Unnamed: 0
int64
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40.3k
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ground_truth
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float64
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100
30,200
A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$ , $BC = 12$ , $CA = 13$ . The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$ .
61
0.78125
30,201
The third quartile of the data $13$, $11$, $12$, $15$, $16$, $18$, $21$, $17$ is ______.
17.5
46.09375
30,202
Determine the value of $x^2 + y^2$ if $x - y = 25$ and $xy = 36$. Additionally, find what $x+y$ equals.
\sqrt{769}
10.9375
30,203
How many integer triples $(x,y,z)$ are there such that $\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}$ where $0\leq x < 13$ , $0\leq y <13$ , and $0\leq z< 13$ .
13
70.3125
30,204
Find the number of rationals $\frac{m}{n}$ such that (i) $0 < \frac{m}{n} < 1$ ; (ii) $m$ and $n$ are relatively prime; (iii) $mn = 25!$ .
256
37.5
30,205
Given the sets \( A = \{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B = \{(x, y) \mid |xy| + 1 = |x| + |y| \} \), if the intersection \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).
2 + \sqrt{2}
2.34375
30,206
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight line from point A to some point C in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point C could possibly be located.
\frac{\pi}{4}
7.03125
30,207
In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other intersection point of line \( AF \) with the circumcircle of \( \triangle ACE \); \( P \) is the other intersection point of line \( AD \) with the circumcircle of \( \triangle AMN \). Find the length of \( AP \).
\sqrt{2}
2.34375
30,208
It is given that $x = -2272$ , $y = 10^3+10^2c+10b+a$ , and $z = 1$ satisfy the equation $ax + by + cz = 1$ , where $a, b, c$ are positive integers with $a < b < c$ . Find $y.$
1987
95.3125
30,209
Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$ . An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer. *2022 CCA Math Bonanza Lightning Round 5.2*
500
51.5625
30,210
A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Groups of 4 random numbers represent the results of 4 shots. After randomly simulating, 20 groups of random numbers were generated: 7527   0293   7140   9857   0347   4373   8636   6947   1417   4698 0371   6233   2616   8045   6011   3661   9597   7424   7610   4281 Estimate the probability that the shooter hits the target at least 3 times out of 4 shots based on the data above.
0.75
13.28125
30,211
In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon.
48
20.3125
30,212
In $\triangle ABC$, $AB = 6$, $BC = 10$, $CA = 8$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$.
40
1.5625
30,213
If $cos2α=-\frac{{\sqrt{10}}}{{10}}$, $sin({α-β})=\frac{{\sqrt{5}}}{5}$, and $α∈({\frac{π}{4},\frac{π}{2}})$, $β∈({-π,-\frac{π}{2}})$, then $\alpha +\beta =$____.
-\frac{\pi}{4}
17.1875
30,214
In triangle \( \triangle ABC \), given that \( \sin A = 10 \sin B \sin C \) and \( \cos A = 10 \cos B \cos C \), find the value of \( \tan A \).
-9
12.5
30,215
Given the following four conclusions: \\((1)\\) The center of symmetry of the function \\(f(x)= \dfrac {x-1}{2x+1}\\) is \\((- \dfrac {1}{2},- \dfrac {1}{2})\\); \\((2)\\) If the equation in \\(x\\), \\(x- \dfrac {1}{x}+k=0\\), has no real roots for \\(x \in (0,1)\\), then the range of \\(k\\) is \\(k \geqslant 2\\); \\((3)\\) Given that point \\(P(a,b)\\) and point \\(Q(1,0)\\) are on opposite sides of the line \\(2x-3y+1=0\\), then \\(3b-2a > 1\\); \\((4)\\) If the graph of the function \\(f(x)=\sin (2x- \dfrac {\pi}{3})\\) is shifted to the right by \\(\varphi (\varphi > 0)\\) units and becomes an even function, then the minimum value of \\(\varphi\\) is \\( \dfrac {\pi}{12}\\), among these conclusions, the correct ones are: \_\_\_\_\_\_ .
(3)(4)
0
30,216
Given the function \( f(x) = \sin^4 x \), 1. Let \( g(x) = f(x) + f\left(\frac{\pi}{2} - x\right) \). Find the maximum and minimum values of \( g(x) \) in the interval \(\left[\frac{\pi}{6}, \frac{3\pi}{8}\right]\). 2. Find the value of \(\sum_{k=1}^{89} f\left(\frac{k\pi}{180}\right)\).
\frac{133}{4}
3.90625
30,217
What is the maximum number of rooks that can be placed in an \(8 \times 8 \times 8\) cube so that they do not attack each other?
64
57.03125
30,218
In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.
\frac{7}{16}
25.78125
30,219
How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows?
6435
44.53125
30,220
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Define $A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50.
42
57.03125
30,221
Given vectors $\overrightarrow{a}=(\sin \theta+\cos \theta,1)$ and $\overrightarrow{b}=(5,1)$, which are orthogonal, and $\theta \in (0,\pi)$, find the value of $\tan \theta$.
-\frac{3}{4}
42.96875
30,222
Simplify: \\( \dfrac {\sin 7 ^{\circ} + \cos 15 ^{\circ} \sin 8 ^{\circ} }{\cos 7 ^{\circ} - \sin 15 ^{\circ} \sin 8 ^{\circ} }= \) \_\_\_\_\_\_ .
2- \sqrt {3}
0
30,223
The graph of the function $y=\sin 2x-\sqrt{3}\cos 2x$ can be obtained by shifting the graph of the function $y=\sin 2x+\sqrt{3}\cos 2x$ to the right by $\frac{\pi}{3}$ units.
\frac{\pi}{3}
86.71875
30,224
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$ .
83
96.875
30,225
How many nonnegative integers can be expressed as \[b_6\cdot4^6+b_5\cdot4^5+b_4\cdot4^4+b_3\cdot4^3+b_2\cdot4^2+b_1\cdot4^1+b_0\cdot4^0,\] where $b_i \in \{-1,0,1\}$ for $0 \leq i \leq 6$?
5462
27.34375
30,226
In an acute-angled triangle, the sides $a$ and $b$ are the roots of the equation $x^{2}-2 \sqrt {3}x+2=0$. The angles $A$ and $B$ satisfy the equation $2\sin (A+B)- \sqrt {3}=0$. Find the value of the side $c$ and the area of $\triangle ABC$.
\dfrac { \sqrt {3}}{2}
0
30,227
Calculate:<br/>$(1)64.83-5\frac{18}{19}+35.17-44\frac{1}{19}$;<br/>$(2)(+2.5)+(-3\frac{1}{3})-(-1)$;<br/>$(3)\frac{(0.125+\frac{3}{5})×\frac{33}{87}}{12.1×\frac{1}{11}$;<br/>$(4)41\frac{1}{3}×\frac{3}{4}+52\frac{1}{2}÷1\frac{1}{4}+63\frac{3}{5}×\frac{5}{6}$;<br/>$(5)3\frac{2}{3}×2\frac{2}{15}+5\frac{2}{3}×\frac{13}{15}-2×\frac{13}{15}$;<br/>$(6)\frac{567+345×566}{567×345+222}$;<br/>$(7)3\frac{1}{8}÷[(4\frac{5}{12}-3\frac{13}{24})×\frac{4}{7}+(3\frac{1}{18}-2\frac{7}{12})×1\frac{10}{17}]$;<br/>$(8)\frac{0.1×0.3×0.9+0.2×0.6×1.8+0.3×0.9×2.7}{0.1×0.2×0.4+0.2×0.4×0.8+0.3×0.6×1.2}$;<br/>$(9)\frac{1^2+2^2}{1×2}+\frac{2^2+3^2}{2×3}+\frac{3^2+4^2}{3×4}+…+\frac{2022^2+2023^2}{2022×2023}$.
4044\frac{2022}{2023}
0
30,228
The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola.
4\sqrt{2}
25
30,229
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$. (I) Find the range of the eccentricity $e$ of the hyperbola $C$. (II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\overrightarrow{PA} = \frac{5}{12}\overrightarrow{PB}$. Find the value of $a$.
\frac{17}{13}
28.90625
30,230
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $\overrightarrow{b}=(4\cos \alpha,-4\sin \alpha)$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$, then $\theta$ equals \_\_\_\_\_\_.
\dfrac {\pi}{3}
92.96875
30,231
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence. (Ⅰ) Find the value of $q$; (Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and explain why.
-\frac{1}{2}
21.09375
30,232
To ensure the safety of property during the Spring Festival holiday, an office needs to arrange for one person to be on duty each day for seven days. Given that there are 4 people in the office, and each person needs to work for either one or two days, the number of different duty arrangements is \_\_\_\_\_\_ . (Answer with a number)
2520
36.71875
30,233
In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$. Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$?
\frac{1}{4}
44.53125
30,234
Given the ellipse $\Gamma$: $\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\left( \dfrac { \sqrt {3}- \sqrt {2}}{2}, \dfrac {1- \sqrt {6}}{2}\right)$. (Ⅰ) Find the equation of the ellipse; (Ⅱ) If the line $l$: $y=kx+m$ ($k,m$ are constants, $k\neq 0$) intersects the ellipse $\Gamma$ at two distinct points $M$ and $N$. (i) When the line $l$ passes through $E(1,0)$, and $\overrightarrow{EM}+2 \overrightarrow{EN}= \overrightarrow{0}$, find the equation of the line $l$; (ii) When the distance from the origin $O$ to the line $l$ is $\dfrac { \sqrt {3}}{2}$, find the maximum area of $\triangle MON$.
\dfrac { \sqrt {3}}{2}
0
30,235
Given the function $f(x) = \cos x \cdot \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}, x \in \mathbb{R}$, $(1)$ Find the smallest positive period of $f(x)$; $(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$.
-\frac{1}{2}
60.15625
30,236
The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD.
25
4.6875
30,237
Given a function $f(x)$ whose domain is $D$, if for any $x_1, x_2 \in D$, it holds that $f(x_1) \leq f(x_2)$ whenever $x_1 < x_2$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on the interval $[0,1]$ and satisfies the following three conditions: 1. $f(0) = 0$; 2. $f\left(\frac{x}{3}\right) = \frac{1}{2}f(x)$; 3. $f(1-x) = 1-f(x)$. Find the value of $f(1) + f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + f\left(\frac{1}{6}\right) + f\left(\frac{1}{7}\right) + f\left(\frac{1}{8}\right)$.
\frac{11}{4}
57.8125
30,238
Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.)
6655
2.34375
30,239
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?
220
1.5625
30,240
Given the differences between the scores of 14 students in a group and the class average score of 85 are 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, find the average score of this group.
87.29
0.78125
30,241
Let $\sigma (n)$ denote the sum and $\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$ ). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$ \frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n} $$
\frac{3 \sqrt{2}}{4}
53.90625
30,242
Suppose that \( a^3 \) varies inversely with \( b^2 \). If \( a = 5 \) when \( b = 2 \), find the value of \( a \) when \( b = 8 \).
2.5
10.9375
30,243
Calculate the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 3 = 0,\] given that all roots are real and nonnegative.
46
14.0625
30,244
Determine for how many different values of $p<2000$, there exists a quadrilateral $ABCD$ with integer side lengths, perimeter $p$, right angles at $D$ and $C$, $AB=3$, and $CD=AD$.
996
12.5
30,245
A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\frac{AP}{AB} =\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circle twice at points $S$ and $T$ . Two more circles $s$ and $t$ are drawn with diameters $SP$ and $ST$ respectively. For any circle $\omega$ let $A(\omega)$ denote the area of the circle. What is $\frac{A(s)+A(t)}{A(a)+A(b)}$ ?
205/1681
0
30,246
In the movie "The Wandering Earth 2," there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not special effects, but are actually designed and modified from cutting-edge domestic equipment. Many of the equipment can find prototypes in reality. A new device has been developed in a workshop of the group. The specific requirement of the group for the new device is: the product with a part inner diameter (unit: mm) within the range of $(199.82, 200.18)$ is qualified, otherwise it is defective; the part inner diameter $X$ follows a normal distribution $X \sim N(200, 0.0036)$. $(1)$ If the workshop installed and debugged the new device and produced 5 parts for trial, with measured inner diameters (unit: mm) of $199.87$, $199.91$, $199.99$, $200.13$, $200.19$, if you are in charge of the workshop, try to determine whether this device needs further debugging based on the $3\sigma$ principle? Explain your reasoning. $(2)$ If the device meets the production requirements of the group, and now 10,000 parts produced by this device are tracked. ① Approximately how many parts out of the 10,000 parts have an inner diameter exceeding $200.12$ mm? ② What is the most likely number of defective parts out of the 10,000 parts? Reference data: If a random variable $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) \approx 0.683$, $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.955$, $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997$, $0.997^4 \approx 0.988$, $0.997^5 \approx 0.985$.
30
32.03125
30,247
Given two-dimensional vectors $\vec{a}$, $\vec{b}$, with $|\vec{a}|=1$, $|\vec{b}|=2$, and $\vec{a} \cdot \vec{b}=1$. If $\vec{e}$ is a two-dimensional unit vector, find the maximum value of $|\vec{a} \cdot \vec{e}| + |\vec{b} \cdot \vec{e}|$.
\sqrt{7}
19.53125
30,248
In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = 5$. What is $\tan X$?
\frac{4}{3}
21.875
30,249
** How many non-similar regular 500-pointed stars are there? **
99
7.8125
30,250
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:
60
64.0625
30,251
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{8}$ be the 8 zeros of the polynomial $z^{8} - 16^8$. For each $j$, let $w_j$ be either $z_j$, $-z_j$, or $iz_j$. Find the maximum possible value of the real part of \[\sum_{j = 1}^{8} w_j.\]
32 + 32 \sqrt{2}
14.84375
30,252
Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\leq4$, calculate the value of $2x+y+z$.
\frac{2}{3}
0
30,253
A natural number $n$ is called a "good number" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a "good number" because $32+33+34$ does not result in a carry-over; however, 23 is not a "good number" because $23+24+25$ does result in a carry-over. The number of "good numbers" less than 1000 is \_\_\_\_\_\_.
48
3.90625
30,254
The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon?
$2:5$
0
30,255
Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3.
58
18.75
30,256
If the function \( y = \sin(w x) \) with \( w > 0 \) attains its maximum value at least 50 times in the interval \([0,1]\), what is the minimum value of \( w \)?
100 \pi
9.375
30,257
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus?
1: 1
0
30,258
If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?
42
12.5
30,259
Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$ ; what is the sum of all possible distances from $P$ to line $AB$ ?
\frac{1 + \sqrt{3}}{2}
10.15625
30,260
All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list?
194
5.46875
30,261
In rectangle $ABCD$, $AB = 4$ cm, $BC = 10$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fourth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.
2\sqrt{10}
66.40625
30,262
Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x$ (1) Find the smallest positive period of the function $f(x)$. (2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and satisfy $f(A) = 0$, $c = 1$, $b = \sqrt{2}$. Find the area of $\triangle ABC$.
\frac{1}{2}
92.96875
30,263
ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.
1/2
8.59375
30,264
Let $c$ and $d$ be real numbers such that \[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\] Find \[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\]
\frac{9}{11}
7.03125
30,265
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, find the maximum value of $(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) \cdot (\overrightarrow{a} + \overrightarrow{c})$.
2 + \sqrt{5}
12.5
30,266
Over the summer, a one-room apartment increased in price by 21%, a two-room apartment by 11%, and the total cost of both apartments by 15%. How many times cheaper is the one-room apartment compared to the two-room apartment?
1.5
25.78125
30,267
\( z_{1}, z_{2}, z_{3} \) are the three roots of the polynomial \[ P(z) = z^{3} + a z + b \] and satisfy the condition \[ \left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \left|z_{3}\right|^{2} = 250 \] Moreover, the three points \( z_{1}, z_{2}, z_{3} \) in the complex plane form a right triangle. Find the length of the hypotenuse of this right triangle.
5\sqrt{15}
1.5625
30,268
Three of the four vertices of a rectangle are \((1, 7)\), \((14, 7)\), and \((14, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 1)^2 + (y + 4)^2 = 16\)?
4\pi
77.34375
30,269
Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who also did not attempt 350 points on the first day, scored a positive integer number of points on each of the two days, and Delta's daily success ratio (points scored divided by points attempted) on each day was less than Gamma's on that day. Gamma's overall success ratio for the two days was $360/600 = 3/5$. Find the largest possible overall success ratio that Delta could have achieved.
\frac{359}{600}
19.53125
30,270
Form five-digit numbers without repeating digits using the numbers \\(0\\), \\(1\\), \\(2\\), \\(3\\), and \\(4\\). \\((\\)I\\()\\) How many of these five-digit numbers are even? \\((\\)II\\()\\) How many of these five-digit numbers are less than \\(32000\\)?
54
6.25
30,271
What is the probability that $abc + ab + a$ is divisible by $4$, where positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 1024\}$.
\frac{7}{16}
21.09375
30,272
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola.
\sqrt{2}+1
0
30,273
Arrange numbers $ 1,\ 2,\ 3,\ 4,\ 5$ in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum of that of the third, fourth and fifth. Note that in each arrangement each number are used one time without overlapping.
1/15
17.96875
30,274
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 45^\circ$, and $AE=28$. Find the length of $CE$, given that $CE$ forms the diagonal of a square $CDEF$.
28
28.90625
30,275
Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse, the minimum value of $|\overrightarrow{b} - t\overrightarrow{a}| (t \in \mathbb{R})$ is $\frac{\sqrt{3}}{2}$, and $(\overrightarrow{c} - \overrightarrow{a}) \cdot (\overrightarrow{c} - \overrightarrow{b}) = 0$, find the maximum value of $\overrightarrow{c} \cdot (\overrightarrow{a} + \overrightarrow{b})$.
\frac{\sqrt{3} + 1}{2}
0
30,276
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.
2017
52.34375
30,277
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?
1236
0.78125
30,278
In rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle JPL=90^{\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy] size(7cm);defaultpen(fontsize(9)); real vd = 7/9 * 12; path extend(pair a, pair b) {return a--(10 * (b - a));} // Rectangle pair j = (0, 0); pair l = (0, 16); pair m = (24 + vd, 0); pair k = (m.x, l.y); draw(j--l--k--m--cycle); label("$J$", j, SW);label("$L$", l, NW);label("$K$", k, NE);label("$M$", m, SE); // Extra points and lines pair q = (24, 7); pair v = (q.x, 0); pair u = (q.x, l.y); pair r = IP(k--m, extend(j, q)); pair p = (12, l.y); draw(q--j--p--m--r--cycle);draw(u--v); label("$R$", r, E); label("$P$", p, N);label("$Q$", q, 1.2 * NE + 0.2 * N);label("$V$", v, S); label("$U$", u, N); // Right angles and tick marks markscalefactor = 0.1; draw(rightanglemark(j, l, p)); draw(rightanglemark(p, u, v)); draw(rightanglemark(q, v, m));draw(rightanglemark(j, p, q)); add(pathticks(l--p, 2, spacing=3.4, s=10));add(pathticks(p--u, 2, spacing=3.5, s=10)); // Number labels label("$16$", midpoint(j--l), W); label("$25$", midpoint(j--p), NW); label("$15$", midpoint(p--q), NE); label("$20$", midpoint(j--q), 0.8 * S + E); [/asy]
\dfrac{28}{3}
11.71875
30,279
Given a finite arithmetic sequence \(\left\{a_{n}\right\}\) with the first term equal to 1 and the last term \(a_{n} = 1997\) (where \(n > 3\)), and the common difference being a natural number, find the sum of all possible values of \(n\).
3501
32.8125
30,280
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.
\frac{6}{35}
28.125
30,281
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $b^2 + c^2 - a^2 = \sqrt{3}bc$. (1) If $\tan B = \frac{\sqrt{6}}{12}$, find $\frac{b}{a}$; (2) If $B = \frac{2\pi}{3}$ and $b = 2\sqrt{3}$, find the length of the median on side $BC$.
\sqrt{7}
64.0625
30,282
A circle with equation $x^{2}+y^{2}=1$ passes through point $P(1, \sqrt {3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$.
\sqrt {3}
0
30,283
What is the smallest whole number $b$ such that 101 can be expressed in base $b$ using only two digits?
10
38.28125
30,284
Let $A$ be a point on the parabola $y = x^2 - 4x + 4,$ and let $B$ be a point on the line $y = 2x - 3.$ Find shortest possible distance $AB.$
\frac{2\sqrt{5}}{5}
17.1875
30,285
If $547\,932\,BC4$ is divisible by $12$, where $B$ and $C$ represent digits, what is the sum of all unique values of $B + C$?
57
5.46875
30,286
The product of the first and third terms of the geometric sequence $\{a_n\}$, given that $a_1$ and $a_4$ are the roots of the equation $x^2-2x-3=0$.
-3
8.59375
30,287
For any real number $x$, the symbol $[x]$ represents the largest integer not greater than $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. The function $y=[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Then, the value of $[\log _{3}1]+[\log _{3}2]+[\log _{3}3]+…+[\log _{3}11]$ is \_\_\_\_\_\_.
12
96.875
30,288
In triangle $\triangle ABC$, $2b\cos A+a=2c$, $c=8$, $\sin A=\frac{{3\sqrt{3}}}{{14}}$. Find: $(Ⅰ)$ $\angle B$; $(Ⅱ)$ the area of $\triangle ABC$.
6\sqrt{3}
27.34375
30,289
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest point on the wall is 400 meters, and from B to the nearest point on the wall is 600 meters.
1803
46.875
30,290
You, your friend, and two strangers are sitting at a table. A standard $52$ -card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.
22/703
10.9375
30,291
If four people, A, B, C, and D, line up in a row, calculate the number of arrangements in which B and C are on the same side of A.
16
17.96875
30,292
Find the number of positive integers \(n \le 500\) that can be expressed in the form \[ \lfloor x \rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor = n \] for some real number \(x\).
248
0
30,293
On a circle, ten points \(A_{1}, A_{2}, A_{3}, \ldots, A_{10}\) are equally spaced. If \(C\) is the center of the circle, what is the size, in degrees, of the angle \(A_{1} A_{5} C\) ?
18
4.6875
30,294
Given that point $P$ is the circumcenter of $\triangle ABC$, and $\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = 0$, $\angle C = 120^{\circ}$, determine the value of the real number $\lambda$.
-1
28.125
30,295
On a grid where each dot is spaced one unit apart both horizontally and vertically, a polygon is described by joining dots at coordinates (0,0), (5,0), (5,2), (3,2), (3,3), (2,3), (2,2), (0,2), and back to (0,0). What is the number of square units enclosed by this polygon?
11
7.03125
30,296
Let $b_n$ be the integer obtained by writing down the integers from 1 to $n$ in reverse, from right to left. Compute the remainder when $b_{39}$ is divided by 125.
21
0
30,297
Calculate the radius of the circle where all the complex roots of the equation $(z - 1)^6 = 64z^6$ lie when plotted in the complex plane.
\frac{2}{3}
0.78125
30,298
Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction.
\dfrac{7}{64}
0
30,299
How many of the smallest 216 positive integers written in base 6 use the digit 3 or 4 (or both) as a digit?
168
0