Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
11,900 |
Ten students (one captain and nine team members) formed a team to participate in a math competition and won first place. The committee decided to award each team member 200 yuan. The captain received 90 yuan more than the average bonus of all ten team members. Determine the amount of bonus the captain received.
|
300
| 65.625 |
11,901 |
A rectangular picture frame is constructed from 1.5-inch-wide pieces of wood. The area of just the frame is \(27\) square inches, and the length of one of the interior edges of the frame is \(4.5\) inches. Determine the sum of the lengths of the four interior edges of the frame.
|
12
| 65.625 |
11,902 |
The mean, median, and mode of the five numbers 12, 9, 11, 16, x are all equal. Find the value of x.
|
12
| 77.34375 |
11,903 |
15 plus 16 equals.
|
31
| 72.65625 |
11,904 |
How, without any measuring tools, can you measure 50 cm from a string that is $2/3$ meters long?
|
50
| 67.1875 |
11,905 |
A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \, \mathrm{m/s^2}$, and the Earth's radius is $R = 6400 \, \mathrm{km}$).
|
6400
| 23.4375 |
11,906 |
The lateral surface area of a regular triangular pyramid is 3 times the area of its base. The area of the circle inscribed in the base is numerically equal to the radius of this circle. Find the volume of the pyramid.
|
\frac{2 \sqrt{6}}{\pi^3}
| 32.8125 |
11,907 |
Four people, A, B, C, and D, participated in an exam. The combined scores of A and B are 17 points higher than the combined scores of C and D. A scored 4 points less than B, and C scored 5 points more than D. How many points higher is the highest score compared to the lowest score among the four?
|
13
| 93.75 |
11,908 |
Find the area of triangle \(ABC\), if \(AC = 3\), \(BC = 4\), and the medians \(AK\) and \(BL\) are mutually perpendicular.
|
\sqrt{11}
| 17.1875 |
11,909 |
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.
|
45
| 31.25 |
11,910 |
Express the given data "$20$ nanoseconds" in scientific notation.
|
2 \times 10^{-8}
| 25.78125 |
11,911 |
For the real numbers \(a\) and \(b\), it holds that \(a^{2} + 4b^{2} = 4\). How large can \(3a^{5}b - 40a^{3}b^{3} + 48ab^{5}\) be?
|
16
| 31.25 |
11,912 |
In a survey of $150$ employees at a tech company, it is found that:
- $90$ employees are working on project A.
- $50$ employees are working on project B.
- $30$ employees are working on both project A and B.
Determine what percent of the employees surveyed are not working on either project.
|
26.67\%
| 88.28125 |
11,913 |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1500$ and $AD = 3000$. Let $\angle A = 30^\circ$, $\angle D = 60^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Determine the length $PQ$.
|
750
| 30.46875 |
11,914 |
Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).
|
1/33
| 77.34375 |
11,915 |
Find the smallest positive real t such that
\[ x_1 + x_3 = 2tx_2, \]
\[ x_2 + x_4 = 2tx_3, \]
\[ x_3 + x_5 = 2tx_4 \]
has a solution \( x_1, x_2, x_3, x_4, x_5 \) in non-negative reals, not all zero.
|
\frac{1}{\sqrt{2}}
| 5.46875 |
11,916 |
Given that the odd function f(x) defined on R satisfies f(x+2) + f(2-x) = 0, and when x ∈ (-2, 0), f(x) = log<sub>2</sub>(x+3) + a. If f(9) = 2f(7) + 1, then the value of the real number a is ( ).
|
-\frac{4}{3}
| 34.375 |
11,917 |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\cos \alpha = \frac{4}{5}\), \(\tan (\alpha - \beta) = -\frac{1}{3}\), find \(\cos \beta\).
|
\frac{9 \sqrt{10}}{50}
| 87.5 |
11,918 |
Given that $S_n$ and $T_n$ represent the sum of the first n terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ respectively, and $\frac{S_n}{T_n} = \frac{2n+1}{n+3}$, determine the value of $\frac{a_7}{b_7}$.
|
\frac{27}{16}
| 66.40625 |
11,919 |
In the complex plane, let the vertices \( A \) and \( B \) of triangle \( \triangle AOB \) correspond to the complex numbers \( \alpha \) and \( \beta \), respectively, and satisfy the conditions: \( \beta = (1 + i)\alpha \) and \( |\alpha - 2| = 1 \). \( O \) is the origin. Find the maximum value of the area \( S \) of the triangle \( \triangle OAB \).
|
9/2
| 60.9375 |
11,920 |
If $a-b=1$ and $ab=-2$, then $\left(a+1\right)\left(b-1\right)=$____.
|
-4
| 89.84375 |
11,921 |
Is there an integer $x$ such that $x \equiv 1 \ (\text{mod} \ 6)$, $x \equiv 9 \ (\text{mod} \ 14)$, and $x \equiv 7 \ (\text{mod} \ 15)$?
|
37
| 32.03125 |
11,922 |
Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=5$, and $|r-s|=7$. What is the sum of all possible values of $|p-s|$?
|
30
| 45.3125 |
11,923 |
Calculate: $|\sqrt{3}-2|+(\pi -\sqrt{10})^{0}-\sqrt{12}$.
|
3-3\sqrt{3}
| 75.78125 |
11,924 |
Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $C: x^{2}-\frac{{y}^{2}}{3}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $\angle PF_{2}Q=120^{\circ}$, find the area of quadrilateral $PF_{1}QF_{2}$.
|
6\sqrt{3}
| 11.71875 |
11,925 |
Given the function $f(x) = 2^x + \ln x$, if $a_n = 0.1n$ ($n \in \mathbb{N}^*$), find the value of $n$ that minimizes $|f(a_n) - 2012|$.
|
110
| 0 |
11,926 |
There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.
How many mushrooms are in the basket?
|
25
| 15.625 |
11,927 |
ABCDEF is a six-digit number. All its digits are different and arranged in ascending order from left to right. This number is a perfect square.
Determine what this number is.
|
134689
| 99.21875 |
11,928 |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}a\cos C=(2b-\sqrt{3}c)\cos A$
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $a=2$, find the maximum area of $\Delta ABC$.
|
2+ \sqrt{3}
| 54.6875 |
11,929 |
Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, and arrived at the airport 15 minutes earlier than planned initially. What is the distance from Victor's home to the airport?
|
150
| 50 |
11,930 |
The base of an inclined parallelepiped is a rhombus with a side length of 60. A diagonal section plane passing through the longer diagonal of the base is perpendicular to the base's plane. The area of this section is 7200. Find the shorter diagonal of the base if the lateral edge is 80 and forms an angle of $60^\circ$ with the base plane.
|
60
| 83.59375 |
11,931 |
It is known that 9 cups of tea cost less than 10 rubles, and 10 cups of tea cost more than 11 rubles. How much does one cup of tea cost?
|
111
| 0 |
11,932 |
Consider a sequence of real numbers \(\{a_n\}\) defined by \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{1 + n a_n}\) for \(n \geq 1\). Find the value of \(\frac{1}{a_{2005}} - 2000000\).
|
9011
| 65.625 |
11,933 |
Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)? If the question allows for multiple solutions, list them all as a set.
|
50
| 28.125 |
11,934 |
If the integer $a$ makes the inequality system about $x$ $\left\{\begin{array}{l}{\frac{x+1}{3}≤\frac{2x+5}{9}}\\{\frac{x-a}{2}＞\frac{x-a+1}{3}}\end{array}\right.$ have at least one integer solution, and makes the solution of the system of equations about $x$ and $y$ $\left\{\begin{array}{l}ax+2y=-4\\ x+y=4\end{array}\right.$ positive integers, find the sum of all values of $a$ that satisfy the conditions.
|
-16
| 28.90625 |
11,935 |
On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle?
|
1:2
| 32.03125 |
11,936 |
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)?
|
35
| 35.15625 |
11,937 |
In triangle \( \triangle ABC \), \( AB = \sqrt{2} \), \( AC = \sqrt{3} \), and \( \angle BAC = 30^\circ \). Let \( P \) be an arbitrary point in the plane containing \( \triangle ABC \). Find the minimum value of \( \mu = \overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA} \).
|
\frac{\sqrt{2}}{2} - \frac{5}{3}
| 0 |
11,938 |
Find the sum of $521_8$ and $146_8$ in base $8$.
|
667_8
| 85.9375 |
11,939 |
In a new game, Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.
|
\frac{5}{6}
| 44.53125 |
11,940 |
Samantha has 10 different colored marbles in her bag. In how many ways can she choose five different marbles such that at least one of them is red?
|
126
| 88.28125 |
11,941 |
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
|
21
| 92.1875 |
11,942 |
What is the smallest eight-digit positive integer that has exactly four digits which are 4?
|
10004444
| 69.53125 |
11,943 |
Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.
|
34
| 3.90625 |
11,944 |
The cafe "Burattino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the cafe worked 18 days" and "from April 10 to April 30, the cafe also worked 18 days." It is known that he made a mistake once. How many days did the cafe work from April 1 to April 27?
|
23
| 32.8125 |
11,945 |
In the convex quadrilateral \(ABCD\),
\[
\angle BAD = \angle BCD = 120^\circ, \quad BC = CD = 10.
\]
Find \(AC.\)
|
10
| 17.96875 |
11,946 |
The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\leq k<2013$ . What is $k$ ? Recall $2013=3\cdot 11\cdot 61$ .
|
27
| 42.96875 |
11,947 |
Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \).
|
\frac{2}{9}
| 10.9375 |
11,948 |
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?
|
79
| 49.21875 |
11,949 |
Let the operation $\#$ be defined as $\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\oplus$ by $\oplus(a, b, c, d) = \#(a, b + d, c) - \#(a, b, c)$. What is the value of $\oplus(2, 4, 1, 3)$?
|
33
| 36.71875 |
11,950 |
Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, calculate the maximum value of the sum of the first $n$ terms, $S_n$.
|
\frac{3}{2}
| 79.6875 |
11,951 |
Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.
|
5.1
| 24.21875 |
11,952 |
If for any real number \( x \), the function
\[ f(x)=x^{2}-2x-|x-1-a|-|x-2|+4 \]
always yields a non-negative real number, then the minimum value of the real number \( a \) is .
|
-2
| 46.09375 |
11,953 |
If $x+y=8$ and $xy=12$, what is the value of $x^3+y^3$ and $x^2 + y^2$?
|
40
| 10.15625 |
11,954 |
Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \).
|
11
| 15.625 |
11,955 |
Let \( ABCD \) be a square with side length \( 5 \), and \( E \) be a point on \( BC \) such that \( BE = 3 \) and \( EC = 2 \). Let \( P \) be a variable point on the diagonal \( BD \). Determine the length of \( PB \) if \( PE + PC \) is minimized.
|
\frac{15 \sqrt{2}}{8}
| 7.8125 |
11,956 |
Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally?
|
82
| 46.09375 |
11,957 |
The diagram shows the ellipse whose equation is \(x^{2}+y^{2}-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). What is the area of the inscribed quadrilateral \(ABCD\)?
|
28
| 35.9375 |
11,958 |
From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form.
|
-\frac{2020}{2017}
| 44.53125 |
11,959 |
On the banks of an island, which has the shape of a circle (viewed from above), there are the cities $A, B, C,$ and $D$. A straight asphalt road $AC$ divides the island into two equal halves. A straight asphalt road $BD$ is shorter than road $AC$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has straight dirt roads $AB, BC, CD,$ and $AD$, on which the cyclist's speed is the same. The cyclist travels from point $B$ to each of points $A, C,$ and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$.
|
450
| 10.9375 |
11,960 |
The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (the intruder remained stationary). Three guards collectively ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
|
150
| 81.25 |
11,961 |
A metallic weight has a mass of 25 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second; the mass of the second metal is related to the mass of the third as \(3: 4\), and the mass of the third metal to the mass of the fourth as \(5: 6\). Determine the mass of the fourth metal. Give the answer in kilograms, rounding to the nearest hundredth if necessary.
|
7.36
| 72.65625 |
11,962 |
Triangle \(ABC\) has \(\angle A = 90^\circ\), side \(BC = 25\), \(AB > AC\), and area 150. Circle \(\omega\) is inscribed in \(ABC\), with \(M\) as its point of tangency on \(AC\). Line \(BM\) meets \(\omega\) a second time at point \(L\). Find the length of segment \(BL\).
|
\frac{45\sqrt{17}}{17}
| 5.46875 |
11,963 |
In \(\triangle ABC\), \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. Given \(a+c=2b\) and \(A-C=\frac{\pi}{3}\), find the value of \(\sin B\).
|
\frac{\sqrt{39}}{8}
| 72.65625 |
11,964 |
Given a parabola $C: y^2 = 3x$ with focus $F$, find the length of segment $AB$ where the line passing through $F$ at a $30^\circ$ angle intersects the parabola at points $A$ and $B$.
|
12
| 49.21875 |
11,965 |
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___
|
-1
| 60.9375 |
11,966 |
A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $m$ and $n$ are relatively prime, that two randomly selected cards from the remaining cards also form a pair. Find $m + n$.
|
299
| 49.21875 |
11,967 |
When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it?
|
\frac{3}{2}
| 62.5 |
11,968 |
The length of one side of the square \(ABCD\) is 4 units. A circle is drawn tangent to \(\overline{BC}\) and passing through the vertices \(A\) and \(D\). Find the area of the circle.
|
\frac{25 \pi}{4}
| 30.46875 |
11,969 |
A crew of workers was tasked with pouring ice rinks on a large and a small field, where the area of the large field is twice the area of the small field. The part of the crew working on the large field had 4 more workers than the part of the crew working on the small field. When the pouring on the large rink was completed, the group working on the small field was still working. What is the maximum number of workers that could have been in the crew?
|
10
| 39.84375 |
11,970 |
Given point $P(2,-1)$,
(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.
(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the maximum distance.
|
\sqrt{5}
| 68.75 |
11,971 |
Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other?
|
17/57
| 6.25 |
11,972 |
Let \( A \) be the set of any 20 points on the circumference of a circle. Joining any two points in \( A \) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords.
|
5036
| 87.5 |
11,973 |
Let \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \).
|
\frac{2012}{2013}
| 84.375 |
11,974 |
If $x \sim N(4, 1)$ and $f(x < 3) = 0.0187$, then $f(x < 5) = \_\_\_\_\_\_$.
|
0.9813
| 2.34375 |
11,975 |
Given the ages of Daisy's four cousins are distinct single-digit positive integers, and the product of two of the ages is $24$ while the product of the other two ages is $35$, find the sum of the ages of Daisy's four cousins.
|
23
| 74.21875 |
11,976 |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases}x=2+t \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases}x=-2+m \\ y= \frac{m}{k}\end{cases}$ (where $m$ is the parameter). Let the intersection point of $l_1$ and $l_2$ be $P$. When $k$ changes, the trajectory of $P$ is curve $C$.
(1) Write the general equation of $C$;
(2) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Let line $l_3: \rho(\cos \theta +\sin \theta)− \sqrt{2} =0$, and $M$ be the intersection point of $l_3$ and $C$. Find the polar radius of $M$.
|
\sqrt{5}
| 42.96875 |
11,977 |
The letters L, K, R, F, O and the digits 1, 7, 8, 9 are "cycled" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time.
|
20
| 36.71875 |
11,978 |
Given that \(\alpha, \beta, \gamma\) satisfy \(0<\alpha<\beta<\gamma<2 \pi\), and for any \(x \in \mathbf{R}\), \(\cos (x+\alpha) + \cos (x+\beta) + \cos (x+\gamma) = 0\), determine the value of \(\gamma - \alpha\).
|
\frac{4\pi}{3}
| 71.09375 |
11,979 |
In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in the spring round?
|
2000
| 17.96875 |
11,980 |
From the 4 digits 0, 1, 2, 3, select 3 digits to form a three-digit number without repetition. How many of these three-digit numbers are divisible by 3?
|
10
| 28.90625 |
11,981 |
A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle.
|
\frac{\sqrt{10}}{5}
| 57.03125 |
11,982 |
In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(N\) respectively. Given that \(\angle MPB = \angle PCN\) and \(\angle NPC = \angle MBP\), find \(\frac{S_{\triangle APC}}{S_{ACDE}}\).
|
1/2
| 64.84375 |
11,983 |
The positive integers are grouped as follows:
\( A_1 = \{1\}, A_2 = \{2, 3, 4\}, A_3 = \{5, 6, 7, 8, 9\} \), and so on.
In which group does 2009 belong?
|
45
| 89.84375 |
11,984 |
There are 150 different cards on the table with the numbers $2, 4, 6, \ldots, 298, 300$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the chosen cards is divisible by $5$?
|
2235
| 77.34375 |
11,985 |
One student has 6 mathematics books, and another has 8. In how many ways can they exchange three books?
|
1120
| 85.15625 |
11,986 |
Let $[ x ]$ denote the greatest integer less than or equal to $x$. For example, $[10.2] = 10$. Calculate the value of $\left[\frac{2017 \times 3}{11}\right] + \left[\frac{2017 \times 4}{11}\right] + \left[\frac{2017 \times 5}{11}\right] + \left[\frac{2017 \times 6}{11}\right] + \left[\frac{2017 \times 7}{11}\right] + \left[\frac{2017 \times 8}{11}\right]$.
|
6048
| 97.65625 |
11,987 |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \sin ^{4}\left(\frac{x}{2}\right) \cos ^{4}\left(\frac{x}{2}\right) d x
$$
|
\frac{3\pi}{8}
| 93.75 |
11,988 |
In the set \(\{1, 2, 3, \cdots, 99, 100\}\), how many numbers \(n\) satisfy the condition that the tens digit of \(n^2\) is odd?
(45th American High School Mathematics Examination, 1994)
|
20
| 100 |
11,989 |
Given $f(x) = e^{-x}$, calculate the limit $$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1 - 2\Delta x)}{\Delta x}$$.
|
-\frac{3}{e}
| 85.9375 |
11,990 |
Given $f\left(x\right)=\left(1+2x\right)^{n}$, where the sum of the binomial coefficients in the expansion is $64$, and ${\left(1+2x\right)^n}={a_0}+{a_1}x+{a_2}{x^2}+…+{a_n}{x^n}$.
$(1)$ Find the value of $a_{2}$;
$(2)$ Find the term with the largest binomial coefficient in the expansion of $\left(1+2x\right)^{n}$;
$(3)$ Find the value of $a_{1}+2a_{2}+3a_{3}+\ldots +na_{n}$.
|
2916
| 85.9375 |
11,991 |
In an election, there are two candidates, A and B, who each have 5 supporters. Each supporter, independent of other supporters, has a \(\frac{1}{2}\) probability of voting for his or her candidate and a \(\frac{1}{2}\) probability of being lazy and not voting. What is the probability of a tie (which includes the case in which no one votes)?
|
63/256
| 35.9375 |
11,992 |
If \( A \) is a positive integer such that \( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} \), find the value of \( A \).
|
22
| 58.59375 |
11,993 |
If two people, A and B, work together on a project, they can complete it in a certain number of days. If person A works alone to complete half of the project, it takes them 10 days less than it would take both A and B working together to complete the entire project. If person B works alone to complete half of the project, it takes them 15 days more than it would take both A and B working together to complete the entire project. How many days would it take for A and B to complete the entire project working together?
|
60
| 62.5 |
11,994 |
A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the building have?
|
13
| 82.03125 |
11,995 |
What is the maximum value of \( N \) such that \( N! \) has exactly 2013 trailing zeros?
|
8069
| 18.75 |
11,996 |
Find the integer \(n\), such that \(-180 < n < 180\), for which \(\tan n^\circ = \tan 276^\circ.\)
|
96
| 46.09375 |
11,997 |
Solve for \(x\): \[\frac{x-60}{3} = \frac{5-3x}{4}.\]
|
\frac{255}{13}
| 93.75 |
11,998 |
Given the sequence $\{a_{n}\}$ where ${a}_{1}=\frac{1}{2}$, ${a}_{4}=\frac{1}{8}$, and $a_{n+1}a_{n}+a_{n-1}a_{n}=2a_{n+1}a_{n-1}$ for $n\geqslant 2$, find the value of $T_{2024}$, where $T_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$ and $b_{n}=a_{n}a_{n+1}$.
|
\frac{506}{2025}
| 35.15625 |
11,999 |
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$ ). The expected value of $N'$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$ .
*Proposed by Yannick Yao*
|
555556
| 9.375 |
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