Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
30,600 | Point $(x, y)$ is randomly selected from the rectangular region defined by vertices $(0, 0), (3013, 0), (3013, 3014),$ and $(0, 3014)$. What is the probability that $x > 8y$? | \frac{3013}{48224} | 25.78125 |
30,601 | Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist? | 12 | 0 |
30,602 | Given that point P is any point on the graph of the function $f(x) = 2\sqrt{2x}$, and a tangent line is drawn from point P to circle D: $x^2 + y^2 - 4x + 3 = 0$, with the points of tangency being A and B, find the minimum value of the area of quadrilateral PADB. | \sqrt{3} | 53.125 |
30,603 | Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make? | 36 | 74.21875 |
30,604 | Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$, and for any real number $x$, it always holds that $f(x)-f(-x)=0$. When $x \in [0,1]$, $f(x)=-\sqrt{1-x^{2}}$. Determine the number of zeros of the function $g(x)=f(x)-e^{x}+1$ in the interval $[-2018,2018]$. | 2018 | 10.15625 |
30,605 | Given that the simplest quadratic radical $\sqrt{m+1}$ and $\sqrt{8}$ are of the same type of quadratic radical, the value of $m$ is ______. | m = 1 | 71.09375 |
30,606 | Suppose $w$ is a complex number such that $w^2 = 45-21i$. Find $|w|$. | \sqrt[4]{2466} | 7.8125 |
30,607 | Given that the vector $\overrightarrow {a} = (3\cos\alpha, 2)$ is parallel to the vector $\overrightarrow {b} = (3, 4\sin\alpha)$, find the value of the acute angle $\alpha$. | \frac{\pi}{4} | 92.1875 |
30,608 | Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$. | 50 | 76.5625 |
30,609 | Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly eight steps to become palindromes? | 187 | 4.6875 |
30,610 | In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly scooped out, the probability that at least one dumpling of each type is scooped out is ______. | \dfrac{48}{91} | 18.75 |
30,611 | Given the complex numbers $z\_1=a^2-2-3ai$ and $z\_2=a+(a^2+2)i$, if $z\_1+z\_2$ is a purely imaginary number, determine the value of the real number $a$. | -2 | 16.40625 |
30,612 | Some bugs are sitting on squares of $10\times 10$ board. Each bug has a direction associated with it **(up, down, left, right)**. After 1 second, the bugs jump one square in **their associated**direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes right, down becomes up, and right becomes left) and the bug moves in that direction. It is observed that it is **never** the case that two bugs are on same square. What is the maximum number of bugs possible on the board? | 40 | 0 |
30,613 | A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers. | 5/2 | 17.1875 |
30,614 | What is \( \frac{1}{4} \) more than 32.5? | 32.75 | 78.125 |
30,615 | Let $I, T, E, S$ be distinct positive integers such that the product $ITEST = 2006$ . What is the largest possible value of the sum $I + T + E + S + T + 2006$ ? | 2086 | 10.15625 |
30,616 | Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{1}{4}$
B) $\frac{57}{64}$
C) $\frac{199}{256}$
D) $\frac{57}{256}$
E) $\frac{63}{64}$ | \frac{199}{256} | 21.09375 |
30,617 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 97.65625 |
30,618 | Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{2}$, and passing through the point $A(\frac{3}{2}, -\frac{1}{2})$.
(1) Find the equation of the ellipse;
(2) Find the coordinates of a point $P$ on the ellipse $C$ such that its distance to the line $l$: $x+y+4=0$ is minimized, and find this minimum distance. | \sqrt{2} | 21.09375 |
30,619 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 66.40625 |
30,620 | Define a function $B(m, n)$ by \[ B(m,n) = \left\{ \begin{aligned} &n+2& \text{ if } m = 0 \\ &B(m-1, 2) & \text{ if } m > 0 \text{ and } n = 0 \\ &B(m-1, B(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $B(2, 2).$ | 12 | 0 |
30,621 | For a rational number $r$ , its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$ . If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$ , find the sum of all elements in $S$ . | 50000000 | 25.78125 |
30,622 | In the diagram, $\triangle ABC$, $\triangle BCD$, and $\triangle CDE$ are right-angled at $B$, $C$, and $D$ respectively, with $\angle ACB=\angle BCD = \angle CDE = 45^\circ$, and $AB=15$. [asy]
pair A, B, C, D, E;
A=(0,15);
B=(0,0);
C=(10.6066,0);
D=(15,0);
E=(21.2132,0);
draw(A--B--C--D--E);
draw(B--C);
draw(C--D);
label("A", A, N);
label("B", B, W);
label("C", C, S);
label("D", D, S);
label("E", E, S);
[/asy] Find the length of $CD$. | \frac{15\sqrt{2}}{2} | 3.90625 |
30,623 | In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\angle BMC = 90^\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ . | 30 | 8.59375 |
30,624 | Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collected at the point nearest to the trapezoid's perimeter. What fraction of the crop is brought to the longest side $AD$? | \frac{1}{2} | 10.9375 |
30,625 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.298 | 0 |
30,626 | In $\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$. | \frac{56}{3} | 1.5625 |
30,627 | On the lateral side \( CD \) of the trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. A perpendicular \( AH \) is dropped from vertex \( A \) to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \). | 18 | 9.375 |
30,628 | Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides? | 30 | 16.40625 |
30,629 | Let \( S = \{1, 2, \cdots, 2005\} \). If in any set of \( n \) pairwise coprime numbers in \( S \) there is at least one prime number, find the minimum value of \( n \). | 16 | 5.46875 |
30,630 | If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$ , what is the least integer $n\geq 2012$ ? | 2014 | 4.6875 |
30,631 | Given $f(x) = \log_a x$, its inverse function is $g(x)$.
(1) Solve the equation for $x$: $f(x-1) = f(a-x) - f(5-x)$;
(2) Let $F(x) = (2m-1)g(x) + \left( \frac{1}{m} - \frac{1}{2} \right)g(-x)$. If $F(x)$ has a minimum value, find the expression for $h(m)$;
(3) Find the maximum value of $h(m)$. | \sqrt{2} | 24.21875 |
30,632 | When \( x^{2} \) is added to the quadratic polynomial \( f(x) \), its maximum value increases by \( \frac{27}{2} \), and when \( 4x^{2} \) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \( f(x) \) change if \( 2x^{2} \) is subtracted from it? | \frac{27}{4} | 3.90625 |
30,633 | Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}+\overrightarrow{b}|+2\overrightarrow{a}\cdot\overrightarrow{b}=0$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 89.84375 |
30,634 | Let $n$ be a fixed integer, $n \geqslant 2$.
1. Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_i x_j (x_i^2 + x_j^2) \leqslant c \left(\sum_{i=1}^n x_i \right)^4
$$
holds for all non-negative real numbers $x_1, x_2, \cdots, x_n$.
2. For this constant $c$, determine the necessary and sufficient conditions for which equality holds. | \frac{1}{8} | 50 |
30,635 | Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0 |
30,636 | Given the function $f(x)=(x-a)^{2}+(2\ln x-2a)^{2}$, where $x > 0, a \in \mathbb{R}$, find the value of the real number $a$ such that there exists $x_{0}$ such that $f(x_{0}) \leqslant \frac{4}{5}$. | \frac{1}{5} | 20.3125 |
30,637 | Determine the smallest odd prime factor of $2021^{10} + 1$. | 61 | 2.34375 |
30,638 | There are 2012 distinct points in the plane, each of which is to be coloured using one of \( n \) colours so that the number of points of each colour are distinct. A set of \( n \) points is said to be multi-coloured if their colours are distinct. Determine \( n \) that maximizes the number of multi-coloured sets. | 61 | 0 |
30,639 | Given that $|\cos\theta|= \frac {1}{5}$ and $\frac {5\pi}{2}<\theta<3\pi$, find the value of $\sin \frac {\theta}{2}$. | -\frac{\sqrt{15}}{5} | 11.71875 |
30,640 | How many roots does the equation
$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$
where $f(x)=|x|-1$ have? | 20 | 7.8125 |
30,641 | Given a finite sequence $D$: $a\_1$, $a\_2$, ..., $a\_n$, where $S\_n$ represents the sum of the first $n$ terms of the sequence $D$, define $\frac{S\_1 + S\_2 + ... + S\_n}{n}$ as the "De-Guang sum" of $D$. If a 99-term sequence $a\_1$, $a\_2$, ..., $a\_99$ has a "De-Guang sum" of $1000$, find the "De-Guang sum" of the 100-term sequence $8$, $a\_1$, $a\_2$, ..., $a\_99$. | 998 | 98.4375 |
30,642 | Points $A, B$ and $C$ lie on the same line so that $CA = AB$ . Square $ABDE$ and the equilateral triangle $CFA$ , are constructed on the same side of line $CB$ . Find the acute angle between straight lines $CE$ and $BF$ . | 75 | 57.8125 |
30,643 | Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____. | 1012 | 54.6875 |
30,644 | Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | 106 | 14.84375 |
30,645 | There are 3 boys and 4 girls, all lined up in a row. How many ways are there for the following situations?
- $(1)$ Person A is neither at the middle nor at the ends;
- $(2)$ Persons A and B must be at the two ends;
- $(3)$ Boys and girls alternate. | 144 | 35.15625 |
30,646 | **p1.** Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly
a) Three other points?
b) Four other points?**p2.** Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it possible to change a note of $25$ marts to exactly $10$ notes of smaller denomination?**p3.** What is bigger: $99 \cdot 99 \cdot .... \cdot 99$ (product of $20$ factors) or $9999 \cdot 9999 \cdot ... \cdot 9999$ (product of $10$ factors)?**p4.** How many whole numbers, from $1$ to $2006$ , are divisible neither by $5$ nor by $7$ ?
PS. You should use hide for answers. | 1376 | 19.53125 |
30,647 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac{b}{a}+\sin({A-B})=\sin C$. Find:<br/>
$(1)$ the value of angle $A$;<br/>
$(2)$ if $a=2$, find the maximum value of $\sqrt{2}b+2c$ and the area of triangle $\triangle ABC$. | \frac{12}{5} | 26.5625 |
30,648 | Given the function \( y = \sqrt{2x^2 + 2} \) with its graph represented as curve \( G \), and the focus of curve \( G \) denoted as \( F \), two lines \( l_1 \) and \( l_2 \) pass through \( F \) and intersect curve \( G \) at points \( A, C \) and \( B, D \) respectively, such that \( \overrightarrow{AC} \cdot \overrightarrow{BD} = 0 \).
(1) Find the equation of curve \( G \) and the coordinates of its focus \( F \).
(2) Determine the minimum value of the area \( S \) of quadrilateral \( ABCD \). | 16 | 0 |
30,649 | In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$. | \sqrt {7} | 0 |
30,650 | Three people jointly start a business with a total investment of 143 million yuan. The ratio of the highest investment to the lowest investment is 5:3. What is the maximum and minimum amount the third person could invest in millions of yuan? | 39 | 16.40625 |
30,651 | An isosceles triangle and a rectangle have the same area. The base of the triangle is equal to the width of the rectangle, and this dimension is 10 units. The length of the rectangle is twice its width. What is the height of the triangle, $h$, in terms of the dimensions of the rectangle? | 40 | 83.59375 |
30,652 | In the hexagonal pyramid $(P-ABCDEF)$, the base is a regular hexagon with side length $\sqrt{2}$, $PA=2$ and is perpendicular to the base. Find the volume of the circumscribed sphere of the hexagonal pyramid. | 4\sqrt{3}\pi | 26.5625 |
30,653 | Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after "Gauss". Let $x\in\mathbb{R}$, use $[x]$ to represent the largest integer not exceeding $x$, and use $\{x\}=x-[x]$ to represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_n\}$ satisfies: $$a_{1}= \sqrt {3}, a_{n+1}=[a_{n}]+ \frac {1}{\{a_{n}\}}, (n\in\mathbb{N}^{*})$$, then $a_{2017}=$ \_\_\_\_\_\_. | 3024+ \sqrt {3} | 0 |
30,654 | Find the minimum value of
\[\sqrt{x^2 + (1 + 2x)^2} + \sqrt{(x - 1)^2 + (x - 1)^2}\]over all real numbers $x.$ | \sqrt{2} \times \sqrt{5} | 0 |
30,655 | What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon? | 2006 | 11.71875 |
30,656 | Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. | \frac{1}{3} | 2.34375 |
30,657 | Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? | \frac{45}{248} | 24.21875 |
30,658 | A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape. | 34 | 10.15625 |
30,659 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi - 18}{2} | 14.0625 |
30,660 | Given $0 < \alpha < \pi$, $\tan\alpha = -2$.
(1) Find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$;
(2) Calculate the value of $$\frac{2\cos\left(\frac{\pi}{2} + \alpha\right) - \cos(\pi - \alpha)}{\sin\left(\frac{\pi}{2} - \alpha\right) - 3\sin(\pi + \alpha)};$$
(3) Simplify $2\sin^2\alpha - \sin\alpha\cos\alpha + \cos^2\alpha$. | \frac{11}{5} | 88.28125 |
30,661 | Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial. | 8\sqrt{6} | 69.53125 |
30,662 | In diagram square $ABCD$, four triangles are removed resulting in rectangle $PQRS$. Two triangles at opposite corners ($SAP$ and $QCR$) are isosceles with each having area $120 \text{ m}^2$. The other two triangles ($SDR$ and $BPQ$) are right-angled at $D$ and $B$ respectively, each with area $80 \text{ m}^2$. What is the length of $PQ$, in meters? | 4\sqrt{15} | 5.46875 |
30,663 | Carefully observe the arrangement pattern of the following hollow circles ($○$) and solid circles ($●$): $○●○○●○○○●○○○○●○○○○○●○○○○○○●…$. If this pattern continues, a series of $○$ and $●$ will be obtained. The number of $●$ in the first $100$ of $○$ and $●$ is $\_\_\_\_\_\_\_$. | 12 | 23.4375 |
30,664 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.) | 100 | 22.65625 |
30,665 | We seek the true statement. There are one hundred statements written in a notebook:
1) There is exactly one false statement in this notebook.
2) There are exactly two false statements in this notebook.
...
100) There are exactly one hundred false statements in this notebook.
Which one of these statements is true, given that only one is true? | 99 | 51.5625 |
30,666 | A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display? | n = 9 | 12.5 |
30,667 | Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline. | 22.7 | 0 |
30,668 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 95.3125 |
30,669 | Given $f(x)= \sqrt {3}\sin x\cos (x+ \dfrac {π}{6})+\cos x\sin (x+ \dfrac {π}{3})+ \sqrt {3}\cos ^{2}x- \dfrac { \sqrt {3}}{2}$.
(I) Find the range of $f(x)$ when $x\in(0, \dfrac {π}{2})$;
(II) Given $\dfrac {π}{12} < α < \dfrac {π}{3}$, $f(α)= \dfrac {6}{5}$, $- \dfrac {π}{6} < β < \dfrac {π}{12}$, $f(β)= \dfrac {10}{13}$, find $\cos (2α-2β)$. | -\dfrac{33}{65} | 20.3125 |
30,670 | A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this lattice. Assume the points are numbered 1 to 7, with 1 to 6 being the peripheral points and 7 being the center. | 12 | 2.34375 |
30,671 | (1) Evaluate: $\sin^2 120^\circ + \cos 180^\circ + \tan 45^\circ - \cos^2 (-330^\circ) + \sin (-210^\circ)$;
(2) Determine the monotonic intervals of the function $f(x) = \left(\frac{1}{3}\right)^{\sin x}$. | \frac{1}{2} | 23.4375 |
30,672 | Diagonals \( AC \) and \( BD \) of the cyclic quadrilateral \( ABCD \) are perpendicular and intersect at point \( M \). It is known that \( AM = 3 \), \( BM = 4 \), and \( CM = 6 \). Find \( CD \). | 10.5 | 0.78125 |
30,673 | Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. How many points did Lily score? | 40 | 2.34375 |
30,674 | Let $m \in \mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$. | 2\sqrt{5} | 10.9375 |
30,675 | Solve the equation \[-2x^2 = \frac{4x + 2}{x + 2}.\] | -1 | 39.84375 |
30,676 | How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? | 343 | 85.15625 |
30,677 | Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$ . Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$ . Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$ , respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$ .
*Proposed by Andrew Wu* | 31 | 3.90625 |
30,678 | Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$? | 60 | 33.59375 |
30,679 | There are two ways of choosing six different numbers from the list \( 1,2,3,4,5,6,7,8,9 \) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \( m^{2} \) and \( n^{2} \), with \( m \) and \( n \) positive integers and \( m \neq n \). What is the value of \( m+n \)? | 108 | 0 |
30,680 | One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has a one in two chance of not finding her. If she is on one of the courts, there is another one in three chance of missing her, but if she went to the cafe, François will definitely find her: he knows which cafe Béatrice usually enjoys her ice cream. François visited all three possible meeting places but still did not find Béatrice.
What is the probability that Béatrice was on the beach, assuming she did not change locations while François was searching for her? | \frac{3}{5} | 66.40625 |
30,681 | Suppose we wish to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. We need to form these groups such that Rex is in the 4-dog group, Buddy is in the 6-dog group, and Bella and Duke are both in the 2-dog group. How many ways can we form these groups under these conditions? | 56 | 51.5625 |
30,682 | Given a sequence of numbers arranged according to a certain rule: $-\frac{3}{2}$, $1$, $-\frac{7}{10}$, $\frac{9}{17}$, $\ldots$, based on the pattern of the first $4$ numbers, the $10$th number is ____. | \frac{21}{101} | 71.875 |
30,683 | Given a circle $C: (x-1)^2+(y-2)^2=25$, and a line $l: (2m+1)x+(m+1)y-7m-4=0$. If the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{4} | 2.34375 |
30,684 | Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$ . Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$ . Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$ ?
[asy]
size(10cm);
draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
draw((1,0)--(25,10));
draw(circle((0,5),5));
draw(circle((26,5),5));
dot((1,0));
dot((25,10));
label(" $E$ ",(1,0),SE);
label(" $F$ ",(25,10),NW);
label(" $A$ ", (0,0), SW);
label(" $B$ ", (0,10), NW);
label(" $C$ ", (26,10), NE);
label(" $D$ ", (26,0), SE);
dot((0,0));
dot((0,10));
dot((26,0));
dot((26,10));
[/asy]
*Proposed by Nathan Xiong* | 24 | 53.125 |
30,685 | Every day, Xiaoming goes to school along a flat road \(AB\), an uphill road \(BC\), and a downhill road \(CD\) (as shown in the diagram). Given that \(AB : BC : CD = 1 : 2 : 1\) and that Xiaoming's speeds on flat, uphill, and downhill roads are in the ratio 3 : 2 : 4, respectively, find the ratio of the time Xiaoming takes to go to school to the time he takes to come home. | 19:16 | 0 |
30,686 | A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$ , $x+ y$ does not divide $z$ . Determine the largest possible size of $M$ . | 1004 | 2.34375 |
30,687 | In triangle $XYZ,$ $\angle X = 60^\circ,$ $\angle Y = 75^\circ,$ and $YZ = 6.$ Find $XZ.$ | 3\sqrt{2} + \sqrt{6} | 58.59375 |
30,688 | What is the maximum number of queens that can be placed on the black squares of an $8 \times 8$ chessboard so that each queen is attacked by at least one of the others? | 16 | 61.71875 |
30,689 | Find the sum of $10_{10} + 23_{10}$, first by converting each number to base 3, performing the addition in base 3, and then converting the result back to base 10. | 33 | 78.90625 |
30,690 | A certain bookstore currently has $7700$ yuan in funds, planning to use all of it to purchase a total of $20$ sets of three types of books, A, B, and C. Among them, type A books cost $500$ yuan per set, type B books cost $400$ yuan per set, and type C books cost $250$ yuan per set. The bookstore sets the selling prices of type A, B, and C books at $550$ yuan per set, $430$ yuan per set, and $310$ yuan per set, respectively. Let $x$ represent the number of type A books purchased by the bookstore and $y$ represent the number of type B books purchased. Answer the following questions:<br/>$(1)$ Find the functional relationship between $y$ and $x$ (do not need to specify the range of the independent variable);<br/>$(2)$ If the bookstore purchases at least one set each of type A and type B books, how many purchasing plans are possible?<br/>$(3)$ Under the conditions of $(1)$ and $(2)$, based on market research, the bookstore decides to adjust the selling prices of the three types of books as follows: the selling price of type A books remains unchanged, the selling price of type B books is increased by $a$ yuan (where $a$ is a positive integer), and the selling price of type C books is decreased by $a$ yuan. After selling all three types of books, the profit obtained is $20$ yuan more than the profit from one of the plans in $(2)$. Write down directly which plan the bookstore followed and the value of $a$. | 10 | 15.625 |
30,691 | 10 pairs of distinct shoes are mixed in a bag. If 4 shoes are randomly drawn, how many possible outcomes are there for the following results?
(1) None of the 4 shoes form a pair;
(2) Among the 4 shoes, there is one pair, and the other two shoes do not form a pair;
(3) The 4 shoes form exactly two pairs. | 45 | 77.34375 |
30,692 | Given an ellipse $C$: $\frac{{x}^{2}}{3}+\frac{{y}^{2}}{{b}^{2}}=1\left(b \gt 0\right)$ with the right focus $F$, two lines passing through $F$ are tangent to the circle $O$: $x^{2}+y^{2}=r^{2}(r \gt 0)$ at points $A$ and $B$, forming a right triangle $\triangle ABF$. It is also known that the maximum distance between a point on ellipse $C$ and a point on circle $O$ is $\sqrt{3}+1$. <br/>$(1)$ Find the equations of ellipse $C$ and circle $O$; <br/>$(2)$ If a line $l: y=kx+m$ ($k \lt 0, m \gt 0$) that does not pass through point $F$ is tangent to circle $O$ and intersects ellipse $C$ at points $P$ and $Q$, find the perimeter of $\triangle FPQ$. | 2\sqrt{3} | 10.9375 |
30,693 | How many integers are there from 1 to 16500 that
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by 5, 3, or 11? | 8000 | 40.625 |
30,694 | Given the ellipse $C$: $\begin{cases}x=2\cos θ \\\\ y=\sqrt{3}\sin θ\end{cases}$, find the value of $\frac{1}{m}+\frac{1}{n}$. | \frac{4}{3} | 10.15625 |
30,695 | A circle of radius $15$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $BC$ extended through $C$ intersects the circle at point $E$. Find the number of degrees in angle $AEC$. | 90 | 0 |
30,696 | Given the vector $A$ with components $(1, -1, -3, -4)$, | 3\sqrt{3} | 28.90625 |
30,697 | Given the function $f(x)=\sin (2x+ \frac {π}{3})$, for any $x\_1$, $x\_2$, $x\_3$, and $0\leqslant x\_1 < x\_2 < x\_3\leqslant π$, the equation $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_3)|\leqslant m$ holds true. Find the minimum value of the real number $m$. | 3+ \frac { \sqrt {3}}{2} | 0 |
30,698 | Given an arithmetic sequence $\{a_{n}\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\{b_{n}\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let the sum of the first $n$ terms of the sequence $\{b_{n}\}$ be $B_{n}$, the set $P=\{n|n\leqslant 100$ and ${B}_{n}≤100,n∈{N}^{*}\}$. Find the sum $S$ of all elements in $P$. | 2520 | 13.28125 |
30,699 | Find the sum of $543_7$, $65_7$, and $6_7$ in base $7$. | 650_7 | 64.84375 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.