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40.3k
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stringlengths 10
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| ground_truth
stringlengths 1
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100
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|---|---|---|---|
26,700 | For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines? | 12 | 0 |
26,701 | **Q8.** Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$ . Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$ | 7/4 | 0 |
26,702 | Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements. | 200 | 26.5625 |
26,703 | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $90^{\circ}$, if vector $\overrightarrow{c}$ satisfies $|\overrightarrow{c}- \overrightarrow{a} \overrightarrow{b}|=2$, calculate the maximum value of $|\overrightarrow{c}|$. | 2 + \sqrt{2} | 0 |
26,704 |
Vasiliy came up with a new chess piece called the "super-bishop." One "super-bishop" ($A$) attacks another ($B$) if they are on the same diagonal, there are no pieces between them, and the next cell along the diagonal after the "super-bishop" $B$ is empty. For example, in the image, piece $a$ attacks piece $b$, but does not attack any of the pieces $c, d, e, f,$ and $g$.
What is the maximum number of "super-bishops" that can be placed on a chessboard such that each of them attacks at least one other? | 32 | 95.3125 |
26,705 | If $a$ is an element randomly selected from the set $\{1, 2, 3, 4, 5, 6, 7\}$, then the probability that the circle $C: x^2 + (y-2)^2 = 1$ is contained inside the circle $O: x^2 + y^2 = a^2$ is ______. | \frac{4}{7} | 10.9375 |
26,706 | Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, where exactly one even number is sandwiched between two odd numbers. How many such five-digit numbers are there? | 28 | 85.9375 |
26,707 | What is the smallest positive integer with exactly 16 positive divisors? | 216 | 0 |
26,708 | Sara baked 60 pies. Of these, one-third contained berries, half contained cream, three-fifths contained nuts, and one-fifth contained coconut. What is the largest possible number of pies that had none of these ingredients? | 24 | 10.15625 |
26,709 | Let QR = x, PR = y, and PQ = z. Given that the area of the square on side QR is 144 = x^2 and the area of the square on side PR is 169 = y^2, find the area of the square on side PQ. | 25 | 5.46875 |
26,710 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $A > B$, $\cos C= \frac {5}{13}$, and $\cos (A-B)= \frac {3}{5}$.
(1) Find the value of $\cos 2A$;
(2) If $c=15$, find the value of $a$. | 2 \sqrt {65} | 0 |
26,711 | In a square table with 2015 rows and columns, positive numbers are placed. The product of the numbers in each row and in each column is equal to 2, and the product of the numbers in any 3x3 square is equal to 1. What number is in the center of the table? | 2^{-2017} | 0 |
26,712 | Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees. | 80 | 40.625 |
26,713 | Given that \(\theta\) is an angle in the third quadrant, and \(\sin^{4}\theta + \cos^{4}\theta = \frac{5}{9}\), determine the value of \(\sin 2\theta\). | -\frac{2\sqrt{2}}{3} | 10.9375 |
26,714 | If $\log 2 = 0.3010$ and $\log 5 = 0.6990$, calculate the value of $x$ for the equation $2^{x+2} = 200$. | 5.64 | 1.5625 |
26,715 | If triangle $PQR$ has sides of length $PQ = 7$, $PR = 8$, and $QR = 6$, then calculate
\[
\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.
\] | \frac{16}{7} | 32.03125 |
26,716 | A bin contains 10 black balls and 9 white balls. 4 balls are drawn at random. What is the probability of drawing 3 balls of one color and 1 ball of another color? | \frac{160}{323} | 10.15625 |
26,717 | The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? | 90 | 0.78125 |
26,718 | Given that Lauren has 4 sisters and 7 brothers, and her brother Lucas has S sisters and B brothers. Find the product of S and B. | 35 | 6.25 |
26,719 | For positive integers \( n \), let \( g(n) \) return the smallest positive integer \( k \) such that \( \frac{1}{k} \) has exactly \( n \) digits after the decimal point in base 6 notation. Determine the number of positive integer divisors of \( g(2023) \). | 4096576 | 46.09375 |
26,720 | Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$ . If $IA = 5, IB = 7, IC = 4, ID = 9$ , find the value of $\frac{AB}{CD}$ . | 35/36 | 6.25 |
26,721 | A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} | 1.5625 |
26,722 | In the xy-plane with a rectangular coordinate system, the terminal sides of angles $\alpha$ and $\beta$ intersect the unit circle at points $A$ and $B$, respectively.
1. If point $A$ is in the first quadrant with a horizontal coordinate of $\frac{3}{5}$ and point $B$ has a vertical coordinate of $\frac{12}{13}$, find the value of $\sin(\alpha + \beta)$.
2. If $| \overrightarrow{AB} | = \frac{3}{2}$ and $\overrightarrow{OC} = a\overrightarrow{OA} + \overrightarrow{OB}$, where $a \in \mathbb{R}$, find the minimum value of $| \overrightarrow{OC} |$. | \frac{\sqrt{63}}{8} | 0 |
26,723 | A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a factor of 3 rather than 4. If this process is repeated indefinitely, calculate the total area of the shaded triangles. | 16 | 39.84375 |
26,724 | Using the digits 0, 1, 2, 3, 4, 5, if repetition of digits is not allowed, the number of different five-digit numbers that can be formed, which are divisible by 5 and do not have 3 as the hundred's digit, is ______. | 174 | 7.03125 |
26,725 | Given $f(x)=\sin (\omega x+\dfrac{\pi }{3})$ ($\omega > 0$), $f(\dfrac{\pi }{6})=f(\dfrac{\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\dfrac{\pi }{6},\dfrac{\pi }{3})$, find the value of $\omega$. | \dfrac{14}{3} | 7.8125 |
26,726 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 39 | 0 |
26,727 | Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ | 16 | 0 |
26,728 | Let $M$ denote the number of $8$-digit positive integers where the digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed, and the digit zero can now be used.) | 310 | 68.75 |
26,729 | How many multiples of 5 are between 100 and 400? | 59 | 0 |
26,730 | Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, let the left and right foci of the ellipse be $F_1$ and $F_2$, respectively. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If the line $AF_2$ intersects the ellipse at another point $C$, and the area of triangle $\triangle ABC$ is three times the area of triangle $\triangle BCF_2$, determine the eccentricity of the ellipse. | \frac{\sqrt{5}}{5} | 0.78125 |
26,731 | What number corresponds to the point \( P \) indicated on the given scale? | 12.50 | 0 |
26,732 | Find the least positive integer $n$ such that there are at least $1000$ unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon. | 28 | 2.34375 |
26,733 | Determine the value of \(x\) if \(x\) is positive and \(x \cdot \lfloor x \rfloor = 90\). Express your answer as a decimal. | 10 | 0.78125 |
26,734 | Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \ge K xy$ . | 8\sqrt{95} | 7.03125 |
26,735 | Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[ \text{Area}(WXYZ) = \gamma \omega - \delta \omega^2. \]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 17 | 38.28125 |
26,736 | A factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ million units, an additional cost of $C(x)$ million is required. When the annual production volume is less than 80 million units, $C(x)= \frac {1}{3}x^{2}+10x$; when the annual production volume is not less than 80 million units, $C(x)=51x+ \frac {10000}{x}-1450$. Assume that each unit of the product is sold for $50 million and all produced units can be sold that year.
1. Write out the annual profit $L(x)$ (million) as a function of the annual production volume $x$ (million units).
2. What is the production volume when the factory's profit from producing this product is the highest? What is the maximum profit? | 1000 | 57.8125 |
26,737 | A tank with a mass of $m_{1}=2$ kg rests on a cart with a mass of $m_{2}=10$ kg, which is accelerated with an acceleration of $a=5 \, \text{m/s}^2$. The coefficient of friction between the tank and the cart is $\mu=0.6$. Determine the frictional force acting on the tank from the cart. | 10 | 53.90625 |
26,738 | A circle of radius \( 2 \) cm is inscribed in \( \triangle ABC \). Let \( D \) and \( E \) be the points of tangency of the circle with the sides \( AC \) and \( AB \), respectively. If \( \angle BAC = 45^{\circ} \), find the length of the minor arc \( DE \). | \pi | 10.15625 |
26,739 | $\triangle GHI$ is inscribed inside $\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle GZC, \triangle HYD, \triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{YI} = \stackrel{\frown}{GZ},\ \stackrel{\frown}{XI} = \stackrel{\frown}{HZ},\ \stackrel{\frown}{XH} = \stackrel{\frown}{GY}$. The length of $GY$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. | 29 | 6.25 |
26,740 | Two distinct positive integers \( x \) and \( y \) are factors of 48. If \( x \cdot y \) is not a factor of 48, what is the smallest possible value of \( x \cdot y \)? | 32 | 6.25 |
26,741 | Eight positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $2107$, what is the sum of the numbers written on the faces? | 57 | 14.84375 |
26,742 | Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{28}/R_8$ when $R_{28}$ is divided by $R_8$. | 21 | 5.46875 |
26,743 | What is the greatest common divisor (GCD) and the sum of the numbers 1729 and 867? | 2596 | 42.96875 |
26,744 | Express \( 0.3\overline{45} \) as a common fraction. | \frac{83}{110} | 0 |
26,745 | Given that $|\vec{a}| = 2$, $|\vec{b}| = 1$, and $(2\vec{a} - 3\vec{b}) \cdot (2\vec{a} + \vec{b}) = 9$.
(I) Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$;
(II) Find $|\vec{a} + \vec{b}|$ and the projection of vector $\vec{a}$ in the direction of $\vec{a} + \vec{b}$. | \frac{5\sqrt{7}}{7} | 19.53125 |
26,746 | I randomly choose an integer \( p \) between \( 1 \) and \( 20 \) inclusive. What is the probability that \( p \) is such that there exists an integer \( q \) so that \( p \) and \( q \) satisfy the equation \( pq - 6p - 3q = 3 \)? Express your answer as a common fraction. | \frac{3}{20} | 14.0625 |
26,747 | As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral triangle is constructed outwards on the middle segment of each side, and the middle segments are then removed (\(k=0,1,2, \cdots\)). Let \(S_{n}\) denote the area enclosed by the curve \(P_{n}\).
1. Find a general formula for the sequence \(\{S_{n}\}\).
2. Evaluate \(\lim _{n \rightarrow \infty} S_{n}\). | \frac{8}{5} | 18.75 |
26,748 | Let point $O$ be the origin of a two-dimensional coordinate system, and let points $A$ and $B$ be located on positive $x$ and $y$ axes, respectively. If $OA = \sqrt[3]{54}$ and $\angle AOB = 45^\circ,$ compute the length of the line segment $AB.$ | 54^{1/3} \sqrt{2} | 0 |
26,749 | Given the sets \( P=\left\{m^{2}-4, m+1,-3\right\} \) and \( Q=\{m-3, 2m-1, 3m+1\} \), if \( P \cap Q = \{-3\} \), find the value of the real number \( m \). | -\frac{4}{3} | 7.03125 |
26,750 | Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. | 20 | 92.1875 |
26,751 | Given point $M(\sqrt{6}, \sqrt{2})$ on the ellipse $G$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$.
1. Find the equation of ellipse $G$.
2. If the line $l$ with a slope of $1$ intersects ellipse $G$ at points $A$ and $B$, and an isosceles triangle is formed with $AB$ as the base and $P(-3, 2)$ as the apex, find the area of $\triangle PAB$. | \frac{9}{2} | 40.625 |
26,752 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 39 | 85.9375 |
26,753 | The young man paid with $100 for a gift that cost $18 and received $79 in change from Mr. Wang. Mr. Wang then returned the counterfeit $100 bill to the neighbor. | 97 | 23.4375 |
26,754 | If the product of the first 2011 positive integers $1 \times 2 \times \ldots \times 2011$ is divisible by $2010^k$, then the maximum value of the positive integer $k$ is. | 30 | 92.96875 |
26,755 | If the non-negative real numbers $x$ and $y$ satisfy $x^{2}+4y^{2}+4xy+4x^{2}y^{2}=32$, find the minimum value of $x+2y$, and the maximum value of $\sqrt{7}(x+2y)+2xy$. | 4\sqrt{7}+4 | 6.25 |
26,756 | Let $\omega$ be a circle with radius $1$ . Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$ . If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$ . | \frac{2 \sqrt{3}}{3} | 7.8125 |
26,757 | The result of the addition shown is
```
300
2020
+10001
``` | 12321 | 78.125 |
26,758 | Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)$ ($m > \frac{3}{4}$) and intersects ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4}, 0)$, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.
(Ⅰ) Find the equation of ellipse $E$;
(Ⅱ) Find the maximum value of the area of $\triangle OAB$. | \frac{\sqrt{2}}{2} | 10.15625 |
26,759 | In triangle \( \triangle ABC \), it is given that \( \angle C=90^\circ \), \( \angle A=60^\circ \), and \( AC=1 \). Points \( D \) and \( E \) are on sides \( BC \) and \( AB \) respectively such that triangle \( \triangle ADE \) is an isosceles right triangle with \( \angle ADE=90^\circ \). Find the length of \( BE \). | 4-2\sqrt{3} | 0 |
26,760 | Equilateral $\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\triangle RDE$ and $\triangle SDE$ form a $150^{\circ}$ dihedral angle. There is a point $M$ whose distance from each of $D$, $E$, $F$, $R$, and $S$ is $k$. Find $k$. | 300 | 3.90625 |
26,761 | A set $\mathcal{T}$ of distinct positive integers has the following property: for every integer $y$ in $\mathcal{T},$ the arithmetic mean of the set of values obtained by deleting $y$ from $\mathcal{T}$ is an integer. Given that 1 belongs to $\mathcal{T}$ and that 1764 is the largest element of $\mathcal{T},$ what is the greatest number of elements that $\mathcal{T}$ can have? | 42 | 10.9375 |
26,762 | Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9. | 51 | 30.46875 |
26,763 | We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. | 2500 | 7.03125 |
26,764 | Let $f(x)$ be an odd function defined on $(-\infty, +\infty)$, and $f(x+2) = -f(x)$. Given that $f(x) = x$ for $0 \leq x \leq 1$, find $f(3\pi)$. | 10 - 3\pi | 3.125 |
26,765 | Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with
every possible height-weight combination.
The USAMTS wants to field a competitive team, so there are some strict requirements.
- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$ ’s height can also be on the team.
Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed? | 128 | 8.59375 |
26,766 | In circle $O$ with radius 10 units, chords $AC$ and $BD$ intersect at right angles at point $P$. If $BD$ is a diameter of the circle, and the length of $PC$ is 3 units, calculate the product $AP \cdot PB$. | 51 | 11.71875 |
26,767 | In our daily life, for a pair of new bicycle tires, the rear tire wears out faster than the front tire. Through testing, it is found that the front tire of a general bicycle is scrapped after traveling 11,000 kilometers, while the rear tire is scrapped after traveling 9,000 kilometers. It is evident that when the rear tire is scrapped after traveling 9,000 kilometers, the front tire can still be used, which inevitably leads to a certain waste. If the front and rear tires are swapped once, allowing the front and rear tires to be scrapped simultaneously, the bicycle can travel a longer distance. How many kilometers can the bicycle travel at most after swapping once? And after how many kilometers should the front and rear tires be swapped? | 4950 | 60.15625 |
26,768 | $Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. | \sqrt{6} | 15.625 |
26,769 | Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*2021 CCA Math Bonanza Individual Round #8* | 21 | 22.65625 |
26,770 | Find the largest natural number from which it is impossible to obtain a number divisible by 11 by deleting some of its digits. | 987654321 | 3.125 |
26,771 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 38 | 1.5625 |
26,772 | A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets \(F\) and \(T\), each with at least 2 elements, such that the elements of \(F\) are consecutive Fibonacci numbers, and the elements of \(T\) are consecutive triangular numbers. Find all positive integers \(n\) such that the set containing all the positive divisors of \(n\) except \(n\) itself is pilak. | 30 | 1.5625 |
26,773 | Given that a normal vector of the straight line $l$ is $\overrightarrow{n} = (1, -\sqrt{3})$, find the size of the inclination angle of this straight line. | \frac{\pi}{6} | 68.75 |
26,774 | In a right triangle $DEF$ where leg $DE = 30$ and leg $EF = 40$, determine the number of line segments with integer length that can be drawn from vertex $E$ to a point on hypotenuse $\overline{DF}$. | 17 | 7.03125 |
26,775 | If $x + x^2 + x^3 + \ldots + x^9 + x^{10} = a_0 + a_1(1 + x) + a_2(1 + x)^2 + \ldots + a_9(1 + x)^9 + a_{10}(1 + x)^{10}$, then $a_9 = \_\_\_\_\_\_\_\_$. | -9 | 7.8125 |
26,776 | Select the shape of diagram $b$ from the regular hexagonal grid of diagram $a$. There are $\qquad$ different ways to make the selection (note: diagram $b$ can be rotated). | 72 | 0 |
26,777 | What is the least positive integer with exactly $12$ positive factors? | 96 | 0 |
26,778 | A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make? | 24 | 0 |
26,779 | A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second time. How many minutes does it take for the frog to climb from the bottom of the well to the top? | 22 | 3.90625 |
26,780 | Jennifer wants to do origami, and she has a square of side length $ 1$ . However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains? | 1 - \frac{\sqrt{2}}{2} | 0 |
26,781 | In triangle $XYZ$, $\cos(2X - Y) + \sin(X + Y) = 2$ and $XY = 6$. What is $YZ$? | 3\sqrt{3} | 10.9375 |
26,782 | For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$.
$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$.
$(2)$ The inequality $t\leqslant \left(m-1\right)^{2}+\left(n-1\right)^{2}+\left(m-n\right)^{2}$ always holds. Find the maximum value of the real number $t$. | \frac{9}{8} | 0 |
26,783 | There is a committee composed of 10 members who meet around a table: 7 women, 2 men, and 1 child. The women sit in indistinguishable rocking chairs, the men on indistinguishable stools, and the child on a bench, also indistinguishable from any other benches. How many distinct ways are there to arrange the seven rocking chairs, two stools, and one bench for a meeting? | 360 | 71.09375 |
26,784 | In a table consisting of $n$ rows and $m$ columns, numbers are written such that the sum of the elements in each row is 1248, and the sum of the elements in each column is 2184. Find the numbers $n$ and $m$ for which the expression $2n - 3m$ takes the smallest possible natural value. In the answer, specify the value of $n + m$. | 143 | 38.28125 |
26,785 | Select three digits from 1, 3, 5, 7, 9, and two digits from 0, 2, 4, 6, 8 to form a five-digit number without any repeating digits. How many such numbers can be formed? | 11040 | 4.6875 |
26,786 | The cross below is made up of five congruent squares. The perimeter of the cross is $72$ . Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy] | 180 | 92.96875 |
26,787 | Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$. | 72 | 0 |
26,788 | In any isosceles triangle $PQR$ with $PQ = PR$, the altitude $PS$ bisects the base $QR$ so that $QS = RS$. Given that the triangle sides are $PQ = PR = 15$ and the entire base length $QR = 20$.
[asy]
draw((0,0)--(20,0)--(10,36)--cycle,black+linewidth(1));
draw((10,36)--(10,0),black+linewidth(1)+dashed);
draw((10,0)--(10,1)--(9,1)--(9,0)--cycle,black+linewidth(1));
draw((4.5,-4)--(0,-4),black+linewidth(1));
draw((4.5,-4)--(0,-4),EndArrow);
draw((15.5,-4)--(20,-4),black+linewidth(1));
draw((15.5,-4)--(20,-4),EndArrow);
label("$P$",(10,36),N);
label("$Q$",(0,0),SW);
label("$R$",(20,0),SE);
label("$S$",(10,0),S);
label("15",(0,0)--(10,36),NW);
label("15",(10,36)--(20,0),NE);
label("20",(10,-4));
[/asy] | 50\sqrt{5} | 9.375 |
26,789 | What is the maximum number of rooks that can be placed on a $300 \times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.) | 400 | 0 |
26,790 | Given that point $P$ is a moving point on the curve $y= \frac {3-e^{x}}{e^{x}+1}$, find the minimum value of the slant angle $\alpha$ of the tangent line at point $P$. | \frac{3\pi}{4} | 23.4375 |
26,791 | Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$ . For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$ , let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). Compute the value of $n$ for which there exists a permutation $\sigma$ of $S_n$ satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017, \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]
*Proposed by Michael Tang* | 53 | 0 |
26,792 | Two companies, A and B, each donated 60,000 yuan to the "See a Just Cause Foundation". It is known that the average donation per person in company B is 40 yuan more than that in company A, and the number of people in company A is 20% more than that in company B. What are the respective number of people in companies A and B? | 250 | 39.84375 |
26,793 | In a class, there are 4 lessons in one morning, and each lesson needs a teacher to teach it. Now, from 6 teachers A, B, C, D, E, F, 4 teachers are to be arranged to teach one lesson each. The first lesson can only be taught by either A or B, and the fourth lesson can only be taught by either A or C. How many different arrangement plans are there? | 36 | 31.25 |
26,794 | Let \( f(n) \) be a function where, for an integer \( n \), \( f(n) = k \) and \( k \) is the smallest integer such that \( k! \) is divisible by \( n \). If \( n \) is a multiple of 20, determine the smallest \( n \) such that \( f(n) > 20 \). | 420 | 0 |
26,795 | Fill in the 3x3 grid with 9 different natural numbers such that for each row, the sum of the first two numbers equals the third number, and for each column, the sum of the top two numbers equals the bottom number. What is the smallest possible value for the number in the bottom right corner? | 12 | 2.34375 |
26,796 | Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\triangle PF_{1}F_{2}$ is $30^{\circ}$, then the eccentricity of $C$ is ______. | \sqrt{3} | 11.71875 |
26,797 | $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . | \sqrt{51} | 0.78125 |
26,798 | We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection? | 120 | 28.90625 |
26,799 | A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to do this (answer with a number)? | 30 | 5.46875 |
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