Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,600 |
Given the function $$f(x)=2\sin x( \sqrt {3}\cos x-\sin x)+1$$, if $f(x-\varphi)$ is an even function, determine the value of $\varphi$.
|
\frac {\pi}{3}
| 54.6875 |
20,601 |
In triangle $ ABC$ , $ 3\sin A \plus{} 4\cos B \equal{} 6$ and $ 4\sin B \plus{} 3\cos A \equal{} 1$ . Then $ \angle C$ in degrees is
|
30
| 84.375 |
20,602 |
How many distinct divisors does the number a) 800; b) 126000 have?
|
120
| 20.3125 |
20,603 |
Given the function $f\left(x\right)=x^{3}+ax^{2}+bx-4$ and the tangent line equation $y=x-4$ at point $P\left(2,f\left(2\right)\right)$.<br/>$(1)$ Find the values of $a$ and $b$;<br/>$(2)$ Find the extreme values of $f\left(x\right)$.
|
-\frac{58}{27}
| 10.15625 |
20,604 |
Given $sin(\alpha-\beta)=\frac{1}{3}$ and $cos\alpha sin\beta=\frac{1}{6}$, find $\cos \left(2\alpha +2\beta \right)$.
|
\frac{1}{9}
| 27.34375 |
20,605 |
In triangle $ABC$, the three internal angles are $A$, $B$, and $C$. Find the value of $A$ for which $\cos A + 2\cos\frac{B+C}{2}$ attains its maximum value, and determine this maximum value.
|
\frac{3}{2}
| 70.3125 |
20,606 |
How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)?
|
36
| 18.75 |
20,607 |
Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°.
|
20 \left(1 + \frac {\sqrt {3}}{3}\right)
| 0 |
20,608 |
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and it satisfies $(2b-a) \cdot \cos C = c \cdot \cos A$.
$(1)$ Find the size of angle $C$;
$(2)$ Let $y = -4\sqrt{3}\sin^2\frac{A}{2} + 2\sin(C-B)$, find the maximum value of $y$ and determine the shape of $\triangle ABC$ when $y$ reaches its maximum value.
|
2-2 \sqrt {3}
| 0 |
20,609 |
Given $u$ and $v$ are complex numbers such that $|u+v|=2$ and $|u^2+v^2|=8,$ find the smallest possible value of $|u^3+v^3|$.
|
20
| 47.65625 |
20,610 |
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$ . Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of $p_{2}+2p_{1}-3p_{0}.$
|
18
| 20.3125 |
20,611 |
Compute \[
\left\lfloor \frac{2017! + 2014!}{2016! + 2015!}\right\rfloor.
\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
|
2016
| 89.84375 |
20,612 |
The quadratic $8x^2 - 48x - 320$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?
|
-387
| 81.25 |
20,613 |
A point \((x, y)\) is randomly selected such that \(0 \leq x \leq 4\) and \(0 \leq y \leq 5\). What is the probability that \(x + y \leq 5\)? Express your answer as a common fraction.
|
\frac{3}{5}
| 21.875 |
20,614 |
The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$.
|
420
| 46.875 |
20,615 |
Let the set $I = \{1, 2, 3, 4, 5\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$?
|
49
| 14.0625 |
20,616 |
Given the parabola $y^{2}=2px$ with its directrix equation $x=-2$, let point $P$ be a point on the parabola. Find the minimum distance from point $P$ to the line $y=x+3$.
|
\frac { \sqrt{2} }{2}
| 89.84375 |
20,617 |
$(100^2-99^2) + (98^2-97^2) + \ldots + (2^2-1^2) = \ $
|
5050
| 82.03125 |
20,618 |
What is the largest possible value of the expression $$ gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 ) $$ for naturals $n$ ?
<details><summary>Click to expand</summary>original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n?</details>
|
13
| 42.1875 |
20,619 |
Mrs. Riley revised her data after realizing that there was an additional score bracket and a special bonus score for one of the brackets. Recalculate the average percent score for the $100$ students given the updated table:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&5\\\hline
95&12\\\hline
90&20\\\hline
80&30\\\hline
70&20\\\hline
60&8\\\hline
50&4\\\hline
40&1\\\hline
\end{tabular}
Furthermore, all students scoring 95% receive a 5% bonus, which effectively makes their score 100%.
|
80.2
| 4.6875 |
20,620 |
Compute the argument of the sum:
\[ e^{2\pi i/40} + e^{6\pi i/40} + e^{10\pi i/40} + e^{14\pi i/40} + e^{18\pi i/40} + e^{22\pi i/40} + e^{26\pi i/40} + e^{30\pi i/40} + e^{34\pi i/40} + e^{38\pi i/40} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \).
|
\frac{\pi}{2}
| 28.90625 |
20,621 |
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\).
|
768
| 42.96875 |
20,622 |
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$ . What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$ ?
*2018 CCA Math Bonanza Lightning Round #1.3*
|
45
| 14.84375 |
20,623 |
Given the function $f(x)=e^{-x}+ \frac {nx}{mx+n}$.
$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$.
$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac {m}{n}$.
|
\frac {1}{2}
| 0 |
20,624 |
At what value of $b$ do the graphs of $y=bx^2+5x+3$ and $y=-2x-3$ intersect at exactly one point?
|
\frac{49}{24}
| 98.4375 |
20,625 |
Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $2\cos C(a\cos C+c\cos A)+b=0$.
$(1)$ Find the magnitude of angle $C$;
$(2)$ If $b=2$ and $c=2\sqrt{3}$, find the area of $\triangle ABC$.
|
\sqrt{3}
| 92.1875 |
20,626 |
A student used the "five-point method" to draw the graph of the function $f(x)=A\sin(\omega x+\varphi)$ ($\omega\ \gt 0$, $|\varphi|＜\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:
| $\omega x+\varphi$ | $0$ | $\frac{π}{2}$ | $\pi$ | $\frac{{3π}}{2}$ | $2\pi$ |
|-------------------|-----|---------------|-------|------------------|-------|
| $x$ | | | $\frac{{3π}}{8}$ | $\frac{{5π}}{8}$ | |
| $A\sin(\omega x+\varphi)$ | $0$ | $2$ | | $-2$ | $0$ |
$(1)$ Please complete the data in the table and write the analytical expression of the function $f(x)$ on the answer sheet.
$(2)$ Move the graph of $f(x)$ to the left by $\theta$ units to obtain the graph of $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\frac{π}{3}$, find the minimum value of $\theta$.
|
\frac{7\pi}{24}
| 21.875 |
20,627 |
What is the greatest four-digit number that is one more than a multiple of 7 and five more than a multiple of 8?
|
9997
| 6.25 |
20,628 |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
*(Proposed by Walther Janous)*
|
16
| 96.875 |
20,629 |
Let $x_1, x_2, \ldots, x_n$ be integers, satisfying:
(1) $-1 \leq x_i \leq 2$, for $i=1, 2, \ldots, n$;
(2) $x_1 + x_2 + \ldots + x_n = 19$;
(3) $x_1^2 + x_2^2 + \ldots + x_n^2 = 99$.
Find the maximum and minimum values of $x_1^3 + x_2^3 + \ldots + x_n^3$.
|
133
| 11.71875 |
20,630 |
A basketball player made the following number of successful free throws in 10 successive games: 8, 17, 15, 22, 14, 12, 24, 10, 20, and 16. He attempted 10, 20, 18, 25, 16, 15, 27, 12, 22, and 19 free throws in those respective games. Calculate both the median number of successful free throws and the player's best free-throw shooting percentage game.
|
90.91\%
| 0 |
20,631 |
Find all integers \( z \) for which exactly two of the following five statements are true, and three are false:
1) \( 2z > 130 \)
2) \( z < 200 \)
3) \( 3z > 50 \)
4) \( z > 205 \)
5) \( z > 15 \)
|
16
| 55.46875 |
20,632 |
The area of the enclosed shape formed by the line $y=x-2$ and the curve $y^2=x$ can be calculated.
|
\frac{9}{2}
| 86.71875 |
20,633 |
Given the imaginary unit $i$, let $z=1+i+i^{2}+i^{3}+\ldots+i^{9}$, then $|z|=$______.
|
\sqrt {2}
| 0 |
20,634 |
Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\pi$ in its simplest form.
|
\frac{9\pi - 18}{2}
| 21.09375 |
20,635 |
The number $2022$ has the following property: it is a multiple of $6$ and the sum of its digits is $6$. Such positive integers are called "auspicious numbers." Among all three-digit positive integers, the number of "auspicious numbers" is ____.
|
12
| 88.28125 |
20,636 |
In three-digit numbers, if the digit in the tens place is smaller than the digits in both the hundreds and ones places, the number is called a "concave number," such as 304, 968, etc. How many distinct three-digit concave numbers are there without any repeated digits?
|
240
| 82.03125 |
20,637 |
In a group photo, 4 boys and 3 girls are to stand in a row such that no two boys or two girls stand next to each other. How many different arrangements are possible?
|
144
| 12.5 |
20,638 |
The sum of 20 consecutive integers is a triangular number. What is the smallest such sum?
|
190
| 94.53125 |
20,639 |
Place the sequence $\{2n+1\}$ in parentheses sequentially, with the first parenthesis containing one number, the second two numbers, the third three numbers, the fourth four numbers, the fifth one number again, and then continuing in this cycle. Determine the sum of the numbers in the 104th parenthesis.
|
2104
| 0.78125 |
20,640 |
There are $7$ students, among whom $A_{1}$, $A_{2}$, $A_{3}$ excel in mathematics, $B_{1}$, $B_{2}$ excel in physics, and $C_{1}$, $C_{2}$ excel in chemistry. One student with excellent performance in mathematics, one in physics, and one in chemistry will be selected to form a team to represent the school in a competition.
$(1)$ Find the probability of selecting $C_{1}$.
$(2)$ Find the probability of at most one of $A_{1}$ and $B_{1}$ being selected.
|
\frac{5}{6}
| 74.21875 |
20,641 |
Given the parabola $y^{2}=2x$ and a point $P(m,2)$ on it, find the value of $m$ and the distance between point $P$ and the focus $F$ of the parabola.
|
\frac{5}{2}
| 67.96875 |
20,642 |
Let $P(x) = x^3 - 6x^2 - 5x + 4$ . Suppose that $y$ and $z$ are real numbers such that
\[ zP(y) = P(y - n) + P(y + n) \]
for all reals $n$ . Evaluate $P(y)$ .
|
-22
| 56.25 |
20,643 |
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$ .
*Proposed by Howard Halim*
|
13\sqrt{13}
| 9.375 |
20,644 |
A spherical balloon collapses into a wet horizontal surface and settles into a shape of a hemisphere while keeping the same volume. The minor radius of the original balloon, when viewed as an ellipsoid due to unequal pressure distribution, was $4\sqrt[3]{3}$ cm. Find the major radius of the original balloon, assuming the major and minor axes were proportional and the proportionality constant is 2 before it became a hemisphere.
|
8\sqrt[3]{3}
| 59.375 |
20,645 |
There are four groups of numbers with their respective averages specified as follows:
1. The average of all multiples of 11 from 1 to 100810.
2. The average of all multiples of 13 from 1 to 100810.
3. The average of all multiples of 17 from 1 to 100810.
4. The average of all multiples of 19 from 1 to 100810.
Among these four averages, the value of the largest average is $\qquad$ .
|
50413.5
| 8.59375 |
20,646 |
Express the given value of $22$ nanometers in scientific notation.
|
2.2\times 10^{-8}
| 58.59375 |
20,647 |
Given $a, b \in \mathbb{R}$, $m = ab + 1$, $n = a + b$.
- $(1)$ If $a > 1$, $b < 1$, compare the sizes of $m$ and $n$.
- $(2)$ If $a > 1$, $b > 1$, and $m - n = 49$, find the minimum value of $a + b$.
|
16
| 86.71875 |
20,648 |
In $\triangle ABC$, $2\sin^2 \frac{A+B}{2}-\cos 2C=1$, and the radius of the circumcircle $R=2$.
$(1)$ Find $C$;
$(2)$ Find the maximum value of $S_{\triangle ABC}$.
|
\sqrt{3}
| 48.4375 |
20,649 |
In base \( R_1 \), the fractional expansion of \( F_1 \) is \( 0.373737 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.737373 \cdots \). In base \( R_2 \), the fractional expansion of \( F_1 \) is \( 0.252525 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.525252 \cdots \). What is the sum of \( R_1 \) and \( R_2 \) (both expressed in decimal)?
|
19
| 56.25 |
20,650 |
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{7}{33}
| 5.46875 |
20,651 |
The volume of a regular triangular prism is $8$, the base edge length that minimizes the surface area of the prism is __________.
|
2\sqrt[3]{4}
| 54.6875 |
20,652 |
In the number sequence $1,1,2,3,5,8,x,21,34,55$, what is the value of $x$?
|
13
| 87.5 |
20,653 |
A rectangular table measures $12'$ in length and $9'$ in width and is currently placed against one side of a rectangular room. The owners desire to move the table to lay diagonally in the room. Determine the minimum length of the shorter side of the room, denoted as $S$, in feet, for the table to fit without tilting or taking it apart.
|
15'
| 0 |
20,654 |
In 2010, the ages of a brother and sister were 16 and 10 years old, respectively. In what year was the brother's age twice that of the sister's?
|
2006
| 96.09375 |
20,655 |
A rectangular prism has dimensions 10 inches by 3 inches by 30 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches?
|
6 \times 900^{2/3}
| 1.5625 |
20,656 |
Given \\((a+b-c)(a+b+c)=3ab\\) and \\(c=4\\), the maximum area of \\(\Delta ABC\\) is \_\_\_\_\_\_\_.
|
4\sqrt{3}
| 69.53125 |
20,657 |
The Big Sixteen Basketball League consists of two divisions, each with eight teams. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled?
|
296
| 90.625 |
20,658 |
A printer prints text pages at a rate of 17 pages per minute and graphic pages at a rate of 10 pages per minute. If a document consists of 250 text pages and 90 graphic pages, how many minutes will it take to print the entire document? Express your answer to the nearest whole number.
|
24
| 14.0625 |
20,659 |
Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$ , $y$ , $z$ are nonnegative real numbers such that $x+y+z=1$ .
|
\frac{7}{18}
| 62.5 |
20,660 |
In the rectangular coordinate system $XOY$, there is a line $l:\begin{cases} & x=t \\ & y=-\sqrt{3}t \\ \end{cases}(t$ is a parameter$)$, and a curve ${C_{1:}}\begin{cases} & x=\cos \theta \\ & y=1+\sin \theta \\ \end{cases}(\theta$ is a parameter$)$. Establish a polar coordinate system with the origin $O$ of this rectangular coordinate system as the pole and the non-negative half-axis of the $X$-axis as the polar axis. The equation of the curve ${C_{2}}$ is $\rho=4\sin (\theta -\frac{\pi }{6})$.
1. Find the polar coordinate equation of the curve ${C_{1}}$ and the rectangular coordinate equation of the curve ${C_{2}}$.
2. Suppose the line $l$ intersects the curve ${C_{1}}$ at points $O$ and $A$, and intersects the curve ${C_{2}}$ at points $O$ and $B$. Find the length of $|AB|$.
|
4- \sqrt{3}
| 10.15625 |
20,661 |
What is the sum of all integer solutions to the inequality $|n| < |n-3| < 10$?
|
-20
| 66.40625 |
20,662 |
Peter Ivanovich, along with 49 other men and 50 women, are seated in a random order around a round table. We call a man satisfied if a woman is sitting next to him. Find:
a) The probability that Peter Ivanovich is satisfied.
b) The expected number of satisfied men.
|
\frac{1250}{33}
| 25.78125 |
20,663 |
Given Jasmine has two types of bottles, one that can hold 45 milliliters and another that can hold 675 milliliters, and a vase that can hold 95 milliliters, determine the total number of small bottles she must buy to fill the large bottle as much as possible and the vase.
|
18
| 26.5625 |
20,664 |
Given the sequence of even counting numbers starting from $0$, find the sum of the first $1500$ terms. Then, given the sequence of odd counting numbers, find the sum of the first $1500$ terms, and calculate their difference.
|
1500
| 47.65625 |
20,665 |
Let $M = 123456789101112\dots4950$ be the $95$-digit number formed by writing integers from $1$ to $50$ in order, one after the other. What is the remainder when $M$ is divided by $45$?
|
15
| 65.625 |
20,666 |
The average years of experience of three employees, David, Emma, and Fiona, at a company is 12 years. Five years ago, Fiona had the same years of experience as David has now. In 4 years, Emma's experience will be $\frac{3}{4}$ of David's experience at that time. How many years of experience does Fiona have now?
|
\frac{183}{11}
| 48.4375 |
20,667 |
Determine the value of the expression $\frac{\log \sqrt{27}+\log 8-3 \log \sqrt{10}}{\log 1.2}$.
|
\frac{3}{2}
| 21.875 |
20,668 |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 8.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\]
|
3\sqrt{10}
| 61.71875 |
20,669 |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
|
115
| 25 |
20,670 |
Five women of different heights are standing in a line at a social gathering. Each woman decides to only shake hands with women taller than herself. How many handshakes take place?
|
10
| 55.46875 |
20,671 |
Two couples each bring one child to visit the zoo. After purchasing tickets, they line up to enter the zoo one by one. For safety reasons, the two fathers must be positioned at the beginning and the end of the line. Moreover, the two children must be positioned together. Determine the number of different ways that these six people can line up to enter the zoo.
|
24
| 34.375 |
20,672 |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $3$. The arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\dots$, $C_8$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Find the product of the lengths of these sixteen chords.
|
387420489
| 17.1875 |
20,673 |
Let $h(x) = x - 3$ and $k(x) = 2x$. Also denote the inverses of these functions as $h^{-1}$ and $k^{-1}$. Compute:
\[ h(k^{-1}(h^{-1}(h^{-1}(k(h(28)))))) \]
|
25
| 74.21875 |
20,674 |
Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(a + b + c = 3.\) Find the minimum value of
\[\frac{a + b}{abc}.\]
|
\frac{16}{9}
| 18.75 |
20,675 |
Let
\[ f(x) = \left\{
\begin{array}{cl}
2ax + 4, & \text{if } x > 2, \\
x^2-2 & \text{if } -2 \le x \le 2, \\
3x - c, & \text{if } x < -2.
\end{array}
\right. \]
Find $a+c$ if the piecewise function is continuous.
|
-\frac{17}{2}
| 98.4375 |
20,676 |
In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube?
|
1/6
| 50 |
20,677 |
Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| = 1$ and $\arg(c) = \arg(a) + \arg(b)$. Suppose that
\[ a z^2 + b z + c = 0. \]
Find the largest possible value of $|z|$.
|
\frac{1 + \sqrt{5}}{2}
| 48.4375 |
20,678 |
Given the function $f(x)= \sqrt {3}\sin 2x+2\cos ^{2}x-1$.
(I) Find the smallest positive period of $f(x)$:
(II) Find the maximum and minimum values of $f(x)$ in the interval $\[- \dfrac {π}{6}, \dfrac {π}{4}\]$.
|
-1
| 89.0625 |
20,679 |
Find the area of triangle $MNP$ given below:
[asy]
unitsize(1inch);
pair M,N,P;
M = (0,0);
N= (sqrt(3),0);
P = (0,1);
draw (M--N--P--M, linewidth(0.9));
draw(rightanglemark(N,M,P,3));
label("$M$",M,S);
label("$N$",N,S);
label("$P$",P,N);
label("$15$",(N+P)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy]
|
28.125\sqrt{3}
| 11.71875 |
20,680 |
Given $α∈({\frac{π}{2},π})$, $sinα=\frac{3}{5}$, find $\tan 2\alpha$.
|
-\frac{24}{7}
| 87.5 |
20,681 |
Given a class of 50 students with exam scores following a normal distribution $N(100,10^2)$, and $P(90 ≤ ξ ≤ 100) = 0.3$, estimate the number of students who scored above 110 points.
|
10
| 27.34375 |
20,682 |
How many paths are there from point $A$ to point $B$ in a $7 \times 6$ grid, if every step must be up or to the right, and you must not pass through the cell at position $(3,3)$?
[asy]size(4cm,4cm);int w=7;int h=6;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label("$A$", (0,0), SW, p);label("$B$", (w-1,h-1), NE, p);fill((3,3)--(4,3)--(4,4)--(3,4)--cycle, grey);[/asy]
|
262
| 7.03125 |
20,683 |
A five-digit number has one of its digits crossed out, and the resulting four-digit number is added to the original number. The sum is 54321. Find the original number.
|
49383
| 89.0625 |
20,684 |
Find the product of all positive integral values of $m$ such that $m^2 - 40m + 399 = q$ for some prime number $q$. Note that there is at least one such $m$.
|
396
| 55.46875 |
20,685 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n = \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. For how many positive integers $b_1 \le 1500$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$?
|
1000
| 35.15625 |
20,686 |
Consider a circle with radius $4$, and there are numerous line segments of length $6$ that are tangent to the circle at their midpoints. Compute the area of the region consisting of all such line segments.
A) $8\pi$
B) $7\pi$
C) $9\pi$
D) $10\pi$
|
9\pi
| 33.59375 |
20,687 |
Compute: $${0.027}^{− \frac{1}{3}}−{(− \frac{1}{7})}^{−2}+{256}^{ \frac{3}{4}}−{3}^{−1}+{( \sqrt{2}−1)}^{0} = $$ \_\_\_\_\_\_.
|
19
| 92.1875 |
20,688 |
Let $\triangle ABC$ be a right triangle at $A$ with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 25$, $BC = 34$, and $TX^2 + TY^2 + XY^2 = 1975$. Find $XY^2$.
|
987.5
| 10.9375 |
20,689 |
Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?
|
\frac{1}{4}
| 96.875 |
20,690 |
Given the digits $1, 3, 7, 8, 9$, find the smallest difference that can be achieved in the subtraction problem
\[\begin{tabular}[t]{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}\]
|
39
| 0.78125 |
20,691 |
A chessboard has its squares labeled according to the rule $\frac{1}{c_i + r_j}$, where $c_i$ is the column number and $r_j$ is the row number. Eight squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of these eight chosen squares.
|
\frac{8}{9}
| 5.46875 |
20,692 |
On a sunny day, 3000 people, including children, boarded a cruise ship. Two-fifths of the people were women, and a third were men. If 25% of the women and 15% of the men were wearing sunglasses, and there were also 180 children on board with 10% wearing sunglasses, how many people in total were wearing sunglasses?
|
530
| 18.75 |
20,693 |
Given a regular quadrilateral pyramid $S-ABCD$, with a base side length of $2$ and a volume of $\frac{{4\sqrt{3}}}{3}$, the length of the lateral edge of this quadrilateral pyramid is ______.
|
\sqrt{5}
| 90.625 |
20,694 |
If \(\frac{a}{b} = 5\), \(\frac{b}{c} = \frac{1}{4}\), and \(\frac{c^2}{d} = 16\), then what is \(\frac{d}{a}\)?
|
\frac{1}{25}
| 23.4375 |
20,695 |
Solve the following equations:
1. $4x=20$
2. $x-18=40$
3. $x\div7=12$
4. $8n\div2=15$
|
\frac{15}{4}
| 49.21875 |
20,696 |
Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as "partial fractions."<br/>Example: Express the fraction $\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\frac{{1-3x}}{{{x^2}-1}}=\frac{M}{{x+1}}+\frac{N}{{x-1}}$, cross multiply on the right side of the equation, we get $\frac{{M(x-1)+N(x+1)}}{{(x+1)(x-1)}}=\frac{{(M+N)x+(N-M)}}{{{x^2}-1}}$. According to the question, we have $\left\{\begin{array}{l}M+N=3\\ N-M=1\end{array}\right.$, solving this system gives $\left\{\begin{array}{l}M=-2\\ N=-1\end{array}\right.$, so $\frac{{1-3x}}{{{x^2}-1}}=\frac{{-2}}{{x+1}}+\frac{{-1}}{{x-1}}$. Please use the method learned above to solve the following problems:<br/>$(1)$ Express the fraction $\frac{{2n+1}}{{{n^2}+n}}$ as partial fractions;<br/>$(2)$ Following the pattern in (1), find the value of $\frac{3}{{1×2}}-\frac{5}{{2×3}}+\frac{7}{{3×4}}-\frac{9}{{4×5}}+⋯+\frac{{39}}{{19×20}}-\frac{{41}}{{20×21}}$.
|
\frac{20}{21}
| 89.84375 |
20,697 |
Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are reciprocals to them, and printed the resulting decimal fractions on 20 infinite ribbons (for example, the last ribbon had printed the number $\frac{1}{99!}=0, \underbrace{00 \ldots 00}_{155 \text { zeros! }} 10715$.. ). Sasha wants to cut out a piece from one ribbon that contains $N$ consecutive digits without a comma. What is the maximum value of $N$ for which Dima will not be able to determine from this piece which ribbon Sasha spoiled?
|
155
| 21.875 |
20,698 |
What is the probability that \(2^{n}\), where \(n\) is a randomly chosen positive integer, ends with the digit 2? What is the probability that \(2^{n}\) ends with the digits 12?
|
0.05
| 0 |
20,699 |
Let
\[g(x) = \left\{
\begin{array}{cl}
x + 5 & \text{if $x < 15$}, \\
3x - 6 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $g^{-1}(10) + g^{-1}(57).$
|
26
| 97.65625 |
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