Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
17,200 |
If \( b \) and \( n \) are positive integers with \( b, n \leq 18 \), what is the greatest number of positive factors \( b^n \) can have?
|
703
| 7.8125 |
17,201 |
Given the function $f(x) = \sin\left(\frac{5\pi}{3}x + \frac{\pi}{6}\right) + \frac{3x}{2x-1}$, then the value of $f\left(\frac{1}{2016}\right) + f\left(\frac{3}{2016}\right) + f\left(\frac{5}{2016}\right) + f\left(\frac{7}{2016}\right) + \ldots + f\left(\frac{2015}{2016}\right) = \_\_\_\_\_\_$.
|
1512
| 22.65625 |
17,202 |
In $\Delta ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin C=\sqrt{3}c\cos A$.
1. Find the size of angle $A$.
2. If $a=\sqrt{13}$ and $c=3$, find the area of $\Delta ABC$.
|
3\sqrt{3}
| 96.875 |
17,203 |
What is the product of the numerator and the denominator when $0.\overline{0012}$ is expressed as a fraction in lowest terms?
|
13332
| 83.59375 |
17,204 |
What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers?
|
-1000
| 28.90625 |
17,205 |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.
How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively.
|
33
| 52.34375 |
17,206 |
Given real numbers $a$ and $b$ satisfying $a^{2}-4\ln a-b=0$, find the minimum value of $\left(a-c\right)^{2}+\left(b+2c\right)^{2}$.
|
\frac{9}{5}
| 4.6875 |
17,207 |
A granary has collected 1536 shi of rice, and upon inspection, it is found that out of 224 grains, 28 are weeds. Determine the approximate amount of weeds in this batch of rice.
|
192
| 22.65625 |
17,208 |
In a pocket, there are several balls of three different colors (enough in quantity), and each time 2 balls are drawn. To ensure that the result of drawing is the same 5 times, at least how many times must one draw?
|
25
| 59.375 |
17,209 |
The inclination angle of the line given by the parametric equations \[
\begin{cases}
x=1+t \\
y=1-t
\end{cases}
\]
calculate the inclination angle.
|
\frac {3\pi }{4}
| 70.3125 |
17,210 |
The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola.
|
\frac{\sqrt{5}}{2}
| 91.40625 |
17,211 |
Given the function $f(x)=\cos ( \sqrt {3}x+\phi)- \sqrt {3}\sin ( \sqrt {3}x+\phi)$, find the smallest positive value of $\phi$ such that $f(x)$ is an even function.
|
\frac{2\pi}{3}
| 75 |
17,212 |
Compute the multiplicative inverse of $217$ modulo $397$. Express your answer as an integer from $0$ to $396$.
|
161
| 6.25 |
17,213 |
A club has 12 members, and wishes to pick a president, a vice-president, a secretary, and a treasurer. However, the president and vice-president must have been members of the club for at least 3 years. If 4 of the existing members meet this criterion, in how many ways can these positions be filled, given that each member can hold at most one office?
|
1080
| 98.4375 |
17,214 |
How many positive integers less than $900$ are either a perfect cube or a perfect square?
|
35
| 38.28125 |
17,215 |
A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$
|
\sqrt{59}
| 34.375 |
17,216 |
Given the function $f(x)=\cos x(\sin x+\cos x)-\frac{1}{2}$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[-\frac{\pi}{4},\frac{\pi}{2}]$.
|
-\frac{1}{2}
| 63.28125 |
17,217 |
A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________.
|
0.032
| 82.03125 |
17,218 |
Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 3$ and $QS = 9$, what is $RS$?
|
27
| 65.625 |
17,219 |
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Calculate the value of $(1+a)(1+b)(1+c)$.
|
51
| 88.28125 |
17,220 |
Consider a matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \in \{0, 1\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$.
|
10
| 51.5625 |
17,221 |
What is the value of the following expression: $3 - 8 + 13 - 18 + 23 - \cdots - 98 + 103 - 108 + 113$ ?
|
58
| 58.59375 |
17,222 |
The constant term in the expansion of $(x+3)\left(2x- \frac{1}{4x\sqrt{x}}\right)^5$ is ______.
|
15
| 60.15625 |
17,223 |
The first $30$ numbers of a similar arrangement are shown below. What would be the value of the $50^{\mathrm{th}}$ number if the arrangement were continued?
$\bullet$ Row 1: $3,$ $3$
$\bullet$ Row 2: $6,$ $6,$ $6,$ $6$
$\bullet$ Row 3: $9,$ $9,$ $9,$ $9,$ $9,$ $9$
$\bullet$ Row 4: $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12$
|
21
| 25 |
17,224 |
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, ignoring additional fees.
|
270.55
| 25 |
17,225 |
Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions?
|
15
| 13.28125 |
17,226 |
In Pascal's triangle, compute the seventh element in Row 20. Afterward, determine how many times greater this element is compared to the third element in the same row.
|
204
| 97.65625 |
17,227 |
A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with point A on line $l$.
(1) Find the polar coordinate equation of curve $C_1$ and the rectangular coordinate equation of line $l$.
(2) After line $l$ is moved 6 units to the left to obtain $l'$, the intersection points of $l'$ and $C_1$ are M and N. Find the polar coordinate equation of $l'$ and the length of $|MN|$.
|
2\sqrt{2}
| 17.1875 |
17,228 |
Given $F$ is a point on diagonal $BC$ of the unit square $ABCD$ such that $\triangle{ABF}$ is isosceles right triangle with $AB$ as the hypotenuse, consider a strip inside $ABCD$ parallel to $AD$ ranging from $y=\frac{1}{4}$ to $y=\frac{3}{4}$ of the unit square, calculate the area of the region $Q$ which lies inside the strip but outside of $\triangle{ABF}$.
|
\frac{1}{2}
| 1.5625 |
17,229 |
Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$
|
\frac{26}{15}
| 55.46875 |
17,230 |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have?
|
432
| 80.46875 |
17,231 |
What is the minimum number of equilateral triangles, each with a side length of 2 units, required to cover an equilateral triangle with a side length of 16 units?
|
64
| 88.28125 |
17,232 |
Two distinct integers $x$ and $y$ are factors of 48. One of these integers must be even. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$?
|
32
| 11.71875 |
17,233 |
A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z^2 \), where \( a \) and \( b \) are real numbers. The function has the property that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Given that \( |a+bi| = 5 \), find \( b^2 \).
|
\frac{99}{4}
| 37.5 |
17,234 |
Determine the number of real solutions to the equation:
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{150}{x - 150} = x^2.
\]
|
151
| 53.90625 |
17,235 |
Calculate the number of different rectangles with sides parallel to the grid that can be formed by connecting four of the dots in a $5\times 5$ square array of dots. (Two rectangles are different if they do not share all four vertices.)
|
100
| 25.78125 |
17,236 |
The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$ for every positive integer $n$ , where $a_0 = a_1 = 1$ . Calculate the sum $$ \frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}} $$ .
|
2009 * 1005
| 0 |
17,237 |
Compute the integrals:
1) \(\int_{0}^{5} \frac{x \, dx}{\sqrt{1+3x}}\)
2) \(\int_{\ln 2}^{\ln 9} \frac{dx}{e^{x}-e^{-x}}\)
3) \(\int_{1}^{\sqrt{3}} \frac{(x^{3}+1) \, dx}{x^{2} \sqrt{4-x^{2}}}\)
4) \(\int_{0}^{\frac{\pi}{2}} \frac{dx}{2+\cos x}\)
|
\frac{\pi}{3 \sqrt{3}}
| 9.375 |
17,238 |
The local library has two service windows. In how many ways can eight people line up to be served if there are two lines, one for each window?
|
40320
| 46.875 |
17,239 |
A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars?
|
75,000,000
| 0 |
17,240 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $\sqrt{3}b\cos A - a\sin B = 0$.
$(1)$ Find the measure of angle $A$;
$(2)$ Given that $c=4$ and the area of $\triangle ABC$ is $6\sqrt{3}$, find the value of side length $a$.
|
2\sqrt{7}
| 96.09375 |
17,241 |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_.
|
\frac{3}{2}
| 53.90625 |
17,242 |
The left focus of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \, (a > 0, b > 0)$ is $F$. If the symmetric point $A$ of $F$ with respect to the line $\sqrt{3}x + y = 0$ is a point on the hyperbola $C$, then the eccentricity of the hyperbola $C$ is \_\_\_\_\_\_.
|
\sqrt{3} + 1
| 47.65625 |
17,243 |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).
|
\frac{16}{3}
| 0 |
17,244 |
A tangent line is drawn from a point on the line $y=x$ to the circle $(x-4)^2 + (y+2)^2 = 1$. Find the minimum length of the tangent line.
|
\sqrt{17}
| 39.0625 |
17,245 |
Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $\triangle ABC$, if $\triangle ABC$ simultaneously satisfies three of the following four conditions:①$a=\sqrt{3}$; ②$b=2$; ③$\frac{{sinB+sinC}}{{sinA}}=\frac{{a+c}}{{b-c}}$; ④${cos^2}({\frac{{B-C}}{2}})-sinBsinC=\frac{1}{4}$.<br/>$(1)$ What are the possible combinations of conditions that have a solution for the triangle?<br/>$(2)$ Among the combinations in $(1)$, choose one and find the area of the corresponding $\triangle ABC$.
|
\frac{\sqrt{3}}{2}
| 43.75 |
17,246 |
Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$.
|
24\pi
| 28.90625 |
17,247 |
Given that $a > 0$, if $f(g(a)) = 18$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$, what is the value of $a$?
|
\sqrt{2\sqrt{2} + 6}
| 0 |
17,248 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of four consecutive positive integers all of which are nonprime?
|
29
| 97.65625 |
17,249 |
A rectangular chocolate bar is made of equal squares. Irena breaks off two complete strips of squares and eats the 12 squares she obtains. Later, Jack breaks off one complete strip of squares from the same bar and eats the 9 squares he obtains. How many squares of chocolate are left in the bar?
A) 72
B) 63
C) 54
D) 45
E) 36
|
45
| 7.03125 |
17,250 |
The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?
A) \(\frac{1}{3}\)
B) \(\frac{1}{2}\)
C) \(\frac{5}{9}\)
D) \(\frac{2}{3}\)
E) \(\frac{5}{6}\)
|
\frac{1}{2}
| 42.96875 |
17,251 |
Find the product of all constants $t$ such that the quadratic $x^2 + tx + 6$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.
|
1225
| 81.25 |
17,252 |
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have any other vertices. How many triangles result from this triangulation?
|
1998
| 77.34375 |
17,253 |
In February 1983, $789$ millimeters of rain fell in Jorhat, India. What was the average rainfall in millimeters per hour during that particular month?
A) $\frac{789}{672}$
B) $\frac{789 \times 28}{24}$
C) $\frac{789 \times 24}{28}$
D) $\frac{28 \times 24}{789}$
E) $789 \times 28 \times 24$
|
\frac{789}{672}
| 41.40625 |
17,254 |
In the parallelepiped $ABCD-{A'}{B'}{C'}{D'}$, the base $ABCD$ is a square with side length $2$, the length of the side edge $AA'$ is $3$, and $\angle {A'}AB=\angle {A'}AD=60^{\circ}$. Find the length of $AC'$.
|
\sqrt{29}
| 14.84375 |
17,255 |
Let $x$ be a real number selected uniformly at random between 100 and 300. If $\lfloor \sqrt{x} \rfloor = 14$, find the probability that $\lfloor \sqrt{100x} \rfloor = 140$.
A) $\frac{281}{2900}$
B) $\frac{4}{29}$
C) $\frac{1}{10}$
D) $\frac{96}{625}$
E) $\frac{1}{100}$
|
\frac{281}{2900}
| 70.3125 |
17,256 |
A merchant's cumulative sales from January to May reached 38.6 million yuan. It is predicted that the sales in June will be 5 million yuan, and the sales in July will increase by x% compared to June. The sales in August will increase by x% compared to July. The total sales in September and October are equal to the total sales in July and August. If the total sales from January to October must reach at least 70 million yuan, then the minimum value of x is.
|
20
| 39.84375 |
17,257 |
Given that $a > 0$, $b > 0$, and $\frac{1}{a}$, $\frac{1}{2}$, $\frac{1}{b}$ form an arithmetic sequence, find the minimum value of $a+9b$.
|
16
| 83.59375 |
17,258 |
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy]
|
2 : 1
| 2.34375 |
17,259 |
Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores.
|
237
| 53.125 |
17,260 |
In the diagram below, $WXYZ$ is a trapezoid where $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. Given $YZ = 15$, $\tan Z = 2$, and $\tan X = 2.5$, what is the length of $XY$?
|
2\sqrt{261}
| 0 |
17,261 |
Rectangle $ABCD$ has sides $AB = 3$ , $BC = 2$ . Point $ P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$ . Find $BP$ .
|
1/3
| 1.5625 |
17,262 |
The constant term in the expansion of \\(\left(x^{2}- \frac{1}{x}+3\right)^{4}\\) is ______.
|
117
| 85.15625 |
17,263 |
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$ , $60^\circ$ , and $75^\circ$ .
|
3\sqrt{2} + 2\sqrt{3} - \sqrt{6}
| 0.78125 |
17,264 |
Given a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+6)=f(x)$. When $-3<x\leq-1$, $f(x)=-(x+2)^2$, and when $-1\leq x\leq 3$, $f(x)=x$. Calculate the value of $f(1) + f(2) + \dots + f(2015)$.
|
1680
| 45.3125 |
17,265 |
Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\).
|
2100
| 10.9375 |
17,266 |
Observe the following equations:<br/>$\frac{1}{1×2}=1-\frac{1}{2}=\frac{1}{2}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}=\frac{2}{3}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=\frac{3}{4}$;<br/>$\ldots $<br/>Based on the pattern you discovered, answer the following questions:<br/>$(1)\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}×\frac{1}{4×5}=$______;<br/>$(2)$If $n$ is a positive integer, then $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\frac{1}{4×5}+…+\frac{1}{n(n+1)}=$______;<br/>$(3)$Calculate $1-\frac{1}{1×2}-\frac{1}{2×3}-\frac{1}{3×4}-…-\frac{1}{99×100}$.
|
\frac{1}{100}
| 89.84375 |
17,267 |
Given that in the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are $ \begin{cases} x=2+2\cos θ \
y=2\sin θ\end{cases} $ for the parameter $(θ)$, and in the polar coordinate system $(rOθ)$ (with the same unit length as the rectangular coordinate system $(xOy)$, and the origin $O$ as the pole, and the positive semi-axis of $x$ as the polar axis), the equation of the line $l$ is $ρ\sin (θ+ \dfrac {π}{4})=2 \sqrt {2}$.
(I) Find the equation of the curve $C$ in the polar coordinate system;
(II) Find the length of the chord cut off by the line $l$ on the curve $C$.
|
2 \sqrt {2}
| 0 |
17,268 |
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin)
|
1990
| 94.53125 |
17,269 |
Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points?
|
1.5
| 7.8125 |
17,270 |
A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle.
|
9\pi - \frac{81\sqrt{3}}{4}
| 48.4375 |
17,271 |
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?
|
126
| 81.25 |
17,272 |
Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \(\pi\).
|
\frac{27}{4}\pi
| 9.375 |
17,273 |
Calculate the following expression:
$$2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))))))$$
|
4094
| 9.375 |
17,274 |
A classroom has 10 chairs arranged in a row. Tom and Jerry choose their seats at random, but they are not allowed to sit on the first and last chairs. What is the probability that they don't sit next to each other?
|
\frac{3}{4}
| 59.375 |
17,275 |
Given the function $f(x)=\sin (ωx+φ)(ω > 0,|φ|\leqslant \dfrac {π}{2})$, $y=f(x- \dfrac {π}{4})$ is an odd function, $x= \dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\dfrac {π}{14}, \dfrac {13π}{84})$, determine the maximum value of $ω$.
|
11
| 15.625 |
17,276 |
Determine how many integer values of $x$ satisfy $\lceil{\sqrt{x}}\rceil = 20$.
|
39
| 82.03125 |
17,277 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $C= \frac{\pi}{3}$, $b=8$, and the area of $\triangle ABC$ is $10\sqrt{3}$.
$(1).$ Find the value of $c$; $(2).$ Find the value of $\cos(B-C)$.
|
\frac{13}{14}
| 76.5625 |
17,278 |
Place several small circles with a radius of 1 inside a large circle with a radius of 11, so that each small circle is tangentially inscribed in the large circle and these small circles do not overlap. What is the maximum number of small circles that can be placed?
|
31
| 50.78125 |
17,279 |
Given that $a$ and $b$ are rational numbers, a new operation is defined as follows: $a$☼$b=a^{3}-2ab+4$. For example, $2$☼$5=2^{3}-2\times 2\times 5+4=-8$. Find $4$☼$\left(-9\right)=\_\_\_\_\_\_$.
|
140
| 96.875 |
17,280 |
Let $p(x)=x^4-4x^3+2x^2+ax+b$ . Suppose that for every root $\lambda$ of $p$ , $\frac{1}{\lambda}$ is also a root of $p$ . Then $a+b=$ [list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list]
|
-3
| 93.75 |
17,281 |
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the problem above. If $AB = 34$ units and $CD = 66$ units, what is the area of square $BCFE$?
|
2244
| 10.9375 |
17,282 |
Given $f(x)=x^{2005}+ax^{3}- \frac {b}{x}-8$, and $f(-2)=10$, find $f(2)$.
|
-26
| 89.0625 |
17,283 |
In the interval \\(\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]\\), a number \\(x\\) is randomly selected. The probability that \\(\sin x + \cos x \in [1, \sqrt{2}]\\) is \_\_\_\_\_\_.
|
\frac{3}{4}
| 86.71875 |
17,284 |
Given $\cos x + \cos y = \frac{1}{2}$ and $\sin x + \sin y = \frac{1}{3}$, find the value of $\cos (x - y)$.
|
$-\frac{59}{72}$
| 0 |
17,285 |
Let $m, n \in \mathbb{N}$, and $f(x) = (1+x)^m + (1+x)^n$.
1. When $m=n=7$, $f(x) = a_7x^7 + a_6x^6 + \ldots + a_1x + a_0$, find $a_0 + a_2 + a_4 + a_6$.
2. If the coefficient of the expanded form of $f(x)$ is 19 when $m, n$ vary, find the minimum value of the coefficient of $x^2$.
|
81
| 20.3125 |
17,286 |
A convex polyhedron \(Q\) has \(30\) vertices, \(70\) edges, and \(40\) faces, of which \(30\) are triangular and \(10\) are pentagonal. Compute the number of space diagonals in polyhedron \(Q\).
|
315
| 4.6875 |
17,287 |
Suppose
\[
3 + \frac{1}{1 + \frac{1}{3 + \frac{3}{4+y}}} = \frac{169}{53}.
\]
Solve for the value of $y$.
|
\frac{-605}{119}
| 0 |
17,288 |
John has 15 marbles of different colors, including two reds, two greens, and two blues. In how many ways can he choose 5 marbles, if exactly one of the chosen marbles must be red and one must be green?
|
660
| 42.1875 |
17,289 |
An arithmetic sequence {a_n} has a sum of the first n terms as S_n, and S_6/S_3 = 4. Find the value of S_9/S_6.
|
\dfrac{9}{4}
| 56.25 |
17,290 |
Given that the terminal side of angle $\alpha$ passes through the point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos\alpha = \frac{m}{6}$, then $\sin\alpha = \_\_\_\_\_\_$.
|
\frac{\sqrt{3}}{2}
| 7.03125 |
17,291 |
Simplify $\frac{\sin 7^{\circ}+\cos 15^{\circ} \cdot \sin 8^{\circ}}{\cos 7^{\circ}-\sin 15^{\circ} \cdot \sin 8^{\circ}}$. The value equals ( ).
|
$2-\sqrt{3}$
| 0 |
17,292 |
If $\frac{8^x}{4^{x+y}}=16$ and $\frac{16^{x+y}}{4^{7y}}=1024$, find $x+y$.
|
13
| 87.5 |
17,293 |
A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th game for their mean number of points-per-game to exceed 22?
|
24
| 57.03125 |
17,294 |
What is the maximum possible product of three different numbers from the set $\{-9, -7, -2, 0, 4, 6, 8\}$, where the product contains exactly one negative number?
|
-96
| 11.71875 |
17,295 |
Simplify $5 \cdot \frac{12}{7} \cdot \frac{49}{-60}$.
|
-7
| 92.1875 |
17,296 |
If the set $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2016}$.
|
-1
| 53.125 |
17,297 |
Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values?
|
\frac{25}{162}
| 17.1875 |
17,298 |
For all positive integers $n > 1$ , let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$ , compute
\[\frac{f(N)}{f(f(f(N)))}.\]
|
25
| 35.15625 |
17,299 |
At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make?
|
126
| 78.90625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.