dice or keep two and reroll the other two. After rerolling, he wins if and only if the sum of the numbers face up on the four dice is exactly $9.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
**A)** $\frac{7}{36}$
**B)** $\frac{1}{18}$
**C)** $\frac{2}{9}$
**D)** $\frac{1}{12}$
**E)** $\frac{1}{4}$ | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( A = \{0, 1, 2, 3, 4, 5, 6, 7\} \), how many mappings \( f \) from \( A \) to \( A \) satisfy the following conditions?
1. For all \( i, j \in A \) with \( i \neq j \), \( f(i) \neq f(j) \).
2. For all \( i, j \in A \) with \( i + j = 7 \), \( f(i) + f(j) = 7 \). | {
"answer": "384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | {
"answer": "172822",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of \( \cos (\angle OBC + \angle OCB) \) in triangle \( \triangle ABC \), where angle \( \angle A \) is an obtuse angle, \( O \) is the orthocenter, and \( AO = BC \). | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:
The value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$? | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function in terms of \\(x\\), \\(f(x)=ax^{2}-4bx+1\\).
\\((1)\\) Let set \\(P=\\{1,2,3\\}\\) and \\(Q=\\{-1,1,2,3,4\\}\\), randomly pick a number from set \\(P\\) as \\(a\\) and from set \\(Q\\) as \\(b\\), calculate the probability that the function \\(y=f(x)\\) is increasing in the interval \\([1,+∞)\\).
\\((2)\\) Suppose point \\((a,b)\\) is a random point within the region defined by \\( \\begin{cases} x+y-8\\leqslant 0 \\\\ x > 0 \\\\ y > 0\\end{cases}\\), denote \\(A=\\{y=f(x)\\) has two zeros, one greater than \\(1\\) and the other less than \\(1\\}\\), calculate the probability of event \\(A\\) occurring. | {
"answer": "\\dfrac{961}{1280}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = |\lg(x+1)|$, where $a$ and $b$ are real numbers, and $a < b$ satisfies $f(a) = f(- \frac{b+1}{b+2})$ and $f(10a + 6b + 21) = 4\lg2$. Find the value of $a + b$. | {
"answer": "- \\frac{11}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_{n}\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \lt k$ holds for all $n$. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in $\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\triangle ABC$ is 1, find the area of $\triangle BMD$. | {
"answer": "\\frac{4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $E$ lies on $\overline{YZ}$ and $G$ lies on $\overline{XY}$. Let $\overline{XE}$ and $\overline{YG}$ intersect at $Q.$
If $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\frac{XG}{GY}.$ | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that A and B can only take on the first three roles, and the other three volunteers (C, D, and E) can take on all four roles, calculate the total number of different selection schemes for four people from five volunteers. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C$: $y^{2}=2px$ with the focus at $F(2,0)$, and points $P(m,0)$ and $Q(-m,n)$, a line $l$ passing through $P$ with a slope of $k$ (where $k\neq 0$) intersects the parabola $C$ at points $A$ and $B$.
(Ⅰ) For $m=k=2$, if $\vec{QA} \cdot \vec{QB} = 0$, find the value of $n$.
(Ⅱ) If $O$ represents the origin and $m$ is constant, for any change in $k$ such that $\vec{OA} \cdot \vec{OB} = 0$ always holds, find the value of the constant $m$.
(Ⅲ) For $k=1$, $n=0$, and $m < 0$, find the maximum area of triangle $QAB$ as $m$ changes. | {
"answer": "\\frac{32\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the average of a set of sample data 4, 5, 7, 9, $a$ is 6, then the variance $s^2$ of this set of data is \_\_\_\_\_\_. | {
"answer": "\\frac{16}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
a) In how many ways can a rectangle $8 \times 2$ be divided into $1 \times 2$ rectangles?
b) Imagine and describe a shape that can be divided into $1 \times 2$ rectangles in exactly 555 ways. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any four-digit number $m$, if the digits of $m$ are all non-zero and distinct, and the sum of the units digit and the thousands digit is equal to the sum of the tens digit and the hundreds digit, then this number is called a "mirror number". If we swap the units digit and the thousands digit of a "mirror number" to get a new four-digit number $m_{1}$, and swap the tens digit and the hundreds digit to get another new four-digit number $m_{2}$, let $F_{(m)}=\frac{{m_{1}+m_{2}}}{{1111}}$. For example, if $m=1234$, swapping the units digit and the thousands digit gives $m_{1}=4231$, and swapping the tens digit and the hundreds digit gives $m_{2}=1324$, the sum of these two four-digit numbers is $m_{1}+m_{2}=4231+1324=5555$, so $F_{(1234)}=\frac{{m_{1}+m_{2}}}{{1111}}=\frac{{5555}}{{1111}}=5$. If $s$ and $t$ are both "mirror numbers", where $s=1000x+100y+32$ and $t=1500+10e+f$ ($1\leqslant x\leqslant 9$, $1\leqslant y\leqslant 9$, $1\leqslant e\leqslant 9$, $1\leqslant f\leqslant 9$, $x$, $y$, $e$, $f$ are all positive integers), define: $k=\frac{{F_{(s)}}}{{F_{(t)}}}$. When $F_{(s)}+F_{(t)}=19$, the maximum value of $k$ is ______. | {
"answer": "\\frac{{11}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$. | {
"answer": "125 - 5\\sqrt{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overset{→}{a\_n}=\left(\cos \frac{nπ}{6},\sin \frac{nπ}{6}\right)$, $n∈ℕ^∗$, $\overset{→}{b}=\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, calculate the value of $y={\left| \overset{→}{{a\_1}}+ \overset{→}{b}\right|}^{2}+{\left| \overset{→}{{a\_2}}+ \overset{→}{b}\right|}^{2}+···+{\left| \overset{→}{{a\_2015}}+ \overset{→}{b}\right|}^{2}$. | {
"answer": "4029",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $x$: $0.04x + 0.05(25 + x) = 13.5$. | {
"answer": "136.\\overline{1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of $\frac{1}{2}$ to win any given game, then calculate the probability of player A becoming the final winner. | {
"answer": "\\frac{11}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-2, -3)$ and $B(5, 3)$ on the $xy$-plane, find the point $C(x, n)$ such that $AC + CB$ is minimized, where $x = 2$. Find the value of $n$.
A) $\frac{6}{7}$
B) $\frac{12}{7}$
C) $6.5$
D) $\frac{25}{6}$
E) $\frac{13}{2}$ | {
"answer": "\\frac{13}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle passes through the midpoints of the hypotenuse $AB$ and the leg $BC$ of the right triangle $ABC$ and touches the leg $AC$. In what ratio does the point of tangency divide the leg $AC$?
| {
"answer": "1 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the number $\sqrt{7200} - 61$ can be expressed in the form $(\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$ | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$ and $|2\overrightarrow{a}+\overrightarrow{b}|=2\sqrt{3}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could end up being rational numbers? | {
"answer": "1250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of a bottle of "Komfort" fabric softener used to be 13.70 Ft, and half a capful was needed for 15 liters of water. The new composition of "Komfort" now costs 49 Ft, and 1 capful is needed for 8 liters of water. By what percentage has the price of the fabric softener increased? | {
"answer": "1240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let the sum of the first $n$ terms of $\{c_{n}\}$ be denoted as $S_{n}$. If $c_{k}=101$, find the value of $k$ and determine $S_{k}$. | {
"answer": "2726",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ? | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and the circle $x^2 + y^2 = a^2 + b^2$ in the first quadrant, find the eccentricity of the hyperbola, where $|PF_1| = 3|PF_2|$. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, three lines meet at the points \( A, B \), and \( C \). If \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), the value of \( x \) is: | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ that satisfies $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $n \in \mathbb{N}^*$.
$(1)$ Find $a_n$.
$(2)$ If the sequence $\{b_n\}$ satisfies $b_n=\frac{1}{(1+\log_3{a_n})(3+\log_3{a_n})}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, and for any positive integer $n$, $T_n < m$, find the minimum value of $m$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The isosceles trapezoid has base lengths of 24 units (bottom) and 12 units (top), and the non-parallel sides are each 12 units long. How long is the diagonal of the trapezoid? | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2005}+1\right)^{11/8}$? | {
"answer": "375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. | {
"answer": "133.\\overline{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \] | {
"answer": "510",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a} = (\cos x, -\sqrt{3}\cos x)$ and $\overrightarrow{b} = (\cos x, \sin x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$.
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;
(Ⅱ) If $f(\theta) = \frac{5}{6}$, where $\theta \in \left( \frac{\pi}{3}, \frac{2\pi}{3} \right)$, find the value of $\sin 2\theta$. | {
"answer": "\\frac{2\\sqrt{3} - \\sqrt{5}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mary typed a six-digit number, but the two $1$ s she typed didn't show. What appeared was $2002$ . How many different six-digit numbers could she have typed? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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