Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
19,000 |
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}.
|
94
| 12.5 |
19,001 |
The radian measure of -150° is equal to what fraction of π.
|
-\frac{5\pi}{6}
| 100 |
19,002 |
Given a function $f(x) = 2\sin x\cos x + 2\cos\left(x + \frac{\pi}{4}\right)\cos\left(x - \frac{\pi}{4}\right)$,
1. Find the interval(s) where $f(x)$ is monotonically decreasing.
2. If $\alpha \in (0, \pi)$ and $f\left(\frac{\alpha}{2}\right) = \frac{\sqrt{2}}{2}$, find the value of $\sin \alpha$.
|
\frac{\sqrt{6} + \sqrt{2}}{4}
| 62.5 |
19,003 |
The side surface of a cylinder unfolds into a rectangle with side lengths of $6\pi$ and $4\pi$. The surface area of the cylinder is ______.
|
24\pi^2 + 8\pi
| 17.96875 |
19,004 |
How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$?
|
49
| 89.84375 |
19,005 |
How many times does the digit 9 appear in the list of all integers from 1 to 700?
|
140
| 35.15625 |
19,006 |
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers?
|
101
| 25 |
19,007 |
Find the area of the triangle with vertices at $(1,4,5)$, $(3,4,1)$, and $(1,1,1)$.
|
\sqrt{61}
| 79.6875 |
19,008 |
How many positive whole numbers have cube roots that are less than $8$?
|
511
| 98.4375 |
19,009 |
Given the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $(a>b>0)$, the “companion point” of a point M$(x_0, y_0)$ on the ellipse C is defined as $$N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$$.
(1) Find the equation of the trajectory of the “companion point” N of point M on the ellipse C;
(2) If the “companion point” of the point $(1, \frac{3}{2})$ on the ellipse C is $\left(\frac{1}{2}, \frac{3}{2b}\right)$, find the range of values for $\overrightarrow{OM} \cdot \overrightarrow{ON}$ for any point M on ellipse C and its “companion point” N;
(3) When $a=2$, $b= \sqrt{3}$, a line l intersects the ellipse C at points A and B. If the “companion points” of A and B are P and Q respectively, and the circle with diameter PQ passes through the origin O, find the area of $\triangle OAB$.
|
\sqrt{3}
| 42.96875 |
19,010 |
If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $
|
11
| 61.71875 |
19,011 |
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?
|
13
| 50.78125 |
19,012 |
John has recorded completion times, in seconds, of 100, 108, 112, 104, and 110 for running a 100-meter dash. After another race, he realized his median time dropped to 106 seconds. What was his time, in seconds, for the latest race?
|
104
| 37.5 |
19,013 |
Let $m$ be the smallest positive, three-digit integer congruent to 6 (mod 13). Let $n$ be the smallest positive, four-digit integer congruent to 7 (mod 17). What is $n-m$?
|
900
| 28.125 |
19,014 |
Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$ , then the area of the original triangle is
|
200
| 53.125 |
19,015 |
How many ways can we arrange 4 math books, 6 English books, and 2 Science books on a shelf if:
1. All books of the same subject must stay together.
2. The Science books can be placed in any order, but cannot be placed next to each other.
(The math, English, and Science books are all different.)
|
207360
| 3.90625 |
19,016 |
Let \( X = \{0, a, b, c\} \) and \( M(X) = \{ f \mid f: X \rightarrow X \} \) be the set of all functions from \( X \) to itself. Define the addition operation \( \oplus \) on \( X \) as given in the following table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\oplus & 0 & a & b & c \\
\hline
0 & 0 & a & b & c \\
\hline
a & a & 0 & c & b \\
\hline
b & b & c & 0 & a \\
\hline
c & c & b & a & 0 \\
\hline
\end{array}
\]
1. Determine the number of elements in the set:
\[
S = \{ f \in M(X) \mid f((x \oplus y) \oplus x) = (f(x) \oplus f(y)) \oplus f(x), \forall x, y \in X \}.
\]
2. Determine the number of elements in the set:
\[
I = \{ f \in M(X) \mid f(x \oplus x) = f(x) \oplus f(x), \forall x \in X \}.
\]
|
64
| 25.78125 |
19,017 |
Each of the $2500$ students at a university studies either Physics or Chemistry, and some study both. The number who study Physics is between $70\%$ and $75\%$ of the university population, and the number who study Chemistry is between $40\%$ and $45\%$. Let $m$ be the smallest number of students who could study both subjects, and let $M$ be the largest number of students who could study both subjects. Find $M-m$.
|
250
| 91.40625 |
19,018 |
A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively.
$(1)$ If the event "$x+y=8$" is denoted as event $A$, find the probability of event $A$ occurring;
$(2)$ If the event "$x^{2}+y^{2} \leqslant 12$" is denoted as event $B$, find the probability of event $B$ occurring.
|
\dfrac{1}{6}
| 43.75 |
19,019 |
Given the function $f(x)=x+ \frac{1}{x}$, and $g(x)=f^{2}(x)-af(x)+2a$ has four distinct zeros $x\_1$, $x\_2$, $x\_3$, $x\_4$, determine the value of $[2-f(x\_1)]\cdot[2-f(x\_2)]\cdot[2-f(x\_3)]\cdot[2-f(x\_4)]$.
|
16
| 67.1875 |
19,020 |
In triangle $ABC$, the lengths of sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively. Given that $(2b - \sqrt{3}c)\cos A = \sqrt{3}a\cos C$.
1. Find the measure of angle $A$.
2. If angle $B = \frac{\pi}{6}$ and the length of the median $AM$ on side $BC$ is $\sqrt{7}$, find the area of triangle $ABC$.
|
\sqrt{3}
| 68.75 |
19,021 |
The diagram shows an \(n \times (n+1)\) rectangle tiled with \(k \times (k+1)\) rectangles, where \(n\) and \(k\) are integers and \(k\) takes each value from 1 to 8 inclusive. What is the value of \(n\)?
|
15
| 71.875 |
19,022 |
Let \(a\) and \(b\) be angles such that
\[\cos (a - b) = \cos a - \cos b.\]
Find the maximum value of \(\cos a\).
|
\sqrt{\frac{3+\sqrt{5}}{2}}
| 0 |
19,023 |
Given the function $f\left( x \right)=\sin\left( 2x+\varphi \right)+\sqrt{3}\cos\left( 2x+\varphi \right)$ $\left( 0 < \varphi < \pi \right)$, its graph is shifted left by $\frac{\pi }{4}$ units, and the shifted graph is symmetric about the point $\left( \frac{\pi }{2},0 \right)$. Find the minimum value of the function $g\left( x \right)=\cos\left( x+\varphi \right)$ on the interval $\left[ -\frac{\pi }{2},\frac{\pi }{6} \right]$.
|
\frac{1}{2}
| 42.96875 |
19,024 |
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & 5 & X & Y_{7}\\
&+& & & 5 & 2_{7}\\
\cline{2-6}
& & & 6 & 4 & X_{7}\\
\end{array} \]
|
10
| 42.96875 |
19,025 |
Given $\frac{cos2α}{sin(α-\frac{π}{4})}=-\frac{\sqrt{6}}{2}$, express $cos(α-\frac{π}{4})$ in terms of radicals.
|
\frac{\sqrt{6}}{4}
| 22.65625 |
19,026 |
Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?
|
105
| 39.84375 |
19,027 |
Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\frac {x^{2}}{16}+ \frac {y^{2}}{9}=1$, and a line passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|AB|=5$, calculate $|AF_{1}|+|BF_{1}|$.
|
11
| 95.3125 |
19,028 |
Find the distance between the foci of the ellipse
\[\frac{x^2}{45} + \frac{y^2}{5} = 9.\]
|
12\sqrt{10}
| 49.21875 |
19,029 |
Calculate the greatest common divisor (GCD) of the numbers 4557, 1953, and 5115.
|
93
| 100 |
19,030 |
Simplify first, then evaluate: $\left(\frac{x}{x-1}-1\right) \div \frac{{x}^{2}-1}{{x}^{2}-2x+1}$, where $x=\sqrt{5}-1$.
|
\frac{\sqrt{5}}{5}
| 82.03125 |
19,031 |
A unit has a total of 620 staff members. To investigate the time workers spend commuting, it was decided to survey 62 workers using a systematic sampling method. The entire staff was divided into 62 segments with equal intervals, and a simple random sampling method was used to determine that the starting number of the first segment was 4. What is the individual number of the worker that should be selected from the 40th segment?
|
394
| 96.09375 |
19,032 |
A square and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction.
|
\frac{130}{1}
| 11.71875 |
19,033 |
Sarah's six assignment scores are 87, 90, 86, 93, 89, and 92. What is the arithmetic mean of these six scores?
|
89.5
| 96.875 |
19,034 |
The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value?
|
45
| 21.09375 |
19,035 |
Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$ {10a+b:1\leq a\leq 5, 1\leq b\leq 5} $$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$ ?
|
47
| 29.6875 |
19,036 |
The sequence 1,3,1,3,3,1,3,3,3,1,3,3,3,3,1,3,... follows a certain rule. What is the sum of the first 44 terms in this sequence?
|
116
| 10.9375 |
19,037 |
Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\frac {x^{ \frac {1}{2}}-y^{ \frac {1}{2}}}{x^{ \frac {1}{2}}+y^{ \frac {1}{2}}}=$ ___.
|
- \frac { \sqrt {3}}{3}
| 0 |
19,038 |
Given the parabola $y^{2}=4x$, and the line $l$: $y=- \frac {1}{2}x+b$ intersects the parabola at points $A$ and $B$.
(I) If the $x$-axis is tangent to the circle with $AB$ as its diameter, find the equation of the circle;
(II) If the line $l$ intersects the negative semi-axis of $y$, find the maximum area of $\triangle AOB$.
|
\frac {32 \sqrt {3}}{9}
| 0 |
19,039 |
Define: In the sequence $\{a_{n}\}$, if $\frac{{a}_{n+2}}{{a}_{n+1}}-\frac{{a}_{n+1}}{{a}_{n}}=d(n∈{N}^{*})$, where $d$ is a constant, then the sequence $\{a_{n}\}$ is called a "geometric difference" sequence. Given a "geometric difference" sequence $\{a_{n}\}$ where $a_{1}=a_{2}=1$ and $a_{3}=3$, find $a_{5}=$______; $\frac{{a}_{31}}{{a}_{29}}=\_\_\_\_\_\_$.
|
3363
| 60.9375 |
19,040 |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(x) = \begin{cases} \log_{2}(x+1), & x \geqslant 0 \\ g(x), & x < 0 \end{cases}$, find $g[f(-7)]$.
|
-2
| 92.96875 |
19,041 |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?
|
10
| 38.28125 |
19,042 |
Given that an ellipse has the equation $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $e = \frac {\sqrt {6}}{3}$. The distance from the origin to the line that passes through points $A(0,-b)$ and $B(a,0)$ is $\frac {\sqrt {3}}{2}$.
$(1)$ Find the equation of the ellipse.
$(2)$ Given the fixed point $E(-1,0)$, if the line $y = kx + 2 \ (k \neq 0)$ intersects the ellipse at points $C$ and $D$, is there a value of $k$ such that the circle with diameter $CD$ passes through point $E$? Please provide an explanation.
|
\frac {7}{6}
| 7.03125 |
19,043 |
A merchant buys $n$ radios for $d$ dollars, where $d$ is a positive integer. The merchant sells two radios at half the cost price to a charity sale, and the remaining radios at a profit of 8 dollars each. If the total profit is 72 dollars, what is the smallest possible value of $n$?
|
12
| 71.875 |
19,044 |
A fair die (a cube toy with 1, 2, 3, 4, 5, 6 dots on each face) is rolled twice. Let 'm' be the number of dots that appear on the first roll, and 'n' be the number of dots that appear on the second roll. The probability that m = kn (k ∈ N*) is _____.
|
\frac{7}{18}
| 61.71875 |
19,045 |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.
|
1936
| 1.5625 |
19,046 |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?
|
43
| 69.53125 |
19,047 |
Given that $θ$ is an angle in the second quadrant, and $\sin θ$ and $\cos θ$ are the two roots of the equation $2x^2 + (\sqrt{3} - 1)x + m = 0$ (where $m \in \mathbb{R}$), find the value of $\sin θ - \cos θ$.
|
\frac{1 + \sqrt{3}}{2}
| 1.5625 |
19,048 |
From the set $\{ -3, 0, 0, 4, 7, 8\}$, find the probability that the product of two randomly selected numbers is $0$.
|
\frac{3}{5}
| 44.53125 |
19,049 |
Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers.
|
26
| 19.53125 |
19,050 |
Let \( x \in \mathbf{R} \). The algebraic expression
$$
(x+1)(x+2)(x+3)(x+4) + 2019
$$
has a minimum value of ( ).
|
2018
| 93.75 |
19,051 |
When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$.
|
338
| 64.84375 |
19,052 |
In her chemistry class, Samantha now needs to prepare a special solution that consists of chemical A, water, and a new substance, chemical B. The proportions in her initial test mix are 40% chemical A, 50% water, and 10% chemical B. She needs to prepare 1.20 liters of this mixture for an experiment. How much water is required for the new solution?
|
0.60
| 71.09375 |
19,053 |
Given that $x=1$ is a root of the quadratic equation $\left(m-2\right)x^{2}+4x-m^{2}=0$, calculate the value(s) of $m$.
|
-1
| 17.1875 |
19,054 |
Calculate the sum of 5739204.742 and -176817.835, and round the result to the nearest integer.
|
5562387
| 92.96875 |
19,055 |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.
|
1009
| 9.375 |
19,056 |
Given a point P on the hyperbola $C_1: \frac{x^2}{16} - \frac{y^2}{9} = 1$, a point Q on the circle $C_2: (x - 5)^2 + y^2 = 1$, and a point R on the circle $C_3: (x + 5)^2 + y^2 = 1$, find the maximum value of $|PQ| - |PR|$.
|
10
| 67.1875 |
19,057 |
A three-digit number is composed of three different non-zero digits in base ten. When divided by the sum of these three digits, the smallest quotient value is what?
|
10.5
| 67.96875 |
19,058 |
(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;
(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \frac{1}{ \sqrt{x}})^{n}$ is $16$, find the coefficient of the $x$ term in the expansion.
|
54
| 6.25 |
19,059 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 216$.
|
\frac{1}{216}
| 77.34375 |
19,060 |
Given that circle $\odot M$ passes through the point $(1,0)$ and is tangent to the line $x=-1$, $S$ is a moving point on the trajectory of the center $M$ of the circle, and $T$ is a moving point on the line $x+y+4=0$. Find the minimum value of $|ST|$.
|
\frac{3\sqrt{2}}{2}
| 41.40625 |
19,061 |
A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?
|
99
| 8.59375 |
19,062 |
A certain school conducted a survey on the daily reading time of students during the summer vacation, as shown in the table below:
| | Mean | Variance | Number of Students |
|---------|------|----------|--------------------|
| Grade 10 | $2.7$ | $1$ | $800$ |
| Grade 11 | $3.1$ | $2$ | $600$ |
| Grade 12 | $3.3$ | $3$ | $600$ |
Find the variance of the daily reading time for all students.
|
1.966
| 14.84375 |
19,063 |
Determine the number of ways to arrange the letters of the word SUCCESS.
|
420
| 36.71875 |
19,064 |
Find the positive value of $x$ that satisfies the equation:
\[\log_2 (x + 2) + \log_{4} (x^2 - 2) + \log_{\frac{1}{2}} (x + 2) = 5.\]
|
\sqrt{1026}
| 14.84375 |
19,065 |
What is the sum of all real numbers \(x\) for which \(|x^2 - 14x + 45| = 3?\)
A) 12
B) 14
C) 16
D) 18
|
14
| 31.25 |
19,066 |
Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$.
|
-2
| 13.28125 |
19,067 |
The price of the jacket was increased and then decreased by a certain percent, and then a 10% discount was applied, resulting in a final price that is 75% of the original price. Determine the percent by which the price was increased and then decreased.
|
40.82
| 42.96875 |
19,068 |
In a bag, there are 2 black balls labeled $1$ and $2$, and 3 white balls labeled $3$, $4$, and $5$. These 5 balls are identical except for their labels and colors.
$(1)$ If two balls are randomly drawn from the bag with replacement, one at a time, what is the probability of drawing a black ball first and then a white ball?
$(2)$ If two balls are randomly drawn from the bag without replacement, denoting the label of a black ball as $x$ and the label of a white ball as $y$, what is the probability that $y-x \gt 2$?
|
\frac{3}{10}
| 20.3125 |
19,069 |
In the plane Cartesian coordinate system \( xO y \), the circle \( \Omega \) and the parabola \( \Gamma: y^{2} = 4x \) share exactly one common point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \).
|
\frac{4 \sqrt{3}}{9}
| 0.78125 |
19,070 |
In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy]
|
13.5
| 68.75 |
19,071 |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = 2x.
\]
|
51
| 73.4375 |
19,072 |
For any two positive integers, define the operation (represented by the operator ⊕): when both $m$ and $n$ are positive even numbers or both are positive odd numbers, $m⊕n=m+n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m⊕n=m×n$. For example, $4⊕6=4+6=10$, $3⊕7=3+7=10$, $3⊕4=3×4=12$. Under the above definition, the number of elements in the set $M=\{(a,b)|a⊕b=12, a,b\in\mathbb{N}^*\}$ is __.
|
15
| 40.625 |
19,073 |
Given the function $$f(x)=\sin^{2}x+2 \sqrt {3}\sin x\cos x- \frac {1}{2}\cos 2x$$, where $x\in\mathbb{R}$.
(I) Find the smallest positive period and the range of $f(x)$.
(II) If $$x_{0}(0\leq x_{0}\leq \frac {\pi}{2})$$ is a zero of $f(x)$, find the value of $\sin 2x_{0}$.
|
\frac { \sqrt {15}- \sqrt {3}}{8}
| 0 |
19,074 |
Within the range of 0° to 360°, find the angle(s) with the same terminal side as -120°.
|
240
| 98.4375 |
19,075 |
At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
*Author: Ray Li*
|
52
| 91.40625 |
19,076 |
Find the smallest positive integer \( n \) such that for any given \( n \) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \( R_{1}, R_{2}, R_{3} \) such that \( R_{1} \) can be nested inside \( R_{2} \) and \( R_{2} \) can be nested inside \( R_{3} \).
|
101
| 12.5 |
19,077 |
Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$
|
397
| 92.1875 |
19,078 |
In a circle with center $O$, the measure of $\angle SIP$ is $48^\circ$ and $OS=12$ cm. Find the number of centimeters in the length of arc $SP$ and also determine the length of arc $SXP$, where $X$ is a point on the arc $SP$ such that $\angle SXP = 24^\circ$. Express your answer in terms of $\pi$.
|
3.2\pi
| 0 |
19,079 |
Given $-\frac{\pi}{2} < x < 0$, $\sin x + \cos x = \frac{1}{5}$.
(1) Find the value of $\sin x - \cos x$.
(2) Find the value of $\tan x$.
|
-\frac{3}{4}
| 79.6875 |
19,080 |
Given a quadratic function $y=-x^{2}+bx+c$ where $b$ and $c$ are constants.
$(1)$ If $y=0$ and the corresponding values of $x$ are $-1$ and $3$, find the maximum value of the quadratic function.
$(2)$ If $c=-5$, and the quadratic function $y=-x^{2}+bx+c$ intersects the line $y=1$ at a unique point, find the expression of the quadratic function in this case.
$(3)$ If $c=b^{2}$, and the maximum value of the function $y=-x^{2}+bx+c$ is $20$ when $b\leqslant x\leqslant b+3$, find the value of $b$.
|
-4
| 74.21875 |
19,081 |
Triangle $ABC$ is isosceles with $AB = AC = 2$ and $BC = 1.5$. Points $E$ and $G$ are on segment $\overline{AC}$, and points $D$ and $F$ are on segment $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$, trapezoid $DFGE$, and trapezoid $FBCG$ all have the same perimeter. Find the sum $DE+FG$.
A) $\frac{17}{6}$
B) $\frac{19}{6}$
C) $\frac{21}{6}$
D) $\frac{11}{3}$
|
\frac{19}{6}
| 42.1875 |
19,082 |
Find the sum of all prime numbers between $1$ and $120$ that are simultaneously $1$ greater than a multiple of $3$ and $1$ less than a multiple of $5$.
|
207
| 7.8125 |
19,083 |
Let $d_1$, $d_2$, $d_3$, $d_4$, $e_1$, $e_2$, $e_3$, and $e_4$ be real numbers such that for every real number $x$, we have
\[
x^8 - 2x^7 + 2x^6 - 2x^5 + 2x^4 - 2x^3 + 2x^2 - 2x + 1 = (x^2 + d_1 x + e_1)(x^2 + d_2 x + e_2)(x^2 + d_3 x + e_3)(x^2 + d_4 x + e_4).
\]
Compute $d_1 e_1 + d_2 e_2 + d_3 e_3 + d_4 e_4$.
|
-2
| 24.21875 |
19,084 |
Five dice with faces numbered 1 through 6 are arranged in a configuration where 14 of the visible faces are showing. The visible numbers are 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 4, 5, 6. What is the total number of dots NOT visible in this view?
|
49
| 15.625 |
19,085 |
Let $x$ be a multiple of $7200$. Determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+5)(4x+7)$ and $x$.
|
30
| 53.90625 |
19,086 |
Let $T$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2011}$. What is $\log_{2}(T)$?
|
1005
| 56.25 |
19,087 |
In $\triangle ABC$, $\overrightarrow {AD}=3 \overrightarrow {DC}$, $\overrightarrow {BP}=2 \overrightarrow {PD}$, if $\overrightarrow {AP}=λ \overrightarrow {BA}+μ \overrightarrow {BC}$, then $λ+μ=\_\_\_\_\_\_$.
|
- \frac {1}{3}
| 18.75 |
19,088 |
Given the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and $P$ is a point on hyperbola $C$ in the second quadrant. If the line $y=\dfrac{b}{a}x$ is exactly the perpendicular bisector of the segment $PF_2$, then find the eccentricity of the hyperbola $C$.
|
\sqrt{5}
| 42.96875 |
19,089 |
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\]
|
\frac{1}{27}
| 28.90625 |
19,090 |
Out of 8 shots, 3 hit the target, and we are interested in the total number of ways in which exactly 2 hits are consecutive.
|
30
| 18.75 |
19,091 |
Given the function $f(x)= \begin{cases} |\ln x|, & (0 < x\leqslant e^{3}) \\ e^{3}+3-x, & (x > e^{3})\end{cases}$, there exist $x\_1 < x\_2 < x\_3$ such that $f(x\_1)=f(x\_2)=f(x\_3)$. Find the maximum value of $\frac{f(x\_3)}{x\_2}$.
|
\frac{1}{e}
| 61.71875 |
19,092 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______.
|
-\frac{1}{2}
| 53.90625 |
19,093 |
Given a geometric sequence $\{a_n\}$, where $a_3$ and $a_7$ are the two roots of the quadratic equation $x^2+7x+9=0$, calculate the value of $a_5$.
|
-3
| 39.84375 |
19,094 |
A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria?
|
20
| 68.75 |
19,095 |
An ellipse and a hyperbola have the same foci $F\_1(-c,0)$, $F\_2(c,0)$. One endpoint of the ellipse's minor axis is $B$, and the line $F\_1B$ is parallel to one of the hyperbola's asymptotes. If the eccentricities of the ellipse and hyperbola are $e\_1$ and $e\_2$, respectively, find the minimum value of $3e\_1^2+e\_2^2$.
|
2\sqrt{3}
| 47.65625 |
19,096 |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{16} w_j.\]
|
16
| 47.65625 |
19,097 |
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a face card $\clubsuit$?
|
\frac{3}{204}
| 0 |
19,098 |
Given a sequence $\{a_{n}\}$ that satisfies ${a}_{n+1}=\frac{1}{3}{a}_{n}$, if $a_{4}+a_{5}=4$, calculate $a_{2}+a_{3}$.
|
36
| 99.21875 |
19,099 |
Let $A=\{0, |x|\}$ and $B=\{1, 0, -1\}$. If $A \subseteq B$, then $x$ equals \_\_\_\_\_\_; The union of sets $A$ and $B$, denoted $A \cup B$, equals \_\_\_\_\_\_; The complement of $A$ in $B$, denoted $\complement_B A$, equals \_\_\_\_\_\_.
|
\{-1\}
| 52.34375 |
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