problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the sets \( M = \{1, 2, 3\} \) and \( N = \{1, 2, 3, 4, 5\} \), define the function \( f: M \rightarrow N \). Let the points \( A(1, f(1)), B(2, f(2)), C(3, f(3)) \) form a triangle \( \triangle ABC \). The circumcenter of \( \triangle ABC \) is \( D \), and it is given that \( \mu DA + DC = \lambda DB (\lambda \in \mathbb{R}) \). Find the number of functions \( f(x) \) that satisfy this condition. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n,$ $1 \le n \le 2000,$ for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$ . The last side of his triangle has side length $\log n$ , where $n$ is an integer. How many possible values are there for $n$ ?
*2018 CCA Math Bonanza Individual Round #6* | {
"answer": "203797",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many even integers are there between $\frac{9}{2}$ and $\frac{24}{1}$? | {
"answer": "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 defined by \( x^2 + 4y^2 = 5 \). The centers of the circles are located along the x-axis. Find the value of \( r \). | {
"answer": "\\frac{\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{P}$ be a regular $2022$ -gon with area $1$ . Find a real number $c$ such that, if points $A$ and $B$ are chosen independently and uniformly at random on the perimeter of $\mathcal{P}$ , then the probability that $AB \geq c$ is $\tfrac{1}{2}$ .
*Espen Slettnes* | {
"answer": "\\sqrt{2/\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jia draws five lines in a plane without any three lines intersecting at one point. For each pair of intersecting lines, Jia gets a candy. If there is a set of parallel lines, Jia also gets a candy. For example, in a diagram with seven intersection points and one set of parallel lines, Jia receives 8 candies. How many candies can Jia receive in different scenarios? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a country, there are 110 cities. Between each pair of cities, there is either a road or no road.
A driver starts in a city with exactly one road leading out of it. After traveling along this road, he arrives at a second city, which has exactly two roads leading out of it. After traveling along one of these roads, he arrives at a third city, which has exactly three roads leading out of it, and so on. At some point, after traveling along one of the roads, he arrives in the $N$-th city, which has exactly $N$ roads leading out of it. At this stage, the driver stops his journey. (For each $2 \leqslant k \leqslant N$, the $k$-th city has exactly $k$ roads, including the one by which the driver arrived in this city.)
What is the maximum possible value of $N$? | {
"answer": "107",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boss and two engineers are to meet at a park. Each of them arrives at a random time between 1:00 PM and 3:00 PM. The boss leaves instantly if not both engineers are present upon his arrival. Each engineer will wait for up to 1.5 hours for the other before leaving. What is the probability that the meeting takes place? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
King Qi and Tian Ji are competing in a horse race. Tian Ji's top horse is better than King Qi's middle horse, worse than King Qi's top horse; Tian Ji's middle horse is better than King Qi's bottom horse, worse than King Qi's middle horse; Tian Ji's bottom horse is worse than King Qi's bottom horse. Now, each side sends one top, one middle, and one bottom horse, forming 3 groups for separate races. The side that wins 2 or more races wins. If both sides do not know the order of the opponent's horses, the probability of Tian Ji winning is ____; if it is known that Tian Ji's top horse and King Qi's middle horse are in the same group, the probability of Tian Ji winning is ____. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the following expression:
\[
\frac{(1 + 15) \left( 1 + \dfrac{15}{2} \right) \left( 1 + \dfrac{15}{3} \right) \dotsm \left( 1 + \dfrac{15}{21} \right)}{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{19} \right)}.
\] | {
"answer": "272",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers. | {
"answer": "1111111110000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(a_1, a_2, a_3, \ldots, a_{15})$ be a permutation of $(1, 2, 3, \ldots, 15)$ for which
$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$
Find the number of such permutations. | {
"answer": "3003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonals of a convex quadrilateral $ABCD$, inscribed in a circle, intersect at point $E$. It is known that diagonal $BD$ is the angle bisector of $\angle ABC$ and that $BD = 25$ and $CD = 15$. Find $BE$. | {
"answer": "15.625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point P $(x, y)$ moves on the circle $x^2 + (y-1)^2 = 1$, the maximum value of $\frac{y-1}{x-2}$ is \_\_\_\_\_\_, and the minimum value is \_\_\_\_\_\_. | {
"answer": "-\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | {
"answer": "2520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the $xy$ -coordinate plane, the $x$ -axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$ -axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$ . They are $(126, 0)$ , $(105, 0)$ , and a third point $(d, 0)$ . What is $d$ ? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.) | {
"answer": "111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We say that a number of 20 digits is *special* if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials. | {
"answer": "10^9 - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$=($\sqrt {3}$sinx, $\sqrt {3}$cos(x+$\frac {\pi}{2}$)+1) and $\overrightarrow {b}$=(cosx, $\sqrt {3}$cos(x+$\frac {\pi}{2}$)-1), define f(x) = $\overrightarrow {a}$$\cdot \overrightarrow {b}$.
(1) Find the minimum positive period and the monotonically increasing interval of f(x);
(2) In △ABC, a, b, and c are the sides opposite to A, B, and C respectively, with a=$2\sqrt {2}$, b=$\sqrt {2}$, and f(C)=2. Find c. | {
"answer": "\\sqrt {10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group consists of 5 girls and 5 boys. The members of the same gender do not know each other, and there are no two girls who have two mutual boy acquaintances. What is the maximum number of acquaintanceships within the group? How does the answer change if the group consists of 7 girls and 7 boys? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac{m}{x} - m + \ln x$ (where $m$ is a constant).
1. Find the monotonic intervals of $f(x)$.
2. For what values of $m$ does $f(x) \geq 0$ hold true? | {
"answer": "m = 1",
"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 $x^2 + 4y^2 = 5$. Find $r$. | {
"answer": "\\frac{\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the island of Zenith has 32500 acres of usable land, each individual requires 2 acres for sustainable living, and the current population of 500 people increases by a factor of 4 every 30 years, calculate the number of years from 2022 when the population reaches its maximum capacity. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $ABC$ with $\angle B = 90^\circ$, we have $AB = 8$ and $AC = 6$. Find $\cos C$. | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor greater than one? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid \(ABCD\), if \(AB = 8\), \(DC = 10\), the area of \(\triangle AMD\) is 10, and the area of \(\triangle BCM\) is 15, then the area of trapezoid \(ABCD\) is \(\quad\). | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron \(ABCD\) has six edges with lengths \(7, 13, 18, 27, 36, 41\) units. If the length of \(AB\) is 41 units, then the length of \(CD\) is | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
People have long been exploring the numerical solution of high-degree equations. Newton gave a numerical solution method for high-degree algebraic equations in his book "Fluxions." This method for finding the roots of equations has been widely used in the scientific community. For example, to find an approximate solution to the equation $x^{3}+2x^{2}+3x+3=0$, first using the existence theorem of function zeros, let $f\left(x\right)=x^{3}+2x^{2}+3x+3$, $f\left(-2\right)=-3 \lt 0$, $f\left(-1\right)=1 \gt 0$, it is known that there exists a zero on $\left(-2,-1\right)$, take $x_{0}=-1$. Newton used the formula ${x}_{n}={x}_{n-1}-\frac{f({x}_{n-1})}{{f}′({x}_{n-1})}$ for iterative calculation, where $x_{n}$ is taken as an approximate solution to $f\left(x\right)=0$. After two iterations, the approximate solution obtained is ______; starting with the interval $\left(-2,-1\right)$, using the bisection method twice, taking the midpoint value of the last interval as the approximate solution to the equation, the approximate solution is ______. | {
"answer": "-\\frac{11}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are five positive integers that are divisors of each number in the list $$48, 144, 24, 192, 216, 120.$$ Find the sum of these positive integers. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | {
"answer": "\\frac{3}{4\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle PQR$, points $X$ and $Y$ lie on $\overline{QR}$ and $\overline{PR}$, respectively. If $\overline{PX}$ and $\overline{QY}$ intersect at $Z$ such that $PZ/ZX = 2$ and $QZ/ZY = 5$, what is $RX/RY$? | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\int_{0}^{1} \frac{\sin x}{x} \, dx$ with an accuracy of 0.01. | {
"answer": "0.94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and every head either always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "On my left is a green dragon."
- 2nd head: "On my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been seated at the table? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many orbitals contain one or more electrons in an isolated ground state iron atom (Z = 26)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 2-dimensional Cartesian coordinate system, there are 16 lattice points \((i, j)\) where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, determine the minimum value of \(n\) such that there always exist four points which are the vertices of a square. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 22. What is the mean of the set of all fifteen numbers? | {
"answer": "18.73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, circles \( C_{1} \) and \( C_{2} \) have centers \( O_{1} \) and \( O_{2} \), respectively. The radii of the circles are \( r_{1} \) and \( r_{2} \), with \( r_{1} = 3r_{2} \). Circle \( C_{2} \) is internally tangent to \( C_{1} \) at point \( P \). Chord \( XY \) of \( C_{1} \) has length 20, is tangent to \( C_{2} \) at point \( Q \), and is parallel to the line segment \( O_{2}O_{1} \). Determine the area of the shaded region, which is the region inside \( C_{1} \) but not \( C_{2} \). | {
"answer": "160\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ali Baba and the thief are dividing a hoard consisting of 100 gold coins, distributed in 10 piles of 10 coins each. Ali Baba chooses 4 piles, places a cup near each of them, and puts a certain number of coins (at least one, but not the entire pile) into each cup. The thief must then rearrange the cups, changing their initial positions, after which the coins are poured from the cups back into the piles they are now next to. Ali Baba then chooses 4 piles from the 10 again, places cups near them, and so on. At any moment, Ali Baba can leave, taking with him any three piles of his choice. The remaining coins go to the thief. What is the maximum number of coins Ali Baba can take with him if the thief also tries to get as many coins as possible? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( A \), \( B \), \( C \), \( D \), and \( E \) are located in 3-dimensional space with \( AB = BC = CD = DE = 3 \) and \( \angle ABC = \angle CDE = 60^\circ \). Additionally, the plane of triangle \( ABC \) is perpendicular to line \( \overline{DE} \). Determine the area of triangle \( BDE \). | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We define five-digit numbers like 31024 and 98567 as "Shenma numbers", where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different. How many such five-digit numbers are there? | {
"answer": "1512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\tan A = 2\tan B$, $b = \sqrt{2}$, and the area of $\triangle ABC$ is at its maximum value, find $a$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the four-digit numbers, the number of four-digit numbers that have exactly 2 digits repeated is. | {
"answer": "3888",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ and $\beta$ are acute angles, and $\cos(\alpha+\beta)=\frac{\sin\alpha}{\sin\beta}$, determine the maximum value of $\tan \alpha$. | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parameterized equation of a line is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$$ (where $t$ is the parameter), determine the angle of inclination of the line. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square flag features a green cross of uniform width with a yellow square in the center on a black background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 50% of the area of the flag, what percent of the area of the flag is yellow? | {
"answer": "6.25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points) | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\frac{m}{n}$ for relatively prime positive integers $m,n$ , compute $m+n$ .
*2020 CCA Math Bonanza Lightning Round #4.1* | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\angle BAF = 18^\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that the value of $\angle FAH$ in degrees is $x$ , find the nearest integer to $x$ .
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy] | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-3}$ is a power function, and when $x\in (0,+\infty)$, $f(x)$ is a decreasing function. Find the real number $m=$____. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Selene has 120 cards numbered from 1 to 120, inclusive, and she places them in a box. Selene then chooses a card from the box at random. What is the probability that the number on the card she chooses is a multiple of 2, 4, or 5? Express your answer as a common fraction. | {
"answer": "\\frac{11}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers are in the list $250, 243, 236, \ldots, 29, 22?$ | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\cos^2x+2\sqrt{3}\sin x\cos x+a$, and when $x\in\left[0, \frac{\pi}{2}\right]$, the minimum value of $f(x)$ is $2$,
$(1)$ Find the value of $a$, and determine the intervals where $f(x)$ is monotonically increasing;
$(2)$ First, transform the graph of the function $y=f(x)$ by keeping the y-coordinates unchanged and reducing the x-coordinates to half of their original values, then shift the resulting graph to the right by $\frac{\pi}{12}$ units to obtain the graph of the function $y=g(x)$. Find the sum of all roots of the equation $g(x)=4$ in the interval $\left[0,\frac{\pi}{2}\right]$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertices of the broken line $A B C D E F G$ have coordinates $A(-1, -7), B(2, 5), C(3, -8), D(-3, 4), E(5, -1), F(-4, -2), G(6, 4)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(\triangle ABC\\) have internal angles \\(A\\), \\(B\\), and \\(C\\) opposite to sides of lengths \\(a\\), \\(b\\), and \\(c\\) respectively, and it satisfies \\(a^{2}+c^{2}-b^{2}= \sqrt {3}ac\\).
\\((1)\\) Find the size of angle \\(B\\);
\\((2)\\) If \\(2b\cos A= \sqrt {3}(c\cos A+a\cos C)\\), and the median \\(AM\\) on side \\(BC\\) has a length of \\(\sqrt {7}\\), find the area of \\(\triangle ABC\\). | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Trapezoid $EFGH$ has sides $EF=86$, $FG=60$, $GH=26$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find $EQ$, expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and compute $p+q$. | {
"answer": "163",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
| {
"answer": "1972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point P moves on the parabola y^2 = 4x with focus F, and point Q moves on the line x-y+5=0. Find the minimum value of ||PF+|PQ||. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has one vertex tangent to the edge of the circle at point $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle. | {
"answer": "9(\\sqrt{3} - \\pi)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain city, license plate numbers consist of 6 digits (ranging from 0 to 9), but it is required that any two plates differ by at least 2 digits (for example, license plates 038471 and 030471 cannot both be used). Determine the maximum number of different license plates that can be issued and provide a proof. | {
"answer": "100000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $\angle A = 60^\circ$ and $\angle B = 45^\circ$. A line $DE$, with $D$ on $AB$ and $\angle ADE = 45^\circ$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{6}}{4}$
B) $\frac{\sqrt{7}}{4}$
C) $\frac{\sqrt{8}}{4}$
D) $\frac{\sqrt{6} + \sqrt{2}}{4\sqrt{2}}$
E) $\frac{\sqrt{3} + 1}{4\sqrt{2}}$ | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{4\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {3}\sin x+\cos x$ $(x\in R)$
(1) Find the value of $f( \frac {5π}{6})$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\[- \frac {π}{2}, \frac {π}{2}\]$ and their respective $x$ values. | {
"answer": "-\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron O-ABC with each edge length equal to 1, if point P satisfies $$\overrightarrow {OP}=x \overrightarrow {OA}+y \overrightarrow {OB}+z \overrightarrow {OC}(x+y+z=1)$$, find the minimum value of $$| \overrightarrow {OP}|$$. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct license plates are possible if the digits and the letter block can appear in any order? | {
"answer": "4,989,000,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive odd integers greater than 1 and less than $150$ are square-free? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of snow for each of the next four days is $\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Express your answer as a common fraction. | {
"answer": "\\dfrac{1023}{1280}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron with edge length $2\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\sqrt{3}$ and the surface of the tetrahedron is ______. | {
"answer": "8\\sqrt{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2510$. Let $N$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $N$? | {
"answer": "1255",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | {
"answer": "10010111_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest $2$-digit prime factor of the integer $n = {180\choose 90}$? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $DEF$ with $\angle D = 90^\circ$, side $DE = 9$ cm and side $EF = 15$ cm. Find $\sin F$. | {
"answer": "\\frac{3\\sqrt{34}}{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boy presses his thumb along a vertical rod that rests on a rough horizontal surface. Then he gradually tilts the rod, keeping the component of the force along the rod constant, which is applied to its end. When the tilt angle of the rod to the horizontal is $\alpha=80^{\circ}$, the rod begins to slide on the surface. Determine the coefficient of friction between the surface and the rod if, in the vertical position, the normal force is 11 times the gravitational force acting on the rod. Round your answer to two decimal places. | {
"answer": "0.17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(\log _{\sqrt{3}} x+\log _{\sqrt{3}} x+\log _{\sqrt[6]{3}} x+\ldots+\log _{\sqrt{3}} x=36\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the spring round of the 2000 City Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, and no two students solved all six problems together. What is the smallest possible number of high school students in country $N$ who participated in the spring round? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$ . Find the largest possible value of $m-n$ .**p3.** Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems?**p4.** Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$ . Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done?**p5.** You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon?**p6.** Let $a, b$ , and $c$ be positive integers such that $gcd(a, b)$ , $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$ , but $gcd(a, b, c) = 1$ . Find the minimum possible value of $a + b + c$ .**p7.** Let $ABC$ be a triangle inscribed in a circle with $AB = 7$ , $AC = 9$ , and $BC = 8$ . Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$ . Find the length of $\overline{BX}$ .**p8.** What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ?**p9.** How many terms are in the simplified expansion of $(L + M + T)^{10}$ ?**p10.** Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$ . What are all possible slopes for a line tangent to both of the circles?
PS. You had better use hide for answers. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A club is creating a special series of membership cards that includes a code consisting of five characters selected from the letters B, E, T, A and the digits in 2023. No character should appear in a code more often than it appears in "BETA" or "2023" (B, E, T, A appear once each; 2 is twice; 0 and 3 appear once each). Calculate the total number of distinct membership cards, $M$, and then determine $\frac{M}{10}$. | {
"answer": "312",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron A-BCD inscribed within sphere O, we have AB=6, AC=10, $\angle ABC = \frac{\pi}{2}$, and the maximum volume of the tetrahedron A-BCD is 200. Find the radius of sphere O. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{v_n\}$ defined by $v_1 = 7$ and the relationship $v_{n+1} - v_n = 2 + 5(n-1)$ for $n=1,2,3,\ldots$, express $v_n$ as a polynomial in $n$ and find the sum of its coefficients. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$, a line $L$ passing through the right focus $F$ of the ellipse intersects the ellipse at points $A$ and $B$, and intersects the $y$-axis at point $P$. Suppose $\overrightarrow{PA} = λ_{1} \overrightarrow{AF}$ and $\overrightarrow{PB} = λ_{2} \overrightarrow{BF}$, then find the value of $λ_{1} + λ_{2}$. | {
"answer": "-\\frac{50}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). A point \(K\) is chosen on the smaller arc \(AC\) of the circumcircle of triangle \(ABC\). Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given that \(KH = BM\) and lines \(MH\) and \(CK\) are parallel. | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $(a + 1)x^2 + (a^2 + 1) + 8x = 9$ is a quadratic equation in terms of $x$, find the value of $a$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the probability of Team A winning a single game is $\frac{2}{3}$, calculate the probability that Team A will win in a "best of three" format, where the first team to win two games wins the match and ends the competition. | {
"answer": "\\frac{16}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ to $B$ were the same as on the section from $B$ to $C$, the entire journey from $A$ to $C$ would take 5.5 hours. How much time would the boat take to travel upstream from $C$ to $A$ under these conditions? The boat's own speed remains unchanged in all cases. | {
"answer": "7.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \( N \geq 5 \) natural numbers written on the board. It is known that the sum of all the numbers is 80, and the sum of any five of them is not more than 19. What is the smallest possible value of \( N \)? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle \(ABCD\) is divided into four parts by \(CE\) and \(DF\). It is known that the areas of three of these parts are \(5\), \(16\), and \(20\) square centimeters, respectively. What is the area of quadrilateral \(ADOE\) in square centimeters? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest prime factor of 1739? | {
"answer": "1739",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many even positive integers $n$ less than or equal to 800 is $$(\sin t - i\cos t)^n = \sin nt - i\cos nt$$ true for all real $t$? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \leq 1000$ satisfy the condition that $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR,$ $PQ = 12,$ $QR = 13,$ $PR = 15,$ and point $H$ is the intersection of the medians. Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $PQR$ and $P'Q'R'?$ | {
"answer": "20\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During what year after 1994 did sales increase the most number of dollars? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,6,Ticks(1.0,NoZero));
yaxis(0,6,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(3,3)--cycle);
fill((5,0)--(6,0)--(6,6)--(0,6)--(0,5)--(5,5)--cycle);
label("$E$",(0,0),SW);
label("$F$",(0,6),N);
label("$G$",(6,6),NE);
label("$H$",(6,0),E);[/asy] | {
"answer": "61.11\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$ , find $p+q$ .
*Proposed by Michael Kural* | {
"answer": "9901",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = (2,0) \) and \( B = (8,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac{x^{2}}{8} + \frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine the slope $k$ of line $BC$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x > 0$, $y > 0$, and $x + 2y = 2$, find the minimum value of $xy$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\phi$ be the smallest acute angle for which $\cos \phi,$ $\cos 2 \phi,$ $\cos 3 \phi$ form an arithmetic progression, in some order. Find $\sin \phi.$ | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculate the value of $\sum _{k=1}^{22}f(k)$. | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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