problem stringlengths 10 5.15k | answer dict |
|---|---|
The sum of 100 numbers is 1000. The largest of these numbers was doubled, while another number was decreased by 10. After these actions, the sum of all numbers remained unchanged. Find the smallest of the original numbers. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars. | {
"answer": "-100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ and its derivative $f'(x)$ have a domain of $R$, and $f(x+2)$ is an odd function, ${f'}(2-x)+{f'}(x)=2$, ${f'}(2)=2$, then $\sum_{i=1}^{50}{{f'}}(i)=$____. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(1+2x)^2(1-x)^5 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, then $a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 =$ ? | {
"answer": "-31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be the number of students in Danny's high school. If Maria's high school has $4$ times as many students as Danny's high school, then Maria's high school has $4x$ students. The difference between the number of students in the two high schools is $1800$, so we have the equation $4x-x=1800$. | {
"answer": "2400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).
Find the number of such sequences \( A \). | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest real number $k$ , such that for any positive real numbers $a,b$ , $$ (a+b)(ab+1)(b+1)\geq kab^2 $$ | {
"answer": "27/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve congruent rectangles are placed together to make a rectangle $PQRS$. What is the ratio $PQ: QR$?
A) $2: 3$
B) $3: 4$
C) $5: 6$
D) $7: 8$
E) $8: 9$ | {
"answer": "8 : 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x) = x^2 + a|x - 1|$ is monotonically increasing on the interval $[-1, +\infty)$, then the set of values for the real number $a$ is ______. | {
"answer": "\\{-2\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 6\}$. What is the probability that at least one number divides another among the selected numbers? Express your answer as a common fraction. | {
"answer": "\\frac{9}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The clock shows $00:00$, with both the hour and minute hands coinciding. Considering this coincidence as number 0, determine after what time interval (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the nearest hundredth. | {
"answer": "1374.55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average of the numbers 47 and $x$ is 53. Besides finding the positive difference between 47 and $x$, also determine their sum. | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).
Find the smallest value of \( n \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an acute angle between them, and satisfying $|\overrightarrow{a}|= \frac{8}{\sqrt{15}}$, $|\overrightarrow{b}|= \frac{4}{\sqrt{15}}$. If for any $(x,y)\in\{(x,y)| |x \overrightarrow{a}+y \overrightarrow{b}|=1, xy > 0\}$, it holds that $|x+y|\leqslant 1$, then the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$ is \_\_\_\_\_\_. | {
"answer": "\\frac{8}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Observe the following equations:
1=1
1-4=-（1+2）=-3
1-4+9=1+2+3=6
1-4+9-16=-（1+2+3+4）=-10
Then, the 5th equation is
The value of the 20th equation is
These equations reflect a certain pattern among integers. Let $n$ represent a positive integer, try to express the pattern you discovered using an equation related to $n$. | {
"answer": "-210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $40\%$ of the birds were geese, $20\%$ were swans, $15\%$ were herons, and $25\%$ were ducks, calculate the percentage of the birds that were not ducks and were geese. | {
"answer": "53.33\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graphs \( y = 2 \cos 3x + 1 \) and \( y = - \cos 2x \) intersect at many points. Two of these points, \( P \) and \( Q \), have \( x \)-coordinates between \(\frac{17 \pi}{4}\) and \(\frac{21 \pi}{4}\). The line through \( P \) and \( Q \) intersects the \( x \)-axis at \( B \) and the \( y \)-axis at \( A \). If \( O \) is the origin, what is the area of \( \triangle BOA \)? | {
"answer": "\\frac{361\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $e^{i \theta} = \frac{1 + i \sqrt{2}}{2}$, then find $\sin 3 \theta.$ | {
"answer": "\\frac{\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \in [\frac{1}{2}, \frac{\sqrt{3}}{2}]$, find the maximum value of $a$. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(b_1,b_2,b_3,\ldots,b_{10})$ be a permutation of $(1,2,3,\ldots,10)$ for which
$b_1>b_2>b_3>b_4 \mathrm{\ and \ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$
Find the number of such permutations. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \( g\left(\frac{1}{1996}\right) + g\left(\frac{2}{1996}\right) + g\left(\frac{3}{1996}\right) + \cdots + g\left(\frac{1995}{1996}\right) \) where \( g(x) = \frac{9x}{3 + 9x} \). | {
"answer": "997",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x^{2}+x-2$, determine the characteristics of the function $f(x)$ in the interval $[-1,1)$. | {
"answer": "-\\frac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set $\mathbf{A}=\{1, 2, 3, 4, 5, 6\}$ and a bijection $f: \mathbf{A} \rightarrow \mathbf{A}$ satisfy the condition: for any $x \in \mathbf{A}$, $f(f(f(x)))=x$. Then the number of bijections $f$ satisfying the above condition is: | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grace is trying to solve the following equation by completing the square: $$36x^2-60x+25 = 0.$$ She successfully rewrites the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the following expression:
$$\frac { \sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$ | {
"answer": "-4 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle, side AB is 8 units, and side BC is 15 units. Calculate $\cos C$ and $\sin C$. | {
"answer": "\\frac{8}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 15 are there between 40 and 240? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the complex number $z$ satisfies the equation $\frac{1-z}{1+z}={i}^{2018}+{i}^{2019}$ (where $i$ is the imaginary unit), find the value of $|2+z|$. | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Definition: If the line $l$ is tangent to the graphs of the functions $y=f(x)$ and $y=g(x)$, then the line $l$ is called the common tangent line of the functions $y=f(x)$ and $y=g(x)$. If the functions $f(x)=a\ln x (a>0)$ and $g(x)=x^{2}$ have exactly one common tangent line, then the value of the real number $a$ is ______. | {
"answer": "2e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they arrived at the destination with the odometer showing another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum possible average speed (in kilometers per hour) that Chester's father could have driven? | {
"answer": "82.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle $ABC$ it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\angle BMC$ .
(Anton Trygub) | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9}{1+2+3+4+5+6+7+8+9+10}$, evaluate the expression. | {
"answer": "6608",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the angle at vertex \(A\) is \(60^{\circ}\). A circle is drawn through points \(B\), \(C\), and point \(D\), which lies on side \(AB\), intersecting side \(AC\) at point \(E\). Find \(AE\) if \(AD = 3\), \(BD = 1\), and \(EC = 4\). Find the radius of the circle. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a non-right triangle $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c=1$, also $$C= \frac {\pi}{3}$$, if $\sin C + \sin(A-B) = 3\sin 2B$, then the area of $\triangle ABC$ is \_\_\_\_\_\_. | {
"answer": "\\frac {3 \\sqrt {3}}{28}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$ Find $p(8).$ | {
"answer": "40328",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( f(x) = x^3 + a x^2 + b x + c \), where \( a \), \( b \), and \( c \) are non-zero integers. If \( f(a) = a^3 \) and \( f(b) = b^3 \), then the value of \( c \) is: | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$ | {
"answer": "\\frac{6\\sqrt{2}}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of four prime numbers $P,$ $Q,$ $P-Q,$ and $P+Q$ must be expressed in terms of a single letter indicating the property of the sum. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assuming that the new demand is given by \( \frac{1}{1+ep} \), where \( p = 0.20 \) and \( e = 1.5 \), calculate the proportionate decrease in demand. | {
"answer": "0.23077",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a, b, c\) are positive real numbers. | {
"answer": "3 \\cdot \\frac{1}{\\sqrt[3]{162}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jun Jun is looking at an incorrect single-digit multiplication equation \( A \times B = \overline{CD} \), where the digits represented by \( A \), \( B \), \( C \), and \( D \) are all different from each other. Clever Jun Jun finds that if only one digit is changed, there are 3 ways to correct it, and if only the order of \( A \), \( B \), \( C \), and \( D \) is changed, the equation can also be corrected. What is \( A + B + C + D = \) ? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$ | {
"answer": "1005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to: | {
"answer": "2.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$. | {
"answer": "100",
"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 25 cents worth of coins come up heads? | {
"answer": "\\dfrac{13}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a square where each side measures 4 units. $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{CD},$ respectively. Find $\sin \phi$ where $\phi$ is the angle $\angle APQ$.
 | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$ , such that this is always possible if: $a)$ The students can be arranged in a circle such that every two students sitting next to one another have different numbers. $b)$ $7$ of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number. | {
"answer": "k = 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | {
"answer": "14.0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $n > 1$ , let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$ . Find the maximum value of $\frac{a_n}{n}$ . | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The absolute value of -1.2 is ____, and its reciprocal is ____. | {
"answer": "-\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One dimension of a cube is increased by $2$, another is decreased by $2$, and the third is increased by $3$. The volume of the new rectangular solid is $7$ less than the volume of the cube. Find the original volume of the cube. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, calculate the value of $m$. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$ . Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$ . If $BC'=30$ , find the sum of all possible side lengths of $\triangle ABC$ .
*Proposed by Connor Gordon* | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) and \(\angle BLN = 45^{\circ}\). In what ratio does point \( N \) divide the side \( AB \)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | {
"answer": "\\frac{7}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows the miles traveled by cyclists Clara and David. After five hours, how many more miles has Clara cycled than David?
[asy]
/* Modified AMC8 1999 #4 Problem */
draw((0,0)--(6,0)--(6,4.5)--(0,4.5)--cycle);
for(int x=0; x <= 6; ++x) {
for(real y=0; y <=4.5; y+=0.9) {
dot((x, y));
}
}
draw((0,0)--(5,3.6)); // Clara's line
draw((0,0)--(5,2.7)); // David's line
label(rotate(36)*"David", (3,1.35));
label(rotate(36)*"Clara", (3,2.4));
label(scale(0.75)*rotate(90)*"MILES", (-1, 2.25));
label(scale(0.75)*"HOURS", (3, -1));
label(scale(0.85)*"90", (0, 4.5), W);
label(scale(0.85)*"72", (0, 3.6), W);
label(scale(0.85)*"54", (0, 2.7), W);
label(scale(0.85)*"36", (0, 1.8), W);
label(scale(0.85)*"18", (0, 0.9), W);
label(scale(0.86)*"1", (1, 0), S);
label(scale(0.86)*"2", (2, 0), S);
label(scale(0.86)*"3", (3, 0), S);
label(scale(0.86)*"4", (4, 0), S);
label(scale(0.86)*"5", (5, 0), S);
label(scale(0.86)*"6", (6, 0), S);
[/asy] | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If an integer $a$ ($a \neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is. | {
"answer": "16",
"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 its upper vertex at $(0,2)$ and an eccentricity of $\frac{\sqrt{5}}{3}$.
(1) Find the equation of ellipse $C$;
(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle $x^{2}+y^{2}=1$, with the tangent points being $A$ and $B$. When the line $AB$ intersects the $x$-axis and $y$-axis at points $N$ and $M$, respectively, find the minimum value of $|MN|$. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos ωx \cdot \sin (ωx- \frac {π}{3})+ \sqrt {3}\cos ^{2}ωx- \frac{ \sqrt {3}}{4}(ω > 0,x∈R)$, and the distance from one symmetry center of the graph of the function $y=f(x)$ to the nearest symmetry axis is $\frac {π}{4}$.
(I) Find the value of $ω$ and the equation of the symmetry axis of $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $f(A)= \frac { \sqrt {3}}{4}, \sin C= \frac {1}{3}, a= \sqrt {3}$, find the value of $b$. | {
"answer": "\\frac {3+2 \\sqrt {6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the broadcast time of the "Midday News" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set. | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x)=\tan (\omega x+ \frac {\pi}{4})$ ($\omega > 0$) has a minimum positive period of $2\pi$, then $\omega=$ ______ ; $f( \frac {\pi}{6})=$ ______ . | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the function $y=\sin^2x+3\sin x\cos x+4\cos^2x$ for $0 \leqslant x \leqslant \frac{\pi}{2}$ and the corresponding value of $x$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____. | {
"answer": "5.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$. | {
"answer": "-\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, point $P\left( \frac{1}{2},\cos^2\theta\right)$ is on the terminal side of angle $\alpha$, and point $Q(\sin^2\theta,-1)$ is on the terminal side of angle $\beta$, and $\overrightarrow{OP}\cdot \overrightarrow{OQ}=-\frac{1}{2}$.
$(1)$ Find $\cos 2\theta$;
$(2)$ Find the value of $\sin(\alpha+\beta)$. | {
"answer": "-\\frac{\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(f(x)\\) be defined on \\((-∞,+∞)\\) and satisfy \\(f(2-x)=f(2+x)\\) and \\(f(7-x)=f(7+x)\\). If in the closed interval \\([0,7]\\), only \\(f(1)=f(3)=0\\), then the number of roots of the equation \\(f(x)=0\\) in the closed interval \\([-2005,2005]\\) is . | {
"answer": "802",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin^2(wx) - \sin^2(wx - \frac{\pi}{6})$ ($x \in \mathbb{R}$, $w$ is a constant and $\frac{1}{2} < w < 1$), the graph of function $f(x)$ is symmetric about the line $x = \pi$.
(I) Find the smallest positive period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a = 1$ and $f(\frac{3}{5}A) = \frac{1}{4}$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.
(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.
(2) Given that among the 3 selected questions, there are 2 Type A questions and 1 Type B question, and the probability of Xiao Ming correctly answering each Type A question is $\dfrac{3}{5}$, and the probability of correctly answering each Type B question is $\dfrac{4}{5}$. Assuming the correctness of answers to different questions is independent, calculate the probability that Xiao Ming correctly answers at least 2 questions. | {
"answer": "\\dfrac{93}{125}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$, when the perimeter of $\triangle APF$ is minimized, the ordinate of point $P$ is ______. | {
"answer": "2 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-10^2)(x-11^3)(x-12^3)\dotsm(x-13^3)}
\]
have for \(x\)? | {
"answer": "190",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$ . | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts? | {
"answer": "27.325\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC below, find the length of side AB.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18\sqrt{2}$",C/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy] | {
"answer": "18\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert the complex number \(1 + i \sqrt{3}\) into its exponential form \(re^{i \theta}\) and find \(\theta\). | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum possible value of \(A\). | {
"answer": "4648",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle. | {
"answer": "2 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x-\frac{1}{x}$ and $g(x)=2a\ln x$.
(1) When $a\geqslant -1$, find the monotonically increasing interval of $F(x)=f(x)-g(x)$;
(2) Let $h(x)=f(x)+g(x)$, and $h(x)$ has two extreme values $({{x}_{1}},{{x}_{2}})$, where ${{x}_{1}}\in (0,\frac{1}{3}]$, find the minimum value of $h({{x}_{1}})-h({{x}_{2}})$. | {
"answer": "\\frac{20\\ln 3-16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all the integers between -25.4 and 15.8, excluding the integer zero? | {
"answer": "-200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rhombus $ABCD$ has side length $3$ and $\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
**A)** $0.81$
**B)** $1.62$
**C)** $2.43$
**D)** $2.16$
**E)** $3.24$ | {
"answer": "2.16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average yield per unit area of a rice variety for five consecutive years was 9.4, 9.7, 9.8, 10.3, and 10.8 (unit: t/hm²). Calculate the variance of this sample data. | {
"answer": "0.244",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In order to cultivate students' financial management skills, Class 1 of the second grade founded a "mini bank". Wang Hua planned to withdraw all the money from a deposit slip. In a hurry, the "bank teller" mistakenly swapped the integer part (the amount in yuan) with the decimal part (the amount in cents) when paying Wang Hua. Without counting, Wang Hua went home. On his way home, he spent 3.50 yuan on shopping and was surprised to find that the remaining amount of money was twice the amount he was supposed to withdraw. He immediately contacted the teller. How much money was Wang Hua supposed to withdraw? | {
"answer": "14.32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle. | {
"answer": "16\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider equilateral triangle $ABC$ with side length $1$ . Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$ . Compute the sum of all possible values of $\kappa$ .
*2018 CCA Math Bonanza Lightning Round #3.4* | {
"answer": "\\frac{18\\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. How many minutes earlier did the Type I car start running compared to the Type II car? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, circle $C$ is centered at point $C\left(2, -\frac{\pi}{6}\right)$ with a radius of $2$.
$(1)$ Find the polar equation of circle $C$;
$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\theta = -\frac{5\pi}{12} (\rho \in \mathbb{R})$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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