Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
30,100 | Given that $\{a_n\}$ is an arithmetic sequence with common difference $d$, and $S_n$ is the sum of the first $n$ terms, if only $S_4$ is the minimum term among $\{S_n\}$, then the correct conclusion(s) can be drawn is/are ______.
$(1) d > 0$ $(2) a_4 < 0$ $(3) a_5 > 0$ $(4) S_7 < 0$ $(5) S_8 > 0$. | (1)(2)(3)(4) | 0 |
30,101 | Let the three internal angles $A$, $B$, and $C$ of $\triangle ABC$ have opposite sides $a$, $b$, and $c$ respectively, and it is given that $b(\sin B-\sin C)+(c-a)(\sin A+\sin C)=0$.
$(1)$ Find the size of angle $A$;
$(2)$ If $a=\sqrt{3}$ and $\sin C=\frac{1+\sqrt{3}}{2}\sin B$, find the area of $\triangle ABC$. | \frac{3+\sqrt{3}}{4} | 14.84375 |
30,102 | Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have? | 62 | 19.53125 |
30,103 | If triangle $PQR$ has sides of length $PQ = 7,$ $PR = 6,$ and $QR = 8,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{12}{7} | 31.25 |
30,104 | For real numbers \( x \) and \( y \), define the operation \( \star \) as follows: \( x \star y = xy + 4y - 3x \).
Compute the value of the expression
$$
((\ldots)(((2022 \star 2021) \star 2020) \star 2019) \star \ldots) \star 2) \star 1
$$ | 12 | 91.40625 |
30,105 | Calculate $8 \cdot 9\frac{2}{5}$. | 75\frac{1}{5} | 76.5625 |
30,106 | The polynomial sequence is defined as follows: \( f_{0}(x)=1 \) and \( f_{n+1}(x)=\left(x^{2}-1\right) f_{n}(x)-2x \) for \( n=0,1,2, \ldots \). Find the sum of the absolute values of the coefficients of \( f_{6}(x) \). | 190 | 0 |
30,107 | Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\] | 1025 | 1.5625 |
30,108 | The diagram shows a regular octagon and a square formed by drawing four diagonals of the octagon. The edges of the square have length 1. What is the area of the octagon?
A) \(\frac{\sqrt{6}}{2}\)
B) \(\frac{4}{3}\)
C) \(\frac{7}{5}\)
D) \(\sqrt{2}\)
E) \(\frac{3}{2}\) | \sqrt{2} | 34.375 |
30,109 | In square ABCD, where AB=2, fold along the diagonal AC so that plane ABC is perpendicular to plane ACD, resulting in the pyramid B-ACD. Find the ratio of the volume of the circumscribed sphere of pyramid B-ACD to the volume of pyramid B-ACD. | 4\pi:1 | 0 |
30,110 | Given the function $f(x) = |\ln x|$, the solution set of the inequality $f(x) - f(x_0) \geq c(x - x_0)$ is $(0, +\infty)$, where $x_0 \in (0, +\infty)$, and $c$ is a constant. When $x_0 = 1$, the range of values for $c$ is \_\_\_\_\_\_; when $x_0 = \frac{1}{2}$, the value of $c$ is \_\_\_\_\_\_. | -2 | 13.28125 |
30,111 | Given a regular hexagon \( A_6 \) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon. | 13 | 3.90625 |
30,112 | Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second. | \frac{\pi}{3} | 1.5625 |
30,113 | Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms. | \frac{1}{5} | 25.78125 |
30,114 | Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$ . | 79 | 42.1875 |
30,115 | I have 7 books, three of which are identical copies of the same novel, and the others are distinct. If a particular book among these must always be placed at the start of the shelf, in how many ways can I arrange the rest of the books? | 120 | 46.09375 |
30,116 | How many lattice points lie on the hyperbola \(x^2 - y^2 = 999^2\)? | 21 | 0 |
30,117 | Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$. | 1641 | 0.78125 |
30,118 | Let $ABC$ be a triangle with $BC = a$ , $CA = b$ , and $AB = c$ . The $A$ -excircle is tangent to $\overline{BC}$ at $A_1$ ; points $B_1$ and $C_1$ are similarly defined.
Determine the number of ways to select positive integers $a$ , $b$ , $c$ such that
- the numbers $-a+b+c$ , $a-b+c$ , and $a+b-c$ are even integers at most 100, and
- the circle through the midpoints of $\overline{AA_1}$ , $\overline{BB_1}$ , and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$ .
| 125000 | 0.78125 |
30,119 | Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .
*Author:Anderson Wang* | 1458 | 0 |
30,120 | Investigate the formula of \\(\cos nα\\) and draw the following conclusions:
\\(2\cos 2α=(2\cos α)^{2}-2\\),
\\(2\cos 3α=(2\cos α)^{3}-3(2\cos α)\\),
\\(2\cos 4α=(2\cos α)^{4}-4(2\cos α)^{2}+2\\),
\\(2\cos 5α=(2\cos α)^{5}-5(2\cos α)^{3}+5(2\cos α)\\),
\\(2\cos 6α=(2\cos α)^{6}-6(2\cos α)^{4}+9(2\cos α)^{2}-2\\),
\\(2\cos 7α=(2\cos α)^{7}-7(2\cos α)^{5}+14(2\cos α)^{3}-7(2\cos α)\\),
And so on. The next equation in the sequence would be:
\\(2\cos 8α=(2\cos α)^{m}+n(2\cos α)^{p}+q(2\cos α)^{4}-16(2\cos α)^{2}+r\\)
Determine the value of \\(m+n+p+q+r\\). | 28 | 23.4375 |
30,121 | Two standard decks of cards are combined, making a total of 104 cards (each deck contains 13 ranks and 4 suits, with all combinations unique within its own deck). The decks are randomly shuffled together. What is the probability that the top card is an Ace of $\heartsuit$? | \frac{1}{52} | 10.15625 |
30,122 | Olya, after covering one-fifth of the way from home to school, realized that she forgot her notebook. If she does not return for it, she will reach school 6 minutes before the bell rings, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take? | 20 | 2.34375 |
30,123 | Given that $F$ is the focus of the parabola $4y^{2}=x$, and points $A$ and $B$ are on the parabola and located on both sides of the $x$-axis. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = 15$ (where $O$ is the origin), determine the minimum value of the sum of the areas of $\triangle ABO$ and $\triangle AFO$. | \dfrac{ \sqrt{65}}{2} | 2.34375 |
30,124 | Given the function $f(x)=2\sqrt{3}\sin^2{x}-\sin\left(2x-\frac{\pi}{3}\right)$,
(Ⅰ) Find the smallest positive period of the function $f(x)$ and the intervals where $f(x)$ is monotonically increasing;
(Ⅱ) Suppose $\alpha\in(0,\pi)$ and $f\left(\frac{\alpha}{2}\right)=\frac{1}{2}+\sqrt{3}$, find the value of $\sin{\alpha}$;
(Ⅲ) If $x\in\left[-\frac{\pi}{2},0\right]$, find the maximum value of the function $f(x)$. | \sqrt{3}+1 | 17.1875 |
30,125 | An ant starts at one vertex of an octahedron and moves along the edges according to a similar rule: at each vertex, the ant chooses one of the four available edges with equal probability, and all choices are independent. What is the probability that after six moves, the ant ends at the vertex exactly opposite to where it started?
A) $\frac{1}{64}$
B) $\frac{1}{128}$
C) $\frac{1}{256}$
D) $\frac{1}{512}$ | \frac{1}{128} | 24.21875 |
30,126 | Given the polynomial expression \( x^4 - 61x^2 + 60 \), for how many integers \( x \) is the expression negative. | 12 | 60.15625 |
30,127 | Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \(C\) to the lower right-hand corner \(D\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagonally. Each domino covers exactly two of the unit squares on the grid. Determine how many distinct arrangements are possible for Jessica to achieve this, assuming the path only moves right or down. | 35 | 7.8125 |
30,128 | The administrator accidentally mixed up the keys for 10 rooms. If each key can only open one room, what is the maximum number of attempts needed to match all keys to their corresponding rooms? | 45 | 53.90625 |
30,129 | Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | 125 | 8.59375 |
30,130 | Given the sequence ${a_n}$ where $a_{1}= \frac {3}{2}$, and $a_{n}=a_{n-1}+ \frac {9}{2}(- \frac {1}{2})^{n-1}$ (for $n\geq2$).
(I) Find the general term formula $a_n$ and the sum of the first $n$ terms $S_n$;
(II) Let $T_{n}=S_{n}- \frac {1}{S_{n}}$ ($n\in\mathbb{N}^*$), find the maximum and minimum terms of the sequence ${T_n}$. | -\frac{7}{12} | 32.03125 |
30,131 | Find the least positive integer $n$ , such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$ .
- There is a real number $\xi$ with $P(\xi)=0$ . | 2014 | 0.78125 |
30,132 | Six people form a circle to play a coin-tossing game (the coin is fair). Each person tosses a coin once. If the coin shows tails, the person has to perform; if it shows heads, they do not have to perform. What is the probability that no two performers (tails) are adjacent? | 9/32 | 3.90625 |
30,133 | What is the ratio of the volume of cone $C$ to the volume of cone $D$? Express your answer as a common fraction. Cone $C$ has a radius of 16.4 and height of 30.5, while cone $D$ has a radius of 30.5 and height of 16.4. | \frac{164}{305} | 62.5 |
30,134 | Consider a sequence of positive real numbers where \( a_1, a_2, \dots \) satisfy
\[ a_n = 9a_{n-1} - n \]
for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{17}{64} | 8.59375 |
30,135 | Given that $x > 0$, $y > 0$, and ${\!\!}^{2x+2y}=2$, find the minimum value of $\frac{1}{x}+\frac{1}{y}$. | 3 + 2\sqrt{2} | 0 |
30,136 | Given that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $|\overrightarrow{a}|=2$, and $\overrightarrow{a} \bot (\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$. | -1 | 0.78125 |
30,137 | Consider the equation $p = 15q^2 - 5$. Determine the value of $q$ when $p = 40$.
A) $q = 1$
B) $q = 2$
C) $q = \sqrt{3}$
D) $q = \sqrt{6}$ | q = \sqrt{3} | 85.9375 |
30,138 | Find the last two digits of $\tbinom{200}{100}$ . Express the answer as an integer between $0$ and $99$ . (e.g. if the last two digits are $05$ , just write $5$ .) | 20 | 52.34375 |
30,139 | Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$. | 19.5 | 3.90625 |
30,140 | Given \( x \in \mathbf{R} \), the sum of the maximum and minimum values of the function \( f(x)=\max \left\{\sin x, \cos x, \frac{\sin x+\cos x}{\sqrt{2}}\right\} \) is equal to? | 1 - \frac{\sqrt{2}}{2} | 14.0625 |
30,141 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 96.875 |
30,142 | In a mathematics competition, there are four problems carrying 1, 2, 3, and 4 marks respectively. For each question, full score is awarded if the answer is correct; otherwise, 0 mark will be given. The total score obtained by a contestant is multiplied by a time bonus of 4, 3, 2, or 1 according to the time taken to solve the problems. An additional bonus score of 20 will be added after multiplying by the time bonus if one gets all four problems correct. How many different final scores are possible? | 25 | 2.34375 |
30,143 |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to \(x\), and the lateral side is equal to 12. On the ray \(AC\), point \(D\) is marked such that \(AD = 24\). From point \(D\), a perpendicular \(DE\) is dropped to the line \(AB\). Find \(x\) given that \(BE = 6\). | 18 | 15.625 |
30,144 | Given the inequality $ax^{2}+bx+c \gt 0$ with the solution set $\{x\left|\right.1 \lt x \lt 2\}$, find the solution set of the inequality $cx^{2}+bx+a \gt 0$ in terms of $x$. When studying the above problem, Xiaoming and Xiaoning respectively came up with the following Solution 1 and Solution 2:
**Solution 1:** From the given information, the roots of the equation $ax^{2}+bx+c=0$ are $1$ and $2$, and $a \lt 0$. By Vieta's formulas, we have $\left\{\begin{array}{c}1+2=-\frac{b}{a},\\ 1×2=\frac{c}{a},\end{array}\right.\left\{\begin{array}{c}b=-3a,\\ c=2a,\end{array}\right.$. Therefore, the inequality $cx^{2}+bx+a \gt 0$ can be transformed into $2ax^{2}-3ax+a \gt 0$, which simplifies to $\left(x-1\right)\left(2x-1\right) \lt 0$. Solving this inequality gives $\frac{1}{2}＜x＜1$, so the solution set of the inequality $cx^{2}+bx+a \gt 0$ is $\{x|\frac{1}{2}＜x＜1\}$.
**Solution 2:** From $ax^{2}+bx+c \gt 0$, we get $c{(\frac{1}{x})}^{2}+b\frac{1}{x}+a＞0$. Let $y=\frac{1}{x}$, then $\frac{1}{2}＜y＜1$. Therefore, the solution set of the inequality $cx^{2}+bx+a \gt 0$ is $\{x|\frac{1}{2}＜x＜1\}$.
Based on the above solutions, answer the following questions:
$(1)$ If the solution set of the inequality $\frac{k}{x+a}+\frac{x+c}{x+b}＜0$ is $\{x\left|\right.-2 \lt x \lt -1$ or $2 \lt x \lt 3\}$, write down the solution set of the inequality $\frac{kx}{ax+1}+\frac{cx+1}{bx+1}＜0$ directly.
$(2)$ If real numbers $m$ and $n$ satisfy the equations $\left(m+1\right)^{2}+\left(4m+1\right)^{2}=1$ and $\left(n+1\right)^{2}+\left(n+4\right)^{2}=n^{2}$, and $mn\neq 1$, find the value of $n^{3}+m^{-3}$. | -490 | 12.5 |
30,145 | Given that $a\in\{0,1,2\}$ and $b\in\{-1,1,3,5\}$, find the probability that the function $f(x)=ax^{2}-2bx$ is increasing on the interval $(1,+\infty)$. | \frac{5}{12} | 35.15625 |
30,146 | What is the sum of the interior numbers of the eighth row of Pascal's Triangle? | 126 | 67.96875 |
30,147 | The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? | \sqrt{3} | 36.71875 |
30,148 | The hypotenuse of a right triangle measures $8\sqrt{2}$ inches and one angle is $45^{\circ}$. Calculate both the area and the perimeter of the triangle. | 16 + 8\sqrt{2} | 92.1875 |
30,149 | In the fictional country of Novaguard, they use the same twelve-letter Rotokas alphabet for their license plates, which are also five letters long. However, for a special series of plates, the following rules apply:
- The plate must start with either P or T.
- The plate must end with R.
- The letter U cannot appear anywhere on the plate.
- No letters can be repeated.
How many such special license plates are possible? | 1440 | 57.03125 |
30,150 | In an Olympic 100-meter final, there are 10 sprinters competing, among which 4 are Americans. The gold, silver, and bronze medals are awarded to first, second, and third place, respectively. Calculate the number of ways the medals can be awarded if at most two Americans are to receive medals. | 588 | 0.78125 |
30,151 | 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 $x = 2$, $g(2) = 4$, find the sum of the values of $f(k)$ from $k=1$ to $k=22$. | -24 | 1.5625 |
30,152 | Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not below the $x$-axis? Express your answer as a common fraction. | \frac{1}{2} | 25 |
30,153 | Luis wrote the sequence of natural numbers, that is,
$$
1,2,3,4,5,6,7,8,9,10,11,12, \ldots
$$
When did he write the digit 3 for the 25th time? | 134 | 8.59375 |
30,154 | Find the coefficient of the third term and the constant term in the expansion of $\left(x^3 + \frac{2}{3x^2}\right)^5$. | \frac{80}{27} | 93.75 |
30,155 | In Tuanjie Village, a cement road of $\frac {1}{2}$ kilometer long is being constructed. On the first day, $\frac {1}{10}$ of the total length was completed, and on the second day, $\frac {1}{5}$ of the total length was completed. What fraction of the total length is still unfinished? | \frac {7}{10} | 74.21875 |
30,156 | What number is formed from five consecutive digits (not necessarily in order) such that the number formed by the first two digits, when multiplied by the middle digit, gives the number formed by the last two digits? (For example, if we take the number 12896, 12 multiplied by 8 gives 96. However, since 1, 2, 6, 8, 9 are not consecutive digits, this example is not suitable as a solution.) | 13452 | 48.4375 |
30,157 | Given triangle $ABC$ with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Points $D$, $E$, $F$ lie on sides $[AB]$, $[AC]$, $[BC]$ respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Find the area of triangle $H_1H_2H_3$, where $H_1$, $H_2$, $H_3$ are the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$. | 96 | 7.03125 |
30,158 | Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, determine the value of $g(x) = f(x) + 2$ when $x = -1$. | -1 | 100 |
30,159 | Jacqueline has 200 liters of a chemical solution. Liliane has 30% more of this chemical solution than Jacqueline, and Alice has 15% more than Jacqueline. Determine the percentage difference in the amount of chemical solution between Liliane and Alice. | 13.04\% | 34.375 |
30,160 | In triangle $PQR$, $\angle Q=90^\circ$, $PQ=9$ and $QR=12$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=6$, then what is the length of $PS$? | 10 | 3.125 |
30,161 | Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs. | 54 | 0 |
30,162 | The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks.
How many penalty kicks are needed to ensure that every player has had a chance to shoot against each of the goalies? | 92 | 21.875 |
30,163 | Given the function $f(x)=(a+ \frac {1}{a})\ln x-x+ \frac {1}{x}$, where $a > 0$.
(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of values for $a$;
(II) Let $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$, does $M(a)$ have a maximum value? If it exists, find its maximum value; if not, explain why. | \frac {4}{e} | 20.3125 |
30,164 | Jack and Jill run a 12 km circuit. First, they run 7 km to a certain point and then the remaining 5 km back to the starting point by different, uneven routes. Jack has a 12-minute head start and runs at the rate of 12 km/hr uphill and 15 km/hr downhill. Jill runs 14 km/hr uphill and 18 km/hr downhill. How far from the turning point are they when they pass each other, assuming their downhill paths are the same but differ in uphill routes (in km)?
A) $\frac{226}{145}$
B) $\frac{371}{145}$
C) $\frac{772}{145}$
D) $\frac{249}{145}$
E) $\frac{524}{145}$ | \frac{772}{145} | 10.9375 |
30,165 | In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $\sin A = \sin B = -\cos C$.
$(1)$ Find the sizes of angles $A$, $B$, and $C$;
$(2)$ If the length of the median $AM$ on side $BC$ is $\sqrt{7}$, find the area of $\triangle ABC$. | \sqrt{3} | 38.28125 |
30,166 | The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | 20 | 14.0625 |
30,167 | A standard die is rolled eight times. What is the probability that the product of all eight rolls is divisible by 4? | \frac{247}{256} | 10.15625 |
30,168 | Given the set $M=\{a, b, -(a+b)\}$, where $a\in \mathbb{R}$ and $b\in \mathbb{R}$, and set $P=\{1, 0, -1\}$. If there is a mapping $f:x \to x$ that maps element $x$ in set $M$ to element $x$ in set $P$ (the image of $x$ under $f$ is still $x$), then the set $S$ formed by the points with coordinates $(a, b)$ has \_\_\_\_\_\_\_\_\_\_\_ subsets. | 64 | 51.5625 |
30,169 | Let $a \star b = \frac{\sqrt{a^2+b}}{\sqrt{a^2 - b}}$. If $y \star 15 = 5$, find $y$. | \frac{\sqrt{65}}{2} | 58.59375 |
30,170 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = \sqrt{2}$, $|\overrightarrow{b}| = 2$, and $(\overrightarrow{a} - \overrightarrow{b}) \perp \overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 98.4375 |
30,171 | Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \(n\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring in the remaining shots. Find the minimum value for \(n\). | 52 | 8.59375 |
30,172 | In triangle $ABC$, $AB = 5$, $BC = 12$, and $AC = 13$. Let $BM$ be the median from vertex $B$ to side $AC$. If $BM = m \sqrt{2}$, then find $m$. | \frac{13}{2} | 4.6875 |
30,173 | In $\triangle ABC$, $A=60^{\circ}$, $AC=4$, $BC=2\sqrt{3}$, the area of $\triangle ABC$ equals \_\_\_\_\_\_. | 2\sqrt{3} | 83.59375 |
30,174 | Let $X=\{1,2,3,...,10\}$ . Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$ . | 2186 | 10.9375 |
30,175 | Let $r(x)$ be a monic quartic polynomial such that $r(1) = 5,$ $r(2) = 8,$ $r(3) = 13,$ and $r(4) = 20.$ Find $r(5).$ | 53 | 0 |
30,176 | A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . | 324 | 60.9375 |
30,177 | A square with side length $x$ is inscribed in a right triangle with sides of length $6$, $8$, and $10$ so that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in the same triangle, but one side of this square lies along the hypotenuse of the triangle. What is $\frac{x}{y}$?
A) $\frac{109}{175}$
B) $\frac{120}{190}$
C) $\frac{111}{175}$
D) $\frac{100}{160}$ | \frac{111}{175} | 16.40625 |
30,178 | Consider the following multiplicative magic square, where each row, column, and diagonal product equals the same value:
$\begin{tabular}{|c|c|c|} \hline 75 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 3 \\ \hline \end{tabular}$
All entries are positive integers. Determine the possible values for $h$ and their sum.
A) 50
B) 75
C) 100
D) 125
E) 150 | 150 | 16.40625 |
30,179 |
A math competition problem has probabilities of being solved independently by person \( A \), \( B \), and \( C \) as \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) respectively, where \( a \), \( b \), and \( c \) are positive integers less than 10. When \( A \), \( B \), and \( C \) work on the problem simultaneously and independently, the probability that exactly one of them solves the problem is \( \frac{7}{15} \). Determine the probability that none of the three persons solve the problem. | 4/15 | 69.53125 |
30,180 | Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$ . Find $\log_a b$ . | -1 | 24.21875 |
30,181 | Given a deck consisting of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning set. | \frac{1}{14} | 42.96875 |
30,182 | On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts the number of vehicles that pass in one hour. Each vehicle is 5 meters long and they can travel at any speed. Let \( N \) be the maximum whole number of vehicles that can pass the sensor in one hour. Find the quotient when \( N \) is divided by 10. | 400 | 7.03125 |
30,183 | When manufacturing a steel cable, it was found that the cable has the same length as the curve given by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=-22
\end{array}\right.
$$
Find the length of the cable. | 4 \pi \sqrt{\frac{83}{3}} | 0 |
30,184 | Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. Given that $\sin B - \sin C = \sin (A-C)$.
$(1)$ Find the value of $A$.
$(2)$ If $AB=2$, $AC=5$, and the medians $AM$ and $BN$ on sides $BC$ and $AC$ intersect at point $P$, find the cosine value of $\angle MPN$. | \frac{4\sqrt{91}}{91} | 15.625 |
30,185 | A four-digit number has the following properties:
(a) It is a perfect square;
(b) Its first two digits are equal
(c) Its last two digits are equal.
Find all such four-digit numbers. | 7744 | 96.875 |
30,186 | Write the digits from 0 to 9 in a line, in any order you choose. On the line below, combine the neighboring digits to form nine new numbers, and sum these numbers as in the example below:
| 2 | | 1 | | 3 | | 7 | | 4 | | 9 | | 5 | | 8 | | 0 | | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| | 21 | | 13 | | 37 | | 74 | | 49 | | 95 | | 58 | | 80 | | 06 | |
| 1 | | | | | | | | | | | | | | | | | | |
What is the maximum sum that can be obtained in this way?
A) 506
B) 494
C) 469
D) 447
E) 432 | 494 | 70.3125 |
30,187 | In the arithmetic sequence $\{a_n\}$, it is given that $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \cdot a_{2013} < 0$, calculate the largest natural number n that satisfies S_n > 0. | 4024 | 11.71875 |
30,188 | Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\] | \sqrt{69} | 3.125 |
30,189 | Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | \dfrac{1}{2} | 59.375 |
30,190 | Suppose a sequence starts with 1254, 2547, 5478, and ends with 4781. Let $T$ be the sum of all terms in this sequence. Find the largest prime factor that always divides $T$. | 101 | 19.53125 |
30,191 | Given the equation $x^2 + y^2 = |x| + 2|y|$, calculate the area enclosed by the graph of this equation. | \frac{5\pi}{4} | 27.34375 |
30,192 | Given the function $f(x) = ax^3 + (a-1)x^2 + 27(a-2)x + b$, its graph is symmetric about the origin. Determine the monotonicity of $f(x)$ on the interval $[-4, 5]$ and find the maximum and minimum values of $f(x)$ on this interval. | -54 | 44.53125 |
30,193 | A company needs to renovate its new office building. If the renovation is done solely by Team A, it would take 18 weeks, and if done solely by Team B, it would take 12 weeks. The result of the bidding is that Team A will work alone for the first 3 weeks, and then both Team A and Team B will work together. The total renovation cost is 4000 yuan. If the payment for the renovation is based on the amount of work completed by each team, how should the payment be distributed? | 2000 | 44.53125 |
30,194 | Hou Yi shot three arrows at each of three targets. On the first target, he scored 29 points, and on the second target, he scored 43 points. How many points did he score on the third target? | 36 | 3.125 |
30,195 | Mrs. Delta's language class has 52 students, each with unique initials, and no two students have initials that are alphabetically consecutive (e.g., AB cannot follow AC directly). Assuming Y is considered a consonant, what is the probability of randomly picking a student whose initials (each first and last name starts with the same letter, like AA, BB) are both vowels? Express your answer as a common fraction. | \frac{5}{52} | 30.46875 |
30,196 | Let \( [x] \) denote the greatest integer less than or equal to the real number \( x \). Consider a sequence \( \{a_n\} \) defined by \( a_1 = 1 \) and \( a_n = \left[\sqrt{n a_{n-1}}\right] \). Find the value of \( a_{2017} \). | 2015 | 14.84375 |
30,197 | Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards in a row. Each leopard has a unique collar color. The two shortest leopards have inferiority complexes and demand to be placed at the ends of the row. Moreover, the two tallest leopards, which have a superiority complex, insist on being placed next to each other somewhere in the row. How many ways can Marius arrange the leopards under these conditions? | 2880 | 18.75 |
30,198 | In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum. | 81380 | 62.5 |
30,199 | Let \( x = 19.\overline{87} \). If \( 19.\overline{87} = \frac{a}{99} \), find \( a \).
If \( \frac{\sqrt{3}}{b \sqrt{7} - \sqrt{3}} = \frac{2 \sqrt{21} + 3}{c} \), find \( c \).
If \( f(y) = 4 \sin y^{\circ} \) and \( f(a - 18) = b \), find \( b \). | 25 | 2.34375 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.