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40.3k
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5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
17,900 |
In the geometric sequence $\{a_n\}$, $(a_1 \cdot a_2 \cdot a_3 = 27)$, $(a_2 + a_4 = 30)$, $(q > 0)$
Find:
$(1)$ $a_1$ and the common ratio $q$;
$(2)$ The sum of the first $6$ terms $(S_6)$.
|
364
| 24.21875 |
17,901 |
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$.
|
1800
| 1.5625 |
17,902 |
In plane $\alpha$, there are four points, and in plane $\beta$, there are five points. From these nine points, the maximum number of planes that can be determined by any three points is ; the maximum number of tetrahedrons that can be determined by any four points is . (Answer with numbers)
|
120
| 27.34375 |
17,903 |
Let \( a > 0 \). The function \( f(x) = x + \frac{100}{x} \) attains its minimum value \( m_{1} \) on the interval \( (0, a] \), and its minimum value \( m_{2} \) on the interval \( [a, +\infty) \). If \( m_{1} m_{2} = 2020 \), find the value of \( a \).
|
100
| 50 |
17,904 |
The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Find the minimum value of \( m \).
|
33
| 2.34375 |
17,905 |
The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?
|
48
| 7.03125 |
17,906 |
Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$.
|
12
| 77.34375 |
17,907 |
Given the function $f(x)=2\sqrt{3}\cos^2\left(\frac{\pi}{2}+x\right)-2\sin(\pi+x)\cos x-\sqrt{3}$.
$(1)$ Find the extreme values of $f(x)$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.
$(2)$ If $f(x_0-\frac{\pi}{6})=\frac{10}{13}$, where $x_0\in\left[\frac{3\pi}{4}, \pi\right]$, find the value of $\sin 2x_0$.
|
-\frac{5+12\sqrt{3}}{26}
| 21.875 |
17,908 |
Given a sequence: $2016$, $2017$, $1$, $-2016$, $-2017$, $…$, the characteristic of this sequence is that starting from the second term, each term is equal to the sum of the term before and after it. Find the sum of the first $2017$ terms of the sequence.
|
2016
| 79.6875 |
17,909 |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \)
Determine the minimum value of the real number \( A \) such that the inequality \( \left| a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \right| \leqslant A ( a_{1} - a_{n} ) \) holds, and provide a justification for this value.
|
1/2
| 71.09375 |
17,910 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=1+\dfrac{\sqrt{2}}{2}t \\ y=2+\dfrac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-negative half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=6\sin\theta$.
- (I) Find the standard equation of circle $C$ in Cartesian coordinates;
- (II) If point $P(l,2)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA| + |PB|$.
|
2\sqrt{7}
| 97.65625 |
17,911 |
Given the expression $\frac{810 \times 811 \times 812 \times \cdots \times 2010}{810^{n}}$ is an integer, find the maximum value of $n$.
|
149
| 5.46875 |
17,912 |
Let $P$ be a moving point on the line $3x+4y+3=0$, and through point $P$, draw two tangents to the circle $C$: $x^{2}+y^{2}-2x-2y+1=0$, with the points of tangency being $A$ and $B$, respectively. Find the minimum value of the area of quadrilateral $PACB$.
|
\sqrt{3}
| 50.78125 |
17,913 |
12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is ______.
|
28
| 0 |
17,914 |
A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded?
|
33.33\%
| 56.25 |
17,915 |
For the set \( \{1, 2, 3, \ldots, 8\} \) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \( \{1, 3, 4, 7, 8\} \) would be \( 8-7+4-3+1=3 \) and for \( \{8\} \) it is \( 8 \). Find the sum of all such alternating sums for \( n=8 \).
|
1024
| 79.6875 |
17,916 |
If $\lceil{\sqrt{x}}\rceil=20$, how many possible integer values of $x$ are there?
|
39
| 85.15625 |
17,917 |
The equation of the line joining the complex numbers $-1 + 2i$ and $2 + 3i$ can be expressed in the form
\[az + b \overline{z} = d\]for some complex numbers $a$, $b$, and real number $d$. Find the product $ab$.
|
10
| 48.4375 |
17,918 |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[ f(x)f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right) \]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s$.
|
\frac{4047}{2}
| 29.6875 |
17,919 |
Given $\overrightarrow{a}=(-3,4)$, $\overrightarrow{b}=(5,2)$, find $|\overrightarrow{a}|$, $|\overrightarrow{b}|$, and $\overrightarrow{a}\cdot \overrightarrow{b}$.
|
-7
| 42.96875 |
17,920 |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate the value of $f(-\frac{{5π}}{{12}})$.
|
\frac{\sqrt{3}}{2}
| 12.5 |
17,921 |
In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the x-axis as the polar axis. Given circle C: ρ = 2cosθ - 2sinθ, and the parametric equation of line l is x = t, y = -1 + 2√2t (t is the parameter). Line l intersects with circle C at points M and N, and point P is any point on circle C that is different from M and N.
1. Write the rectangular coordinate equation of C and the general equation of l.
2. Find the maximum area of triangle PMN.
|
\frac{10\sqrt{5}}{9}
| 29.6875 |
17,922 |
Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd.
|
\frac{7}{16}
| 62.5 |
17,923 |
Given a sector with a central angle of 120° and an area of $3\pi$, which is used as the lateral surface area of a cone, find the surface area and volume of the cone.
|
\frac{2\sqrt{2}\pi}{3}
| 27.34375 |
17,924 |
Given that the ellipse $C\_2$ passes through the two foci and the two endpoints of the minor axis of the ellipse $C\_1$: $\frac{x^{2}}{14} + \frac{y^{2}}{9} = 1$, find the eccentricity of the ellipse $C\_2$.
|
\frac{2}{3}
| 71.09375 |
17,925 |
What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$?
|
110 + \sqrt{1818}
| 0 |
17,926 |
Given vectors $\overrightarrow {a}=(\sqrt{3}\sin x,\sin x)$ and $\overrightarrow {b}=(\cos x,\sin x)$, where $x \in (0, \frac{\pi}{2})$,
1. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x$;
2. Let the function $f(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{b}$, determine the maximum value of $f(x)$.
|
\frac{5}{2}
| 77.34375 |
17,927 |
The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$.
|
136
| 67.96875 |
17,928 |
Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$.
|
\sqrt{26}-3
| 31.25 |
17,929 |
Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February.
|
142.00
| 7.03125 |
17,930 |
If $$ \sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c} $$ where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$
|
80
| 72.65625 |
17,931 |
Patrick tosses four four-sided dice, each numbered $1$ through $4$ . What's the probability their product is a multiple of four?
|
\frac{13}{16}
| 24.21875 |
17,932 |
A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$
|
\frac{85}{7}
| 50.78125 |
17,933 |
Solve the following equations:
(1) $\frac {1}{2}x - 2 = 4 + \frac {1}{3}x$
(2) $\frac {x-1}{4} - 2 = \frac {2x-3}{6}$
(3) $\frac {1}{3}[x - \frac {1}{2}(x-1)] = \frac {2}{3}(x - \frac {1}{2})$
(4) $\frac {x}{0.7} - \frac {0.17-0.2x}{0.03} = 1$
|
\frac {14}{17}
| 4.6875 |
17,934 |
Given $$f(x)= \begin{cases} f(x+1), x < 4 \\ ( \frac {1}{2})^{x}, x\geq4\end{cases}$$, find $f(\log_{2}3)$.
|
\frac {1}{24}
| 76.5625 |
17,935 |
The slope angle of the tangent line to the curve y=\frac{1}{3}x^{3} at x=1 is what value?
|
\frac{\pi}{4}
| 75.78125 |
17,936 |
Given that 30 balls are put into four boxes A, B, C, and D, such that the sum of the number of balls in A and B is greater than the sum of the number of balls in C and D, find the total number of possible ways.
|
2600
| 0 |
17,937 |
The area enclosed by the curves $y=e^{x}$, $y=e^{-x}$, and the line $x=1$ is $e^{1}-e^{-1}$.
|
e+e^{-1}-2
| 3.125 |
17,938 |
If $x$ is a real number and $\lceil x \rceil = 9,$ how many possible values are there for $\lceil x^2 \rceil$?
|
17
| 89.0625 |
17,939 |
A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?
|
21
| 89.84375 |
17,940 |
In the sequence $\{{a_{n}}\}$, the adjacent terms ${a_{n}}$ and ${a_{n+1}}$ are the roots of the equation ${x^{2}}+3nx+{{b_{n}}=0}$. Given that ${{a_{10}}=-17}$, find the value of ${{b_{51}}}$.
|
5840
| 4.6875 |
17,941 |
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(x+3) = -f(1-x)$. If $f(3) = 2$, then $f(2013) = \ $.
|
-2
| 30.46875 |
17,942 |
You are standing at the edge of a river which is $1$ km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is $1$ km . You can swim at $2$ km/hr and walk at $3$ km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel).
|
\frac{2 + \sqrt{5}}{6}
| 11.71875 |
17,943 |
On a standard dice, the sum of the numbers of pips on opposite faces is always 7. Four standard dice are glued together as shown. What is the minimum number of pips that could lie on the whole surface?
A) 52
B) 54
C) 56
D) 58
E) 60
|
58
| 8.59375 |
17,944 |
Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:
1. The Tigers won more than the Eagles.
2. The Patriots won more than the Cubs, but fewer than the Mounties.
3. The Cubs won more than 15 games.
4. The Falcons won more games than the Eagles but fewer than the Mounties.
How many games did the Patriots win?
Graph data:
Teams (in unknown order): 10, 18, 22, 27, 33 games.
|
27
| 29.6875 |
17,945 |
In right triangle $ABC$, $\angle C=90^{\circ}$, $AB=9$, $\cos B=\frac{2}{3}$, calculate the length of $AC$.
|
3\sqrt{5}
| 69.53125 |
17,946 |
A box contains 4 cards, each with one of the following functions defined on \\(R\\): \\(f_{1}(x)={x}^{3}\\), \\(f_{2}(x)=|x|\\), \\(f_{3}(x)=\sin x\\), \\(f_{4}(x)=\cos x\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new function, the probability that the resulting function is an odd function is \_\_\_\_\_.
|
\dfrac{2}{3}
| 85.9375 |
17,947 |
A basketball team has 20 players. The coach needs to choose a starting lineup consisting of one point guard and 7 other players (these players are interchangeable). How many different starting lineups can the coach choose?
|
1007760
| 0 |
17,948 |
A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone?
|
\frac{1}{3}
| 47.65625 |
17,949 |
Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$
|
128
| 53.125 |
17,950 |
Let set $A=\{x \mid |x-2| \leq 2\}$, and $B=\{y \mid y=-x^2, -1 \leq x \leq 2\}$, then $A \cap B=$ ?
|
\{0\}
| 91.40625 |
17,951 |
In the ancient Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art," there is a problem called "Division of Grains in a Granary": When a granary is opened to receive grain, a person brings 1534 stones of rice. Upon inspection, it is found that there are grains of wheat mixed in the rice. After taking a handful of rice as a sample, it is found that out of 254 grains in the sample, 28 are grains of wheat. What is the approximate amount of wheat mixed in this batch of rice in stones (rounded to one decimal place)?
|
169.1
| 0 |
17,952 |
Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible.
|
90
| 27.34375 |
17,953 |
In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$.
|
\frac{2\sqrt{5}}{5}
| 13.28125 |
17,954 |
Two different natural numbers are selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.
|
\frac{3}{4}
| 5.46875 |
17,955 |
Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \(10 \, \text{m/s}\) and Betty runs at \(15 \, \text{m/s}\). Betty gives Alice a 40 meter head start before they both start running. After 15 minutes, how many times will they have passed each other?
(a) 9
(b) 10
(c) 11
(d) 12
|
11
| 57.03125 |
17,956 |
The sequence $1, 4, 5, 16, 17, 20, 21 \cdots$ consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $150^{\mbox{th}}$ term of this sequence.
|
16660
| 7.8125 |
17,957 |
Given that $0 < α < \dfrac {π}{2}$, and $\cos ( \dfrac {π}{3}+α)= \dfrac {1}{3}$, find the value of $\cos α$.
|
\dfrac {2 \sqrt {6}+1}{6}
| 0 |
17,958 |
In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively, and $\frac{\cos C}{\cos B} = \frac{3a-c}{b}$.
(1) Find the value of $\sin B$;
(2) If $b = 4\sqrt{2}$ and $a = c$, find the area of $\triangle ABC$.
|
8\sqrt{2}
| 63.28125 |
17,959 |
Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$ . The minimal possible area of triangle $ABC$ can be expressed as $\frac{\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2$ dividing $m$ . Compute $100m+n$ .
*Proposed by Yannick Yao*
|
8302
| 2.34375 |
17,960 |
What is the least possible value of \((x+1)(x+2)(x+3)(x+4)+2023\) where \(x\) is a real number?
|
2022
| 87.5 |
17,961 |
A circle $U$ has a circumference of $18\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $UAV$ is $45^{\circ}$, what is the length, in inches, of segment $AV$?
|
9\sqrt{2 - \sqrt{2}}
| 7.03125 |
17,962 |
Given a sequence $\{a_n\}$ with $a_1=1$ and $a_{n+1}= \frac{2a_n}{a_n+2}$, find the value of $a_{10}$.
|
\frac{2}{11}
| 99.21875 |
17,963 |
Let the function \( g : \mathbb{R} \to \mathbb{R} \) satisfy the equation
\[ g(x) + 2g(2 - x) = 4x^3 - x^2 \]
for all \( x \). Find \( g(5) \).
|
-\frac{709}{3}
| 20.3125 |
17,964 |
For any real number $x$, $\lfloor x \rfloor$ represents the largest integer not exceeding $x$, for example: $\lfloor 2 \rfloor = 2$, $\lfloor 3.2 \rfloor = 3$. Then $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \ldots + \lfloor \log_{2}64 \rfloor =$ ?
|
264
| 57.03125 |
17,965 |
Some students are required to stand in lines: On June 1st, each column has 15 people; on June 2nd, everyone stands in one column; on June 3rd, each column has one person; on June 4th, each column has 6 people; and so on until June 12th, with a different number of people per column each day. However, from June 13th onwards, it is impossible to have a different number of people per column than previously. How many students are there in total?
|
60
| 27.34375 |
17,966 |
John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the number of $6 notebooks. Solve for x.
|
14
| 4.6875 |
17,967 |
Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m.
|
\frac{3}{5}
| 3.125 |
17,968 |
For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$?
|
125
| 69.53125 |
17,969 |
The function $g(x)$ satisfies
\[ 3g(x) + 4g(1 - x) = 6x^2 \] for all real numbers \(x\). Find $g(5)$.
|
\frac{-66}{7}
| 0 |
17,970 |
Circles with centers at $(5,5)$ and $(20,15)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?
|
5 \sqrt{13} - 20
| 43.75 |
17,971 |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p + q \).
|
2734
| 1.5625 |
17,972 |
When three numbers are added two at a time, the sums are 35, 47, and 52. What is the sum of all three numbers, and what is their product?
|
9600
| 54.6875 |
17,973 |
If $$log_{4}(a+4b)=log_{2}2 \sqrt {ab}$$, find the minimum value of $a+b$.
|
\frac{9}{4}
| 33.59375 |
17,974 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{\cos C}{\cos B}= \frac{2a-c}{b}$.
(1) Find $B$;
(2) If $\tan \left(A+ \frac{π}{4}\right) =7$, find the value of $\cos C$.
|
\frac{-4+3\sqrt{3}}{10}
| 0 |
17,975 |
Distinct ways to distribute $7$ identical balls into $4$ distinct boxes such that no box is empty.
|
20
| 79.6875 |
17,976 |
Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$.
|
10
| 0 |
17,977 |
A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-digit numbers is 945. What is this three-digit number?
|
327
| 19.53125 |
17,978 |
In the diagram, all rows, columns, and diagonals have the sum 12. What is the sum of the four corner numbers?
|
16
| 14.84375 |
17,979 |
Triangle $ABC$ is a right triangle with $AB = 6$, $BC = 8$, and $AC = 10$. Point $D$ is on line $\overline{BC}$ such that $\overline{AD}$ bisects angle $BAC$. The inscribed circles of $\triangle ADB$ and $\triangle ADC$ have radii $r_1$ and $r_2$, respectively. What is $r_1/r_2$?
A) $\frac{24}{35}$
B) $\frac{35}{24}$
C) $\frac{15}{28}$
D) $\frac{7}{15}$
|
\frac{24}{35}
| 36.71875 |
17,980 |
Three male students and two female students stand in a row. The total number of arrangements where the female students do not stand at the ends is given by what total count.
|
36
| 50.78125 |
17,981 |
A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (both horizontal and vertical) and containing at least 7 black squares, can be drawn on the checkerboard?
|
140
| 14.0625 |
17,982 |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\]
|
\frac{89}{9}
| 4.6875 |
17,983 |
Given the function $y=\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$.
|
\frac{\pi}{12}
| 7.03125 |
17,984 |
Rectangle $EFGH$ has an area of $4032$. An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of rectangle $EFGH$.
|
8\sqrt{2016}
| 0 |
17,985 |
Determine the largest value of $m$ such that $5x^2 + mx + 45$ can be factored as the product of two linear factors with integer coefficients.
|
226
| 50 |
17,986 |
What is the sum of all two-digit positive integers whose squares end with the digits 25?
|
495
| 17.96875 |
17,987 |
The coefficient of $x^3$ in the expansion of $(x^2-x-2)^4$ is __________ (fill in the answer with a number).
|
-40
| 32.03125 |
17,988 |
A stock investment increased by 30% in 2006. Starting at this new value, what percentage decrease is needed in 2007 to return the stock to its original price at the beginning of 2006?
|
23.077\%
| 0 |
17,989 |
Let \( f \) be a function on \([0,1]\) satisfying:
1. \( f(0) = 0, f(1) = 1 \).
2. For all \( x, y \in [0, 1] \), with \( x \leq y \), \( f\left(\frac{x+y}{2}\right) = (1-a) f(x) + a f(y) \), where \( a \) is a real number and \( 0 \leq a \leq 1 \). Find \( f\left(\frac{1}{7}\right) \).
|
\frac{1}{7}
| 78.125 |
17,990 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C= \dfrac {\pi}{6}$, $a=1$, $b= \sqrt {3}$, find the measure of $B$.
|
\dfrac {2\pi}{3}
| 40.625 |
17,991 |
If $y=\left(m-1\right)x^{|m|}$ is a direct proportion function, then the value of $m$ is ____.
|
-1
| 91.40625 |
17,992 |
If \( n \) is any integer, \( n^{2}(n^{2}-1) \) is divisible by \( x \). What is \( x \)?
|
12
| 62.5 |
17,993 |
In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given vectors $\overrightarrow{m} = (\cos A, \sin A)$ and $\overrightarrow{n} = (\cos B, -\sin B)$, and $|\overrightarrow{m} - \overrightarrow{n}| = 1$.
(1) Find the degree measure of angle $C$;
(2) If $c=3$, find the maximum area of triangle $ABC$.
|
\frac{3\sqrt{3}}{4}
| 81.25 |
17,994 |
Find the coefficient of $x^3$ in the expansion of $(1-x)^5(3+x)$.
|
-20
| 50.78125 |
17,995 |
Find the smallest solution to the equation \[\frac{1}{x-3} + \frac{1}{x-5} = \frac{5}{x-4}.\]
|
4 - \frac{\sqrt{15}}{3}
| 78.125 |
17,996 |
A school uses a systematic sampling method to conduct a vision test on 50 out of the 800 students in the first year. The 800 students are numbered from 1 to 800 and are evenly divided into 50 groups in ascending order of their numbers, with group numbers from 1 to 50. It is known that the number drawn in the first group is $m$, and the number drawn in the eighth group is $9m$. Find the number drawn in the sixth group.
|
94
| 92.1875 |
17,997 |
Evaluate: $81^2 - (x+9)^2$ where $x=45$.
|
3645
| 57.03125 |
17,998 |
Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder.
|
4\pi
| 27.34375 |
17,999 |
Given the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then shift all points on the graph to the right by $\frac{\pi}{6}$ units, and find the equation of one of the axes of symmetry for the resulting function $g(x)$.
|
\frac{\pi}{3}
| 74.21875 |
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