Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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float64 0
100
|
|---|---|---|---|
19,200 |
The probability that B sits exactly in the middle in a row of three seats, given that three people A, B, and C are randomly seated.
|
\frac{1}{3}
| 92.96875 |
19,201 |
If \( m \) and \( n \) are the roots of the quadratic equation \( x^2 + 1994x + 7 = 0 \), then the value of \((m^2 + 1993m + 6)(n^2 + 1995n + 8)\) is
|
1986
| 89.0625 |
19,202 |
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ?
|
10
| 48.4375 |
19,203 |
Given the function $f(x)= \begin{cases} ax^{2}-2x-1, & x\geqslant 0\\ x^{2}+bx+c, & x < 0\end{cases}$ is an even function, and the line $y=t$ intersects the graph of $y=f(x)$ from left to right at four distinct points $A$, $B$, $C$, $D$. If $AB=BC$, then the value of the real number $t$ is \_\_\_\_\_\_.
|
- \dfrac {7}{4}
| 30.46875 |
19,204 |
Given $\frac{\sin \alpha \cos \alpha }{1-\cos 2\alpha }=\frac{1}{4}$, $\tan (\alpha-\beta)=2$, calculate $\tan (\beta-2\alpha)$.
|
\frac{4}{3}
| 85.9375 |
19,205 |
If $x=1$ is an extremum point of the function $f(x)=(x^{2}+ax-1)e^{x-1}$, determine the maximum value of $f(x)$.
|
5e^{-3}
| 28.125 |
19,206 |
A novel is to be recorded onto compact discs and takes 528 minutes to read aloud. Each disc can now hold up to 45 minutes of reading. Assuming that the smallest possible number of discs is used and that each disc contains the same length of reading, calculate the number of minutes of reading that will each disc contain.
|
44
| 75.78125 |
19,207 |
On the set of solutions to the system of constraints
$$
\left\{\begin{array}{l}
2-2 x_{1}-x_{2} \geqslant 0 \\
2-x_{1}+x_{2} \geqslant 0 \\
5-x_{1}-x_{2} \geqslant 0 \\
x_{1} \geqslant 0, \quad x_{2} \geqslant 0
\end{array}\right.
$$
find the minimum value of the function $F = x_{2} - x_{1}$.
|
-2
| 31.25 |
19,208 |
Calculate:
(1) $2\log_{2}{10}+\log_{2}{0.04}$
(2) $(\log_{4}{3}+\log_{8}{3})\cdot(\log_{3}{5}+\log_{9}{5})\cdot(\log_{5}{2}+\log_{25}{2})$
|
\frac{15}{8}
| 96.09375 |
19,209 |
Given a geometric sequence $\{a_n\}$ satisfies $a_1=3$, and $a_1+a_3+a_5=21$. Calculate the value of $a_3+a_5+a_7$.
|
42
| 100 |
19,210 |
Given the quadratic function $f(x) = mx^2 - 2x - 3$, if the solution set of the inequality $f(x) < 0$ is $(-1, n)$.
(1) Solve the inequality about $x$: $2x^2 - 4x + n > (m + 1)x - 1$;
(2) Determine whether there exists a real number $a \in (0, 1)$, such that the minimum value of the function $y = f(a^x) - 4a^{x+1}$ ($x \in [1, 2]$) is $-4$. If it exists, find the value of $a$; if not, explain why.
|
\frac{1}{3}
| 22.65625 |
19,211 |
The positive numbers \( x, y, \) and \( z \) are such that \( x + y + z = 5 \). What is the minimum value of the expression \( x^{2} + y^{2} + 2z^{2} - x^{2} y^{2} z \)?
|
-6
| 2.34375 |
19,212 |
Given the fraction $\frac{987654321}{2^{24}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point needed to express it as a decimal.
|
24
| 52.34375 |
19,213 |
Given an arithmetic sequence $\{a_n\}$, if $\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms $S_n$ has a maximum value, find the maximum value of $n$ for which $S_n > 0$.
|
19
| 64.84375 |
19,214 |
Given the function $f(x)=\sqrt{3}\sin^{2}(x+\frac{\pi}{4})-\cos^{2}x-\frac{1+\sqrt{3}}{2} (x\in\mathbb{R})$.
(1) Find the minimum value and the minimum positive period of the function $f(x)$;
(2) If $A$ is an acute angle, and vector $\overrightarrow{m}=(1,5)$ is perpendicular to vector $\overrightarrow{n}=(1,f(\frac{\pi}{4}-A))$, find $\cos 2A$.
|
\frac{4\sqrt{3}+3}{10}
| 10.9375 |
19,215 |
Let ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a>b>0)$ pass through the point $\left(0,4\right)$, with eccentricity $\frac{3}{5}$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If a line $l$ passing through the point $\left(3,0\right)$ with a slope of $\frac{4}{5}$ intersects the ellipse $C$ at points $A$ and $B$, find the length of the chord $|AB|$.
|
\frac{41}{5}
| 21.09375 |
19,216 |
Two sectors of a circle of radius $10$ overlap as shown, with centers at points $A$ and $B$. Each sector subtends an angle of $45^\circ$. Determine the area of the overlapping region.
[asy]
draw((0,0)--(7.07,-7.07)--(14.14,0)--(7.07,7.07)--cycle,black+linewidth(1));
filldraw((7.07,7.07)..(10,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
filldraw((7.07,7.07)..(4.14,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
label("$A$",(0,0),W);
label("$C$",(7.07,7.07),N);
label("$B$",(14.14,0),E);
label("$D$",(7.07,-7.07),S);
label("$45^\circ$",(0,0),2E);
label("$45^\circ$",(14.14,0),2W);
[/asy]
|
25\pi - 50\sqrt{2}
| 36.71875 |
19,217 |
A triangle has one side of length $13$, and the angle opposite this side is $60^{\circ}$. The ratio of the other two sides is $4:3$. Calculate the area of this triangle.
|
39 \sqrt{3}
| 81.25 |
19,218 |
Compute $\sqrt{(41)(40)(39)(38) + 1}$.
|
1559
| 50.78125 |
19,219 |
In the Cartesian coordinate plane $(xOy)$, a moving point $P$ passes through the chords $PA$ and $PB$ of the circles $C_{1}$: $x^{2}+y^{2}+2x+2y+1=0$ and $C_{2}$: $x^{2}+y^{2}-4x-6y+9=0$ respectively, where $A$ and $B$ are the points of intersection. If $|PA|=|PB|$, find the minimum value of $|OP|$.
|
\frac{4}{5}
| 72.65625 |
19,220 |
In the series of activities with the theme "Labor Creates a Better Life" carried out at school, Class 1 of Grade 8 is responsible for the design, planting, and maintenance of a green corner on campus. The students agreed that each person would take care of one pot of green plants. They planned to purchase a total of 46 pots of two types of green plants, green lily and spider plant, with the number of green lily pots being at least twice the number of spider plant pots. It is known that each pot of green lily costs $9, and each pot of spider plant costs $6.
- $(1)$ The purchasing team plans to use the entire budget of $390 to purchase green lily and spider plants. How many pots of green lily and spider plants can be purchased?
- $(2)$ The planning team believes that there is a cheaper purchasing plan than $390 and requests the minimum total cost of purchasing the two types of green plants.
|
$369
| 0 |
19,221 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. Given the equation $$b^{2}- \frac {2 \sqrt {3}}{3}bcsinA+c^{2}=a^{2}$$.
(I) Find the measure of angle $A$;
(II) If $b=2$, $c=3$, find the values of $a$ and $sin(2B-A)$.
|
\frac{3\sqrt{3}}{14}
| 43.75 |
19,222 |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits?
|
90
| 92.96875 |
19,223 |
Given the domain of the function $f(x)$ is $(4a-3,3-2a^{2})$, where $a\in \mathbb{R}$, and $y=f(2x-3)$ is an even function. If $B_{n}=1\times a^{1}+4\times a^{2}+7\times a^{3}+\cdots +(3n-2)a^{n}$, then $B_{50}=$ ?
|
75
| 17.96875 |
19,224 |
Let $r$ and $s$ be positive integers such that\[\frac{5}{11} < \frac{r}{s} < \frac{4}{9}\]and $s$ is as small as possible. What is $s - r$?
|
11
| 8.59375 |
19,225 |
A reservoir is equipped with an inlet tap A and outlet taps B and C. It is known that opening taps A and B together for 8 hours or opening taps A, B, and C together for a whole day can fill an empty reservoir, and opening taps B and C together for 8 hours can empty a full reservoir. Determine how many hours it would take to fill the reservoir with only tap A open, how many hours it would take to empty the reservoir with only tap B open, and how many hours it would take to empty the reservoir with only tap C open.
|
12
| 9.375 |
19,226 |
Define a $\textit{better word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many seven-letter $\textit{better words}$ are there?
|
2916
| 43.75 |
19,227 |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $2\sin A=\sin C$.
$(1)$ Find the length of $c$;
$(2)$ If $\cos C=\frac{1}{4}$, find the area of $\triangle ABC$.
|
\sqrt{15}
| 96.875 |
19,228 |
Given $f(x)=x^{3}+3x^{2}+6x$, $f(a)=1$, $f(b)=-9$, the value of $a+b$ is \_\_\_\_\_\_.
|
-2
| 63.28125 |
19,229 |
If the real number sequence: -1, $a_1$, $a_2$, $a_3$, -81 forms a geometric sequence, determine the eccentricity of the conic section $x^2+ \frac{y^2}{a_2}=1$.
|
\sqrt{10}
| 49.21875 |
19,230 |
Calculate $500,000,000,000 - 3 \times 111,111,111,111$.
|
166,666,666,667
| 0.78125 |
19,231 |
In the 2017 Shanghai college entrance examination reform plan, it is required that each candidate must choose 3 subjects from 6 subjects including Physics, Chemistry, Biology, Politics, History, and Geography to take the level examination. Xiaoming decided to choose at most one subject from Biology, Politics, and History. There are several possible subject combinations for Xiaoming.
|
10
| 60.9375 |
19,232 |
If the area of the triangle formed by the tangent line to the curve $y=x^{-\frac{1}{2}}$ at the point $(a,a^{-\frac{1}{2}})$ and the two coordinate axes is $18$, find the real number $a=\_\_\_\_\_\_\_\_.$
|
64
| 90.625 |
19,233 |
Suppose $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 3^2 \pmod{2^4}, \\
3+x &\equiv 2^3 \pmod{3^4}, \\
4+x &\equiv 3^3 \pmod{2^3}.
\end{align*}
What is the remainder when $x$ is divided by $24$?
|
23
| 19.53125 |
19,234 |
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 3, 4, 5, 6, 7, 8, 9, and 11?
|
27722
| 66.40625 |
19,235 |
A given integer Fahrenheit temperature $F$ is first converted to Kelvin using the formula $K = \frac{5}{9}(F - 32) + 273.15$, rounded to the nearest integer, then converted back to Fahrenheit using the inverse formula $F' = \frac{9}{5}(K - 273.15) + 32$, and rounded to the nearest integer again. Find how many integer Fahrenheit temperatures between 100 and 500 inclusive result in the original temperature equaling the final temperature after these conversions and roundings.
|
401
| 0 |
19,236 |
There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.
What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)?
|
13
| 71.09375 |
19,237 |
A factory has two branches, one in location A and the other in location B, producing 12 and 6 machines respectively. Now, they need to distribute 10 machines to area A and 8 machines to area B. It is known that the transportation cost for moving one machine from location A to area A and B is 400 and 800 yuan respectively, and from location B to area A and B is 300 and 500 yuan respectively.
(Ⅰ) Assume x machines are transported from location B to area A, derive the function expression of the total cost y in terms of the number of machines x;
(Ⅱ) If the total transportation cost does not exceed 9000 yuan, how many transportation plans are there?
(Ⅲ) Find the transportation plan with the lowest total cost and the lowest cost.
|
8600
| 65.625 |
19,238 |
How many times do you have to subtract 8 from 792 to get 0?
|
99
| 97.65625 |
19,239 |
In the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$.
|
-6
| 89.84375 |
19,240 |
Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$ .
[i]Proposed by James Lin
|
11
| 62.5 |
19,241 |
A teacher received 10, 6, 8, 5, and 6 letters from Monday to Friday, respectively. The variance $s^2$ of this set of data is ______.
|
3.2
| 63.28125 |
19,242 |
Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the decreasing interval of the function $g(x)$;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0,\frac{1}{2}\right)$, find the minimum value of $a$.
|
2-4\ln 2
| 70.3125 |
19,243 |
In the number \(2016 * * * * 02 * *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
|
5184
| 51.5625 |
19,244 |
On the beach, there was a pile of apples belonging to 3 monkeys. The first monkey came, divided the apples into 3 equal piles with 1 apple remaining, then it threw the remaining apple into the sea and took one pile for itself. The second monkey came, divided the remaining apples into 3 equal piles with 1 apple remaining again, it also threw the remaining apple into the sea and took one pile. The third monkey did the same. How many apples were there originally at least?
|
25
| 82.03125 |
19,245 |
Calculate: $\sqrt{27} - 2\cos 30^{\circ} + \left(\frac{1}{2}\right)^{-2} - |1 - \sqrt{3}|$
|
\sqrt{3} + 5
| 17.96875 |
19,246 |
The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
what is the value of \(a + c + f\)?
|
73
| 16.40625 |
19,247 |
Evaluate the integral $\int_{2}^{3}{\frac{x-2}{\left( x-1 \right)\left( x-4 \right)}dx}=$
A) $-\frac{1}{3}\ln 2$
B) $\frac{1}{3}\ln 2$
C) $-\ln 2$
D) $\ln 2$
|
-\frac{1}{3}\ln 2
| 85.15625 |
19,248 |
Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\angle CAD = 30^{\circ}$, find $\sin(2\angle BAD)$.
|
\frac{1}{2}
| 19.53125 |
19,249 |
Place 5 balls numbered 1, 2, 3, 4, 5 into 5 boxes also numbered 1, 2, 3, 4, 5.
(1) How many ways are there to do this?
(2) If each box can hold at most one ball, how many ways are there?
(3) If exactly one box is to remain empty, how many ways are there?
(4) If each box contains one ball, and exactly one ball's number matches its box's number, how many ways are there?
(5) If each box contains one ball, and at least two balls' numbers match their boxes' numbers, how many ways are there?
(6) If the 5 distinct balls are replaced with 5 identical balls, and exactly one box is to remain empty, how many ways are there?
(Note: For all parts, list the formula before calculating the value, otherwise, points will be deducted.)
|
20
| 8.59375 |
19,250 |
Find the dimensions of the cone that can be formed from a $300^{\circ}$ sector of a circle with a radius of 12 by aligning the two straight sides.
|
12
| 2.34375 |
19,251 |
How many integers $-15 \leq n \leq 15$ satisfy $(n-3)(n+5)(n+9) < 0$?
|
13
| 67.1875 |
19,252 |
Using the six digits 0, 1, 2, 3, 4, 5 to form a six-digit number without repeating any digit, how many such numbers are there where the unit digit is less than the ten's digit?
|
300
| 51.5625 |
19,253 |
A labor employment service center has 7 volunteers ready to provide labor employment guidance on the streets over the weekend, with 6 people to be arranged for duty on Saturday and Sunday. If 3 people are arranged for each day, there are a total of _______ different arrangements. (Answer with a number)
|
140
| 44.53125 |
19,254 |
Lilian has two older twin sisters, and the product of their three ages is 162. Find the sum of their three ages.
|
20
| 17.96875 |
19,255 |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto $AC$ . Denote by $X$ the intersection point of the lines $FH$ and $QO$ . Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of $1000a + 100b + 10c + d$ .
|
1132
| 35.9375 |
19,256 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]
|
75
| 100 |
19,257 |
Given the vectors $\overrightarrow{m}=(2\sin \omega x, \cos ^{2}\omega x-\sin ^{2}\omega x)$ and $\overrightarrow{n}=( \sqrt {3}\cos \omega x,1)$, where $\omega > 0$ and $x\in R$. If the minimum positive period of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ is $\pi$,
(I) Find the value of $\omega$.
(II) In $\triangle ABC$, if $f(B)=-2$, $BC= \sqrt {3}$, and $\sin B= \sqrt {3}\sin A$, find the value of $\overrightarrow{BA}\cdot \overrightarrow{BC}$.
|
-\frac{3}{2}
| 27.34375 |
19,258 |
Given the function $f(x)=1-2\sin (x+ \frac {π}{8})[\sin (x+ \frac {π}{8})-\cos (x+ \frac {π}{8})]$, $x\in R$.
(I) Find the smallest positive period of the function $f(x)$;
(II) Find the maximum and minimum values of the function $f(x+ \frac {π}{8})$ on the interval $[- \frac {π}{2},0]$.
|
-1
| 53.90625 |
19,259 |
Find the number of positive integers $n,$ $1 \le n \le 2600,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients.
|
50
| 56.25 |
19,260 |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}⊥(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$.
|
-\frac{1}{2}
| 2.34375 |
19,261 |
1. Given $\sin\alpha + \cos\alpha = \frac{7}{13}$, with $\alpha \in (0, \pi)$, find the value of $\tan\alpha$.
2. Find the minimum value for $y=\sin 2x + 2\sqrt{2}\cos\left(\frac{\pi}{4}+x\right)+3$.
|
2 - 2\sqrt{2}
| 7.8125 |
19,262 |
Given the functions $f(x)=x+e^{x-a}$ and $g(x)=\ln (x+2)-4e^{a-x}$, where $e$ is the base of the natural logarithm. If there exists a real number $x_{0}$ such that $f(x_{0})-g(x_{0})=3$, find the value of the real number $a$.
|
-\ln 2-1
| 0 |
19,263 |
Given points $A(2,-1,1)$, $B(1,-2,1)$, $C(0,0,-1)$, the distance from $A$ to $BC$ is ______.
|
\frac{\sqrt{17}}{3}
| 57.03125 |
19,264 |
Given that $a > 0$, $b > 0$, and $4a - b \geq 2$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$.
|
\frac{1}{2}
| 43.75 |
19,265 |
Add $2175_{9} + 1714_{9} + 406_9$. Express your answer in base $9$.
|
4406_{9}
| 0.78125 |
19,266 |
The whole numbers from 1 to 1000 are written. How many of these numbers have at least two 7's appearing side-by-side?
|
19
| 72.65625 |
19,267 |
Given that there are 5 cards of the same size and shape, each marked with the numbers 1, 2, 3, 4, and 5 respectively. If two cards are drawn at random, the probability that the larger number on these two cards is 3 is ______.
|
\dfrac {1}{5}
| 100 |
19,268 |
A trapezium is given with parallel bases having lengths $1$ and $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.
|
15
| 42.1875 |
19,269 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. The vectors $\overrightarrow{m} = (a-b,\sin A+\sin C)$ and $\overrightarrow{n} = (a-c, \sin(A+C))$ are collinear.
(1) Find the value of angle $C$;
(2) If $\overrightarrow{AC} \cdot \overrightarrow{CB} = -27$, find the minimum value of $|\overrightarrow{AB}|$.
|
3\sqrt{6}
| 74.21875 |
19,270 |
Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$ . Determine the area of triangle $ AMC$ .
|
\frac{1}{2}
| 98.4375 |
19,271 |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
*2019 CCA Math Bonanza Individual Round #5*
|
40
| 21.875 |
19,272 |
Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$ .
|
23
| 90.625 |
19,273 |
What is the largest three-digit multiple of 8 whose digits' sum is 24?
|
888
| 1.5625 |
19,274 |
In triangle $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \frac{2\pi}{3}$ and $a = 6$:
(Ⅰ) If $c = 14$, find the value of $\sin A$;
(Ⅱ) If the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $c$.
|
2\sqrt{13}
| 96.875 |
19,275 |
For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.
|
184320
| 98.4375 |
19,276 |
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ .
|
23
| 35.9375 |
19,277 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]
|
75
| 100 |
19,278 |
Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$ with the sums of the first $n$ terms being $S_{n}$ and $T_{n}$, respectively, if $\frac{S_n}{T_n}=\frac{3n+4}{n+2}$, find $\frac{a_3+a_7+a_8}{b_2+b_{10}}$.
|
\frac{111}{26}
| 39.0625 |
19,279 |
A bouncy ball is dropped from a height of 100 meters. After each bounce, it reaches a height that is half of the previous one. What is the total distance the ball has traveled when it hits the ground for the 10th time? (Round the answer to the nearest whole number)
|
300
| 8.59375 |
19,280 |
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$ .
*Proposed by Harun Khan*
|
15
| 6.25 |
19,281 |
Given the parabola $y=x^{2}+2x-3$ intersects the line $l_{1}$: $y=-x+m$ at points $A$ and $C$, and the line $l_{2}$ is symmetric with respect to the axis of symmetry of the parabola. Line $l_{2}$ intersects the parabola at points $B$ and $D$, where points $A$ and $D$ are above the $x$-axis, and points $B$ and $C$ are below the $x$-axis. If $AC \cdot BD = 26$, then the value of $m$ is ______.
|
-2
| 35.15625 |
19,282 |
Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$.
|
\sqrt{6}
| 83.59375 |
19,283 |
The coefficient of the $x$ term in the expansion of $(x^{2}-x-2)^{3}$ is what value?
|
-12
| 18.75 |
19,284 |
A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?
|
75
| 37.5 |
19,285 |
Positive integers less than 900 that can be written as a product of two or more consecutive prime numbers. Find their count.
|
14
| 1.5625 |
19,286 |
I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of being correct. What is the least number of words I should learn to ensure I score at least $90\%$ on the exam?
|
716
| 47.65625 |
19,287 |
Given the function $f(x)=2\ln x+8x$, find the value of $\lim_{n\to\infty} \frac{f(1-2\Delta x)-f(1)}{\Delta x}$ ( ).
|
-20
| 77.34375 |
19,288 |
Regions I, II, and III are bounded by shapes. The perimeter of region I is 16 units and the perimeter of region II is 36 units. Region III is a triangle with a perimeter equal to the average of the perimeters of regions I and II. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction.
|
\frac{144}{169\sqrt{3}}
| 0 |
19,289 |
Given the vectors $\overrightarrow {a}$ = (1, x) and $\overrightarrow {b}$ = (2x+3, -x) in the plane, where x ∈ R, they are parallel to each other. Find the magnitude of $\overrightarrow {a}$ - 2$\overrightarrow {b}$.
|
3\sqrt{5}
| 82.03125 |
19,290 |
Given that the asymptote equation of the hyperbola $y^{2}+\frac{x^2}{m}=1$ is $y=\pm \frac{\sqrt{3}}{3}x$, find the value of $m$.
|
-3
| 56.25 |
19,291 |
The asymptotes of a hyperbola are \(y = 2x + 3\) and \(y = -2x + 1\). The hyperbola also passes through the point \((2, 1)\). Find the distance between the foci of the hyperbola.
|
2\sqrt{30}
| 41.40625 |
19,292 |
Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of % (rounded to one decimal place).
|
20.3
| 3.90625 |
19,293 |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\varphi$.
|
\frac{\pi}{3}
| 91.40625 |
19,294 |
An amusement park has a series of miniature buildings and landscape models from various places in the United States, scaled at $1: 20$. If the height of the United States Capitol Building is 289 feet, what is the nearest integer height of its model in feet?
|
14
| 61.71875 |
19,295 |
Given a $4 \times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\,^{\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation.
|
\frac{1}{65536}
| 35.15625 |
19,296 |
Let \( XYZ \) be an acute-angled triangle. Let \( s \) be the side length of the square which has two adjacent vertices on side \( YZ \), one vertex on side \( XY \), and one vertex on side \( XZ \). Let \( h \) be the distance from \( X \) to the side \( YZ \) and \( b \) be the distance from \( Y \) to \( Z \).
(a) If the vertices have coordinates \( X=(2,4), Y=(0,0) \), and \( Z=(4,0) \), find \( b, h \), and \( s \).
(b) Given the height \( h=3 \) and \( s=2 \), find the base \( b \).
(c) If the area of the square is 2017, determine the minimum area of triangle \( XYZ \).
|
4034
| 27.34375 |
19,297 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $\sin\left(A+ \frac{\pi}{3}\right) = 4\sin \frac{A}{2}\cos \frac{A}{2}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $\sin B= \sqrt{3}\sin C$ and $a=1$, find the area of $\triangle ABC$.
|
\frac{\sqrt{3}}{4}
| 89.84375 |
19,298 |
Given the real sequence $-1$, $a$, $b$, $c$, $-2$ forms a geometric sequence, find the value of $abc$.
|
-2\sqrt{2}
| 51.5625 |
19,299 |
Solve the quadratic equation $(x-h)^2 + 4h = 5 + x$ and find the sum of the squares of its roots. If the sum is equal to $20$, what is the absolute value of $h$?
**A)** $\frac{\sqrt{22}}{2}$
**B)** $\sqrt{22}$
**C)** $\frac{\sqrt{44}}{2}$
**D)** $2$
**E)** None of these
|
\frac{\sqrt{22}}{2}
| 95.3125 |
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