Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,800 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C,\sin A+\sin B)$, $\overrightarrow{n}=(\sin B-\sin C,\sin A)$, and $\overrightarrow{m}\perp \overrightarrow{n}$.
(1) Find the measure of angle $C$;
(2) If $\triangle ABC$ is an isosceles triangle and its circumcircle is a unit circle, find the perimeter $L$ of $\triangle ABC$.
|
2+\sqrt{3}
| 85.9375 |
20,801 |
There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candies is 2010, then the sum of all possible values of $a$ is.
|
105
| 5.46875 |
20,802 |
Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\lim_{n \to \infty} a_n$ .
|
1.7
| 0 |
20,803 |
Equilateral triangle $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$. Determine the sum $\sum_{k=1}^4 (CE_k)^2$.
|
576
| 10.9375 |
20,804 |
Calculate \(7 \cdot 9\frac{2}{5}\).
|
65\frac{4}{5}
| 77.34375 |
20,805 |
Given that line $l$ intersects the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ at points $A$ and $B$, and the midpoint of segment $AB$ has coordinates $(6, 2)$, determine the slope of line $l$.
|
\frac{4}{3}
| 76.5625 |
20,806 |
The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $3$ at two points $C$ and $D$. Find $(CD)^2$.
|
\frac{224}{9}
| 71.09375 |
20,807 |
The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map?
|
30:1
| 7.8125 |
20,808 |
Given vectors $\overrightarrow{a}=(2\sin x,-\cos x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x,2\cos x)$, and function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.
(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\in\left[\dfrac{\pi}{12},\dfrac{2\pi}{3}\right]$;
(II) Translate the graph of function $f(x)$ to the left by $\dfrac{\pi}{3}$ unit to obtain the graph of function $g(x)$. In triangle $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. If $g\left(\dfrac{A}{2}\right)=1$, $a=2$, and $b+c=4$, find the area of $\triangle ABC$.
|
\sqrt{3}
| 95.3125 |
20,809 |
Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$. Let $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}$ for $k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$.
|
2000
| 21.09375 |
20,810 |
Given that 5 students are to be distributed into two groups, A and B, with at least one person in each group, and student A cannot be in group A, calculate the number of different distribution schemes.
|
15
| 10.9375 |
20,811 |
Given the line $l: x+ \sqrt {2}y=4 \sqrt {2}$ and the ellipse $C: mx^{2}+ny^{2}=1$ ($n>m>0$) have exactly one common point $M[2 \sqrt {2},2]$.
(1) Find the equation of the ellipse $C$;
(2) Let the left and right vertices of the ellipse $C$ be $A$ and $B$, respectively, and $O$ be the origin. A moving point $Q$ satisfies $QB \perp AB$. Connect $AQ$ and intersect the ellipse at point $P$. Find the value of $\overrightarrow {OQ} \cdot \overrightarrow {OP}$.
|
16
| 35.15625 |
20,812 |
In triangle $ABC$, $AB = 6$, $AC = 8$, and $BC = 10$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
|
\frac{94}{15}
| 26.5625 |
20,813 |
Given that 60% of all students in Ms. Hanson's class answered "Yes" to the question "Do you love science" at the beginning of the school year, 40% answered "No", 80% answered "Yes" and 20% answered "No" at the end of the school year, calculate the difference between the maximum and the minimum possible values of y%, the percentage of students that gave a different answer at the beginning and end of the school year.
|
40\%
| 34.375 |
20,814 |
In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$.
|
\frac{15}{17}
| 67.96875 |
20,815 |
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$ , find $100a+b$ .
*Proposed by Michael Tang*
|
300
| 47.65625 |
20,816 |
Due to a snow and ice disaster, a citrus orchard suffered severe damage. To address this, experts proposed a rescue plan for the fruit trees, which needs to be implemented over two years and is independent each year. The plan estimates that in the first year, the probability of the citrus yield recovering to 1.0, 0.9, and 0.8 times the pre-disaster level is 0.2, 0.4, and 0.4, respectively. In the second year, the probability of the citrus yield reaching 1.5, 1.25, and 1.0 times the first year's yield is 0.3, 0.3, and 0.4, respectively. Calculate the probability that the citrus yield will exactly reach the pre-disaster level after two years.
|
0.2
| 15.625 |
20,817 |
What is the maximum number of consecutive positive integers starting from 10 that can be added together before the sum exceeds 500?
|
23
| 41.40625 |
20,818 |
Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$ taking into consideration all possible choices of triangle $ABC$ and of point $P$ .
by Elton Bojaxhiu, Albania
|
\frac{2}{\sqrt{3}}
| 0 |
20,819 |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{b_n} \right\}$ is $T_n$. If $T_n < M$ holds for all positive integers $n$, then the minimum value of $M$ is ______.
|
10
| 27.34375 |
20,820 |
Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$.
|
20\pi
| 21.875 |
20,821 |
Given that point M $(3n-2, 2n+7)$ is on the angle bisector of the second and fourth quadrants, then $n=$ .
|
-1
| 100 |
20,822 |
Given that the terminal side of angle $\alpha$ passes through the point $(3a, 4a)$ ($a < 0$), then $\sin\alpha=$ ______, $\tan(\pi-2\alpha)=$ ______.
|
\frac{24}{7}
| 100 |
20,823 |
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, and $a_{n+2} - a_n = 1 + (-1)^n$ $(n \in \mathbb{N}^*)$, then $S_{100} = \_\_\_\_\_\_\_\_$.
|
2600
| 95.3125 |
20,824 |
Given the function $f(x)=2|x|+|2x-m|$ where $m>0$, and the graph of the function is symmetric about the line $x=1$.
$(Ⅰ)$ Find the minimum value of $f(x)$.
$(Ⅱ)$ Let $a$ and $b$ be positive numbers such that $a+b=m$. Find the minimum value of $\frac{1}{a}+\frac{4}{b}$.
|
\frac{9}{4}
| 35.9375 |
20,825 |
Given that the terminal side of $\alpha$ passes through the point $(a, 2a)$ (where $a < 0$),
(1) Find the values of $\cos\alpha$ and $\tan\alpha$.
(2) Simplify and find the value of $$\frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\sin(-\alpha+ \frac {3\pi}{2})}{\tan(-\alpha-\pi)\sin(-\pi-\alpha)}$$.
|
\frac{1}{10}
| 71.875 |
20,826 |
An academy has $200$ students and $8$ teachers. The class sizes are as follows: $80, 40, 40, 20, 10, 5, 3, 2$. Calculate the average number of students per class as seen by a randomly picked teacher, represented by $t$, and the average number of students per class from the perspective of a randomly selected student, denoted as $s$, and compute the value of $t-s$.
|
-25.69
| 4.6875 |
20,827 |
Given two quadratic functions $y=x^{2}-2x+2$ and $y=-x^{2}+ax+b$ $(a > 0,b > 0)$, if their tangent lines at one of their intersection points are perpendicular to each other, find the maximum value of $ab$.
|
\frac{25}{16}
| 11.71875 |
20,828 |
Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$?
|
\frac{35}{132}
| 3.125 |
20,829 |
Let $\triangle PQR$ be a right triangle with angle $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $192$ and $PR = 32$, what is the length of $QS$?
|
12
| 76.5625 |
20,830 |
If a 5-digit number $\overline{x a x a x}$ is divisible by 15, calculate the sum of all such numbers.
|
220200
| 11.71875 |
20,831 |
If a point $(-4,a)$ lies on the terminal side of an angle of $600^{\circ}$, determine the value of $a$.
|
-4 \sqrt{3}
| 96.09375 |
20,832 |
In the triangular pyramid $P-ABC$, $PA \perp$ the base $ABC$, $AB=1$, $AC=2$, $\angle BAC=60^{\circ}$, the volume is $\frac{\sqrt{3}}{3}$, then the volume of the circumscribed sphere of the triangular pyramid is $\_\_\_\_\_\_\_\_\_\_.$
|
\frac{8 \sqrt{2}}{3} \pi
| 59.375 |
20,833 |
Given $a-b=4$ and $b+c=2$, determine the value of $a^2+b^2+c^2-ab+bc+ca$.
|
28
| 60.9375 |
20,834 |
What is the ratio of the volume of a cube with edge length four inches to the volume of a cube with edge length two feet? Additionally, calculate the ratio of their surface areas.
|
\frac{1}{36}
| 80.46875 |
20,835 |
Any type of nature use affects at least one of the natural resources, including lithogenic base, soil, water, air, plant world, and animal world. Types that affect the same set of resources belong to the same type. Research has shown that types of nature use developed in the last 700 years can be divided into 23 types. How many types remain unused?
|
40
| 32.8125 |
20,836 |
Given the function $f(x) = \sqrt{3}\cos x\sin x - \frac{1}{2}\cos 2x$.
(1) Find the smallest positive period of $f(x)$.
(2) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding values of $x$.
|
-\frac{1}{2}
| 14.0625 |
20,837 |
Let $a$, $n$, and $l$ be real numbers, and suppose that the roots of the equation \[x^4 - 10x^3 + ax^2 - nx + l = 0\] are four distinct positive integers. Compute $a + n + l.$
|
109
| 99.21875 |
20,838 |
Find the value of cos $$\frac {π}{11}$$cos $$\frac {2π}{11}$$cos $$\frac {3π}{11}$$cos $$\frac {4π}{11}$$cos $$\frac {5π}{11}$$\=\_\_\_\_\_\_．
|
\frac {1}{32}
| 39.0625 |
20,839 |
Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?
|
20
| 46.875 |
20,840 |
James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins?
|
\frac{2}{3}
| 28.90625 |
20,841 |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{b_n} \right\}$ is $T_n$. If $T_n < M$ holds for all positive integers $n$, then the minimum value of $M$ is ______.
|
10
| 24.21875 |
20,842 |
Within a triangular piece of paper, there are 100 points, along with the 3 vertices of the triangle, making it a total of 103 points, and no three of these points are collinear. If these points are used as vertices to create triangles, and the paper is cut into small triangles, then the number of such small triangles is ____.
|
201
| 82.03125 |
20,843 |
In $\triangle ABC$, if $BC=4$, $\cos B= \frac{1}{4}$, then $\sin B=$ _______, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is: _______.
|
-\frac{1}{4}
| 0.78125 |
20,844 |
Find the gradient of the function \( z = x^2 - xy + y^3 \) at the point \( A(1, -1) \) and the derivative in the direction of the vector \( \bar{a} = 3\bar{i} - 4\bar{j} \).
|
\frac{1}{5}
| 43.75 |
20,845 |
Solve for $x$ in the equation $\frac{4}{7} \cdot \frac{1}{9} \cdot x = 14$.
|
220.5
| 70.3125 |
20,846 |
If point A $(3,1)$ lies on the line $mx+ny+1=0$, where $mn>0$, then the maximum value of $\frac {3}{m}+ \frac {1}{n}$ is \_\_\_\_\_.
|
-16
| 28.90625 |
20,847 |
The teacher and two boys and two girls stand in a row for a photo, with the requirement that the two girls must stand together and the teacher cannot stand at either end. Calculate the number of different arrangements.
|
24
| 45.3125 |
20,848 |
Given a hyperbola \( H: x^{2}-y^{2}=1 \) with a point \( M \) in the first quadrant, and a line \( l \) tangent to the hyperbola \( H \) at point \( M \), intersecting the asymptotes of \( H \) at points \( P \) and \( Q \) (where \( P \) is in the first quadrant). If point \( R \) is on the same asymptote as \( Q \), then the minimum value of \( \overrightarrow{R P} \cdot \overrightarrow{R Q} \) is ______.
|
-\frac{1}{2}
| 9.375 |
20,849 |
In triangle $\triangle ABC$, $a$, $b$, and $c$ are the opposite sides of angles $A$, $B$, and $C$ respectively, and $\dfrac{\cos B}{\cos C}=-\dfrac{b}{2a+c}$.
(1) Find the measure of angle $B$;
(2) If $b=\sqrt {13}$ and $a+c=4$, find the area of $\triangle ABC$.
|
\dfrac{3\sqrt{3}}{4}
| 85.15625 |
20,850 |
Find the square root of $\dfrac{9!}{126}$.
|
12.648
| 0 |
20,851 |
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$.
|
\frac{\pi}{12}
| 78.125 |
20,852 |
We call a pair of natural numbers $(a, p)$ good if the number $a^{3} + p^{3}$ is divisible by $a^{2} - p^{2}$, and $a > p$.
(a) (1 point) Give any possible value of $a$ for which the pair $(a, 11)$ is good.
(b) (3 points) Find the number of good pairs, where $p$ is a prime number less than 16.
|
18
| 15.625 |
20,853 |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. What is the percentage increase in the area of the rectangle?
|
49.5\%
| 85.15625 |
20,854 |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?
|
32
| 7.03125 |
20,855 |
Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ .
|
40
| 39.84375 |
20,856 |
Let $ABC$ be a triangle with $AB=9$ , $BC=10$ , $CA=11$ , and orthocenter $H$ . Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$ . Compute $AD$ .
|
\sqrt{102}
| 8.59375 |
20,857 |
A ball was floating in a lake when the lake froze. The ball was removed, leaving a hole $32$ cm across at the top and $16$ cm deep. What was the radius of the ball (in centimeters)?
|
16
| 0.78125 |
20,858 |
Given the function $f(x)=\frac{x}{ax+b}(a≠0)$, and its graph passes through the point $(-4,4)$, and is symmetric about the line $y=-x$, find the value of $a+b$.
|
\frac{3}{2}
| 41.40625 |
20,859 |
Let \( a \), \( b \), and \( c \) be the roots of \( x^3 - x + 2 = 0 \). Find \( \frac{1}{a+2} + \frac{1}{b+2} + \frac{1}{c+2} \).
|
\frac{11}{4}
| 79.6875 |
20,860 |
Suppose $\cos S = 0.5$ in a right triangle where $SP = 10$. What is $SR$?
[asy]
pair S,P,R;
S = (0,0);
P = (10,0);
R = (0,10*tan(acos(0.5)));
draw(S--P--R--S);
draw(rightanglemark(S,P,R,18));
label("$S$",S,SW);
label("$P$",P,SE);
label("$R$",R,N);
label("$10$",P/2,S);
[/asy]
|
20
| 75.78125 |
20,861 |
Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers.
|
240
| 35.9375 |
20,862 |
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. What is the ratio of the speed of $A$ to the speed of $B$?
|
2: 3
| 0 |
20,863 |
Round to the nearest thousandth and then subtract 0.005: 18.48571.
|
18.481
| 62.5 |
20,864 |
A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?
A) 12
B) 15
C) 18
D) 20
E) 24
|
20
| 83.59375 |
20,865 |
Find the product of the divisors of \(72\).
|
72^6
| 0 |
20,866 |
Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$ . It is given that $PO=3 \text{cm}$ and $SO=4 \text{cm}$ . Moreover, the area of the triangle $POR$ is $7 \text{cm}^2$ . Find the area of the triangle $QOS$ .
|
112/9
| 86.71875 |
20,867 |
Points with integer coordinates (including zero) are called lattice points (or grid points). Find the total number of lattice points (including those on the boundary) in the region bounded by the x-axis, the line \(x=4\), and the parabola \(y=x^2\).
|
35
| 93.75 |
20,868 |
In $\triangle ABC$, $\sin ^{2}A-\sin ^{2}C=(\sin A-\sin B)\sin B$, then angle $C$ equals to $\dfrac {\pi}{6}$.
|
\dfrac {\pi}{3}
| 50 |
20,869 |
There are two rows of seats, with 11 seats in the front row and 12 seats in the back row. Now, we need to arrange for two people, A and B, to sit down. It is stipulated that the middle 3 seats of the front row cannot be occupied, and A and B cannot sit next to each other. How many different arrangements are there?
|
346
| 2.34375 |
20,870 |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$.
|
413.5
| 0 |
20,871 |
A geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\frac{1}{2}$. Let $\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\Pi_{n}$ maximized?
|
11
| 20.3125 |
20,872 |
Consider the decimal function denoted by $\{ x \} = x - \lfloor x \rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\{x\} = \frac{1}{\lfloor x \rfloor}$. Express your answer as a mixed number.
|
21\frac{9}{20}
| 23.4375 |
20,873 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.
|
\frac{670}{2011}
| 0.78125 |
20,874 |
In $\triangle ABC$, $\angle A= \frac {\pi}{3}$, $BC=3$, $AB= \sqrt {6}$, find $\angle C=$ \_\_\_\_\_\_ and $AC=$ \_\_\_\_\_\_.
|
\frac{\sqrt{6} + 3\sqrt{2}}{2}
| 58.59375 |
20,875 |
Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$.
|
4\pi
| 47.65625 |
20,876 |
Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars.
|
\frac{300}{13}
| 5.46875 |
20,877 |
Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men.
|
100
| 71.09375 |
20,878 |
Given $\alpha \in \left(0,\pi \right)$, $tan2\alpha=\frac{sin\alpha}{2+cos\alpha}$, find the value of $\ tan \alpha$.
|
-\sqrt{15}
| 7.8125 |
20,879 |
Evaluate: \( \frac {\tan 150^{\circ} \cos (-210^{\circ}) \sin (-420^{\circ})}{\sin 1050^{\circ} \cos (-600^{\circ})} \).
|
-\sqrt{3}
| 87.5 |
20,880 |
If $a^{2}-4a+3=0$, find the value of $\frac{9-3a}{2a-4} \div (a+2-\frac{5}{a-2})$ .
|
-\frac{3}{8}
| 14.0625 |
20,881 |
(1) Given that $x < 3$, find the maximum value of $f(x) = \frac{4}{x - 3} + x$;
(2) Given that $x, y \in \mathbb{R}^+$ and $x + y = 4$, find the minimum value of $\frac{1}{x} + \frac{3}{y}$.
|
1 + \frac{\sqrt{3}}{2}
| 68.75 |
20,882 |
The constant term in the expansion of the binomial \\((x \sqrt {x}- \dfrac {1}{x})^{5}\\) is \_\_\_\_\_\_ . (Answer with a number)
|
-10
| 96.875 |
20,883 |
Let $p$, $q$, and $r$ be the roots of $x^3 - 2x^2 - x + 3 = 0$. Find $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$.
|
-3
| 44.53125 |
20,884 |
In the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; (a > b > 0)$, a line with slope $k(k > 0)$ intersects the ellipse at the left vertex $A$ and another point $B$. The projection of point $B$ on the $x$-axis is exactly the right focus $F$. If the eccentricity of the ellipse $e= \frac{1}{3}$, then the value of $k$ is _____.
|
\frac{2}{3}
| 81.25 |
20,885 |
Three people, A, B, and C, stand on a staircase with 7 steps. If each step can accommodate at most 2 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is.
|
336
| 45.3125 |
20,886 |
In a box, there are two red balls, two yellow balls, and two blue balls. If a ball is randomly drawn from the box, at least how many balls need to be drawn to ensure getting balls of the same color? If one ball is drawn at a time without replacement until balls of the same color are obtained, let $X$ be the number of different colors of balls drawn during this process. Find $E(X)=$____.
|
\frac{11}{5}
| 20.3125 |
20,887 |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$?
|
59
| 76.5625 |
20,888 |
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center of mass) of $\triangle ABC$?
|
18\pi
| 8.59375 |
20,889 |
Draw a perpendicular line from the left focus $F_1$ of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ to the $x$-axis meeting the ellipse at point $P$, and let $F_2$ be the right focus. If $\angle F_{1}PF_{2}=60^{\circ}$, calculate the eccentricity of the ellipse.
|
\frac{\sqrt{3}}{3}
| 66.40625 |
20,890 |
Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$.
|
54.6428571
| 0 |
20,891 |
The mathematical giant Euler in history was the first to represent polynomials in terms of $x$ using the notation $f(x)$. For example, $f(x) = x^2 + 3x - 5$, and the value of the polynomial when $x$ equals a certain number is denoted by $f(\text{certain number})$. For example, when $x = -1$, the value of the polynomial $x^2 + 3x - 5$ is denoted as $f(-1) = (-1)^2 + 3 \times (-1) - 5 = -7$. Given $g(x) = -2x^2 - 3x + 1$, find the values of $g(-1)$ and $g(-2)$ respectively.
|
-1
| 50 |
20,892 |
When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young.
|
18
| 7.8125 |
20,893 |
Simplify and find the value of:<br/>$(1)$ If $a=2$ and $b=-1$, find the value of $(3{a^2}b+\frac{1}{4}a{b^2})-(\frac{3}{4}a{b^2}-{a^2}b)$.<br/>$(2)$ If the value of the algebraic expression $(2x^{2}+ax-y+6)-(2bx^{2}-3x+5y-1)$ is independent of the variable $x$, find the value of the algebraic expression $5ab^{2}-[a^{2}b+2(a^{2}b-3ab^{2})]$.
|
-60
| 54.6875 |
20,894 |
An urn initially contains two red balls and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation five times: he draws a ball from the urn at random and then takes a ball of the same color from the box and adds those two matching balls to the urn. After the five iterations, the urn contains eight balls. What is the probability that the urn contains three red balls and five blue balls?
A) $\frac{1}{10}$
B) $\frac{1}{21}$
C) $\frac{4}{21}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$
|
\frac{4}{21}
| 22.65625 |
20,895 |
Given $0 \leq x_0 < 1$, for all integers $n > 0$, let
$$
x_n = \begin{cases}
2x_{n-1}, & \text{if } 2x_{n-1} < 1,\\
2x_{n-1} - 1, & \text{if } 2x_{n-1} \geq 1.
\end{cases}
$$
Find the number of initial values of $x_0$ such that $x_0 = x_6$.
|
64
| 16.40625 |
20,896 |
Given that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$.
|
-170
| 0 |
20,897 |
Given $\cos \left(\alpha- \frac {\beta}{2}\right)=- \frac {1}{9}$ and $\sin \left( \frac {\alpha}{2}-\beta\right)= \frac {2}{3}$, with $0 < \beta < \frac {\pi}{2} < \alpha < \pi$, find $\sin \frac {\alpha+\beta}{2}=$ ______.
|
\frac {22}{27}
| 57.03125 |
20,898 |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000$ with $S(n)$ even. Find $|a-b|.$
|
54
| 1.5625 |
20,899 |
For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4?
|
34
| 35.9375 |
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