problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Given \( \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+} \) and \( \theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = \pi \), find the minimum value of \( \left(2 \sin^{2} \theta_{1} + \frac{1}{\sin^{2} \theta_{1}}\right)\left(2 \sin^{2} \theta_{2} + \frac{1}{\sin^{2} \theta_{2}}\right)\left(2 \sin^{... | 81 | 0.125 | 7,672.25 | 4,034 | 8,192 | |
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$ | \frac{1}{9} | 0 | 8,192 | -1 | 8,192 | |
Calculate $500,000,000,000 - 3 \times 111,111,111,111$. | 166,666,666,667 | 0.1875 | 5,508.75 | 439 | 6,678.692308 | |
On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they are lit simultaneously? | 20/3 | 0.4375 | 6,606.4375 | 6,907.714286 | 6,372.111111 | |
A triangle has one side of length 5 cm, another side of length 12 cm, and includes a right angle. What is the shortest possible length of the third side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. | 10.91 | 1 | 3,813.125 | 3,813.125 | -1 | |
The points $(-1,4)$ and $(2,-3)$ are adjacent vertices of a square. What is the area of the square? | 58 | 0.8125 | 3,399.4375 | 2,293.461538 | 8,192 | |
What is the value of $$\frac{1}{2}\times4\times\frac{1}{8}\times16\times\frac{1}{32}\times64\times\frac{1}{128}\times256\times\frac{1}{512}\times1024?$$ | 32 | 1 | 2,494.1875 | 2,494.1875 | -1 | |
Given that 28×15=420, directly write out the results of the following multiplications:
2.8×1.5=\_\_\_\_\_\_、0.28×1.5=\_\_\_\_\_\_、0.028×0.15=\_\_\_\_\_\_. | 0.0042 | 0.8125 | 817.9375 | 840.153846 | 721.666667 | |
Rhombus $ABCD$ has side length $3$ and $\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
**A)** $0.81$
**B)** $1.62$
**C)** $2.43$
**D)** $2.16$
**E)** $3.24$ | 2.16 | 0 | 8,192 | -1 | 8,192 | |
Simplify first, then evaluate: $(\frac{{2x}^{2}+2x}{{x}^{2}-1}-\frac{{x}^{2}-x}{{x}^{2}-2x+1})÷\frac{x}{x+1}$, where $x=|\sqrt{3}-2|+(\frac{1}{2})^{-1}-(π-3.14)^0-\sqrt[3]{27}+1$. | -\frac{2\sqrt{3}}{3} + 1 | 0 | 5,356.0625 | -1 | 5,356.0625 | |
Formulas for shortened multiplication (other).
Common fractions | 198719871987 | 0 | 505.625 | -1 | 505.625 | |
All positive odd numbers are arranged in the following table (the number of numbers in the next row is twice the number of numbers in the previous row)
First row 1
Second row 3 5
Third row 7 9 11 13
…
Then, the third number in the sixth row is ． | 67 | 0.5625 | 6,369.25 | 5,011.777778 | 8,114.571429 | |
A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour? | 32 | 0.75 | 4,425.1875 | 3,969.833333 | 5,791.25 | |
In $\triangle ABC$, $a=3 \sqrt {3}$, $c=2$, $B=150^{\circ}$, find the length of side $b$ and the area of $\triangle ABC$. | \frac{3 \sqrt {3}}{2} | 0 | 4,121.5625 | -1 | 4,121.5625 | |
Find $y$ so that the vectors $\begin{pmatrix} 1 \\ -3 \\ -4 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ y \\ -1 \end{pmatrix}$ are orthogonal. | \frac{2}{3} | 1 | 1,194.0625 | 1,194.0625 | -1 | |
Suppose $x, y$, and $z$ are real numbers greater than 1 such that $$\begin{aligned} x^{\log _{y} z} & =2, \\ y^{\log _{z} x} & =4, \text { and } \\ z^{\log _{x} y} & =8 \end{aligned}$$ Compute $\log _{x} y$. | \sqrt{3} | Taking $\log _{2}$ both sides of the first equation gives $$\begin{aligned} & \log _{2} x \log _{y} z=1 \\ & \frac{\log _{2} x \log _{2} z}{\log _{2} y}=1 \end{aligned}$$ Performing similar manipulations on other two equations, we get $$\begin{aligned} & \frac{\log _{2} x \log _{2} z}{\log _{2} y}=1 \\ & \frac{\log _{2... | 0.8125 | 5,782.5625 | 5,226.538462 | 8,192 |
The letter T is formed by placing a $2\:\text{inch}\!\times\!6\:\text{inch}$ rectangle vertically and a $3\:\text{inch}\!\times\!2\:\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?
```
[No graphic input required]
``` | 22 | 0.0625 | 8,020.125 | 5,442 | 8,192 | |
A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. | 20 | 0.0625 | 8,022.5 | 6,912 | 8,096.533333 | |
Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \in R, a \neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$. | \frac{1}{100} | 0.1875 | 8,114.4375 | 7,778.333333 | 8,192 | |
An arithmetic sequence consists of positive terms, with the sum of the first $n$ terms denoted by $S_n$, satisfying $2S_2 = a_2(a_2 + 1)$, and given that $a_1 = 1$, find the minimum value of $\frac{2S_n + 13}{n}$. | \frac{33}{4} | 0.9375 | 6,003.9375 | 5,858.066667 | 8,192 | |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $\frac{S_6}{S_3} = 3$, then calculate $\frac{S_{12}}{S_{9}}$. | \frac{5}{3} | 1 | 4,508.5 | 4,508.5 | -1 | |
A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) | 7 | To solve this problem, we use Burnside's Lemma, which is a powerful tool in combinatorics for counting distinct configurations under group actions, such as rotations in this case. The lemma states that the number of distinct configurations, up to symmetry, is the average number of points fixed by each group element.
#... | 0 | 8,192 | -1 | 8,192 |
The difference between the larger root and the smaller root of $x^2 - px + \frac{p^2 - 1}{4} = 0$ is: | 1 | 1. **Identify the roots**: Let the roots of the quadratic equation $x^2 - px + \frac{p^2 - 1}{4} = 0$ be $r$ and $s$, where $r \geq s$.
2. **Apply Vieta's formulas**: Vieta's formulas tell us that the sum of the roots ($r+s$) is equal to the coefficient of $x$ with the opposite sign, and the product of the roots ($rs$... | 1 | 2,283.6875 | 2,283.6875 | -1 |
A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the st... | 198 | We set up a coordinate system, with the starting point of the car at the origin. At time $t$, the car is at $\left(\frac 23t,0\right)$ and the center of the storm is at $\left(\frac{t}{2}, 110 - \frac{t}{2}\right)$. Using the distance formula,
\begin{eqnarray*} \sqrt{\left(\frac{2}{3}t - \frac 12t\right)^2 + \left(110-... | 0.5625 | 6,366.25 | 4,946.222222 | 8,192 |
Suppose $a, b$, and $c$ are real numbers such that $$\begin{aligned} a^{2}-b c & =14 \\ b^{2}-c a & =14, \text { and } \\ c^{2}-a b & =-3 \end{aligned}$$ Compute $|a+b+c|$. | \frac{17}{5} | Subtracting the first two equations gives $(a-b)(a+b+c)=0$, so either $a=b$ or $a+b+c=0$. However, subtracting first and last equations gives $(a-c)(a+b+c)=17$, so $a+b+c \neq 0$. This means $a=b$. Now adding all three equations gives $(a-c)^{2}=25$, so $a-c= \pm 5$. Then $a+b+c= \pm \frac{17}{5}$. | 0.875 | 5,110.25 | 4,670 | 8,192 |
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? | 6 | 1. **Identify the minimum and maximum scores**: The player can score a minimum of 2 points per basket and a maximum of 3 points per basket. If all baskets were 2-pointers, the minimum score would be:
\[
5 \times 2 = 10
\]
If all baskets were 3-pointers, the maximum score would be:
\[
5 \times 3 = 15
... | 1 | 2,049.3125 | 2,049.3125 | -1 |
The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. What is the probability that they will get ex... | 0.243 | 0.0625 | 5,764.875 | 7,592 | 5,643.066667 | |
Compute
\[\sum_{k = 1}^\infty \frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\] | 2 | 0.0625 | 7,297.3125 | 3,430 | 7,555.133333 | |
There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running... | 6 | To solve this problem, we need to find the least time $T > 0$ such that at least $5$ of the horses are again at the starting point. Each horse $k$ returns to the starting point at multiples of $k$ minutes. Therefore, we are looking for the smallest time $T$ that is a common multiple of the running times of any $5$ hors... | 0 | 8,132.125 | -1 | 8,132.125 |
Given that \(\frac{810 \times 811 \times 812 \times \cdots \times 2010}{810^{n}}\) is an integer, find the maximum value of \(n\). | 150 | 0.625 | 6,474.5 | 5,820.1 | 7,565.166667 | |
How many 12 step paths are there from point $A$ to point $C$ which pass through point $B$ on a grid, where $A$ is at the top left corner, $B$ is 5 steps to the right and 2 steps down from $A$, and $C$ is 7 steps to the right and 4 steps down from $A$? | 126 | 0.625 | 6,011.5625 | 5,392.4 | 7,043.5 | |
Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ where $a,b \in \mathbb{R}$. If the function $f(x)$ has an extremum of $10$ at $x=1$, then the value of $b$ is \_\_\_\_\_\_. | -11 | 1 | 3,736.25 | 3,736.25 | -1 | |
Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is strictly greater than its leftmost digit. | 99 | We will work in base 11, so let $x=\overline{\operatorname{def}}_{11}$ such that $d>0$. Then, based on the first two conditions, we aim to find multiples of 5 between $100_{11}$ and $1000_{11}$. We note that $$\overline{d e f}_{11} \equiv 11^{2} \cdot d+11 \cdot e+f \equiv d+e+f \quad(\bmod 5)$$ Hence, $x$ a multiple o... | 0 | 8,076.375 | -1 | 8,076.375 |
Given that $\theta=\arctan \frac{5}{12}$, find the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$. | \frac{\pi}{4} | 0.625 | 6,558.875 | 5,579 | 8,192 | |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order.
$(1)$ If the vectors $\overrightarrow{m}=(3,\sin B)$ and $\overrightarrow{n}=(2,\sin C)$ are collinear, find the value of $\cos A$;
$(2)$ If $ac=... | 2 \sqrt {3} | 0 | 7,966.9375 | -1 | 7,966.9375 | |
Find $h(x)$, with terms in order of decreasing degree, if \[3x^4+2x-1+h(x)=5x^2-6x-1.\] | -3x^4+5x^2-8x | 0.9375 | 1,932.8125 | 1,995.733333 | 989 | |
Find all values of $r$ such that $\lfloor r \rfloor + r = 16.5$. | 8.5 | 1 | 2,785.5625 | 2,785.5625 | -1 | |
The number of rounds of golf played by each golfer of an amateur golf association is shown in the chart below. What is the average number of rounds played by each golfer? Express your answer to the nearest whole number. [asy]
size(150);
draw((0,7)--(0,0)--(10,0));
for(int i = 1; i <= 5; ++i){
label((string)i,(2*i,0),S... | 3 | 0.8125 | 3,182.6875 | 2,026.692308 | 8,192 | |
What is the smallest positive integer that is neither prime nor a cube and that has an even number of prime factors, all greater than 60? | 3721 | 0.3125 | 7,414.75 | 6,578.4 | 7,794.909091 | |
Simplify $\dfrac{3+4i}{1+2i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). | \dfrac{11}{5} - \dfrac{2}{5}i | 0.8125 | 3,136.25 | 1,969.538462 | 8,192 | |
The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation
\[f(a) + f(b) = n^2\]holds. What is $f(2002)$? | 96 | 0.25 | 7,945.9375 | 7,207.75 | 8,192 | |
Find $AB$ in the triangle below.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0.5,sqrt(3)/2);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18$", (A+C)/2,W);
label("$30^\circ$", (0.3,0),N);
[/asy] | 18\sqrt{3} | 0.3125 | 4,716 | 3,614.4 | 5,216.727273 | |
In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to | 2x+1 | Given that $M$ is the midpoint of arc $CAB$, segment $MP$ is perpendicular to chord $AB$ at $P$, and $M$ is the midpoint of the chord $AB$. This implies that $AP = PB$.
1. **Identify the relationship between $AP$ and $PB$:**
Since $MP$ is perpendicular to $AB$ at $P$ and $M$ is the midpoint of arc $CAB$, $P$ is al... | 0 | 7,803.5 | -1 | 7,803.5 |
Let $A B C$ be an equilateral triangle with side length 1. Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$ on side $\overline{B ... | \frac{1}{1+\sqrt[3]{2}} | First, note that point $X$ can be constructed from intersection of $\odot(D O F)$ and side $\overline{B C}$. Thus, if there is a unique equilateral triangle, then we must have that $\odot(D O F)$ is tangent to $\overline{B C}$. Furthermore, $\odot(D O F)$ is tangent to $D E$, so by equal tangents, we have $C D=C X$. We... | 0 | 8,192 | -1 | 8,192 |
On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour? | 14 | 0.5625 | 6,668.5625 | 6,005 | 7,521.714286 | |
In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006? | 55.2 | 0.1875 | 3,277.1875 | 4,947.333333 | 2,891.769231 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$. | 3 + \sqrt{3} | 0.625 | 6,379.6875 | 5,292.3 | 8,192 | |
Given that 2 students exercised for 0 days, 4 students exercised for 1 day, 2 students exercised for 2 days, 5 students exercised for 3 days, 4 students exercised for 4 days, 7 students exercised for 5 days, 3 students exercised for 6 days, and 2 students exercised for 7 days, find the mean number of days of exercise, ... | 3.66 | 1 | 3,925.25 | 3,925.25 | -1 | |
Given the function $f(x)=2\sqrt{3}\sin x\cos x+2\cos^{2}x-1$.
(I) Find the axis of symmetry and the center of symmetry of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi }{6}, \frac{\pi }{4}\right]$. | -1 | 0.4375 | 7,453.1875 | 6,503.285714 | 8,192 | |
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,
\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]
Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)? | 90 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest $n$ such that $n$! ends in 290 zeroes. | 1170 | Each 0 represents a factor of $10=2 \cdot 5$. Thus, we wish to find the smallest factorial that contains at least 290 2's and 290 5's in its prime factorization. Let this number be $n$!, so the factorization of $n$! contains 2 to the power $p$ and 5 to the power $q$, where $$p=\left\lfloor\frac{n}{2}\right\rfloor+\left... | 0.5 | 6,851.125 | 5,896.25 | 7,806 |
What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps? | 6 | 1 | 4,930.375 | 4,930.375 | -1 | |
An electric car is charged 3 times per week for 52 weeks. The cost to charge the car each time is $0.78. What is the total cost to charge the car over these 52 weeks? | \$121.68 | Since the car is charged 3 times per week for 52 weeks, it is charged \(3 \times 52 = 156\) times. Since the cost per charge is $0.78, then the total cost is \(156 \times 0.78 = 121.68\). | 0.875 | 312.8125 | 314.714286 | 299.5 |
Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$. | -2 | We can substitute $x=y-1$ to obtain a polynomial having roots $a+1, b+1, c+1$, namely, $(y-1)^{3}-(y-1)+1=y^{3}-3 y^{2}+2 y+1$. The sum of the reciprocals of the roots of this polynomial is, by Viete's formulas, $\frac{2}{-1}=-2$. | 0.875 | 4,881.5 | 4,408.571429 | 8,192 |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | \frac{89}{9} | 0.0625 | 8,094.4375 | 8,192 | 8,087.933333 | |
The sum of the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}- \frac{\pi}{3}\right)$ where $(0\leqslant x\leqslant 9)$ is to be determined. | 2-\sqrt{3} | 0.625 | 5,483.125 | 5,147.9 | 6,041.833333 | |
Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. | \frac{16}{3} | 1 | 2,689.5625 | 2,689.5625 | -1 | |
A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
$\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$
| 3 \sec \frac{1}{2} \theta | 0 | 4,728.5 | -1 | 4,728.5 | |
How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? | 4 |
1. **Rearrange and Complete the Square**:
Start by rearranging the given equation:
\[
x^{2020} + y^2 = 2y.
\]
We can rewrite this as:
\[
x^{2020} + y^2 - 2y = 0.
\]
Completing the square for \(y\), we have:
\[
x^{2020} + (y-1)^2 - 1 = 0 \implies x^{2020} + (y-1)^2 = 1.
\]
2. **Anal... | 0.9375 | 3,278.875 | 2,951.333333 | 8,192 |
Let $A B C$ be an isosceles triangle with apex $A$. Let $I$ be the incenter. If $A I=3$ and the distance from $I$ to $B C$ is 2 , then what is the length of $B C$ ? | 4\sqrt{5} | Let $X$ and $Y$ be the points where the incircle touches $A B$ and $B C$, respectively. Then $A X I$ and $A Y B$ are similar right triangles. Since $I$ is the incenter, we have $I X=I Y=2$. Using the Pythagorean theorem on triangle $A X I$, we find $A X=\sqrt{5}$. By similarity, $A Y / A X=B Y / I X$. Plugging in the n... | 0.875 | 4,791.6875 | 4,305.928571 | 8,192 |
Given $\overrightarrow{a}=(1+\cos \omega x,-1)$, $\overrightarrow{b}=( \sqrt {3},\sin \omega x)$ ($\omega > 0$), and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\pi$.
(1) Find the expression of the function $f(x)$.
(2) Let $\theta\in(0, \frac ... | \frac {3 \sqrt {3}+4}{10} | 0 | 4,925.5 | -1 | 4,925.5 | |
The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. | -1 | 0.25 | 7,089.25 | 7,168 | 7,063 | |
In trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the diagonals intersect at point \(E\). The areas of \(\triangle ADE\) and \(\triangle BCE\) are given as 12 and 3, respectively. Find the area of the trapezoid. | 27 | 0.375 | 6,551.5 | 5,123.5 | 7,408.3 | |
The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\sqrt{10}$ units. What is the product of all possible values for $a$? | -3 | 1 | 2,291 | 2,291 | -1 | |
Compute: $5^2-3(4)+3^2$. | 22 | 1 | 340.5 | 340.5 | -1 | |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, and $a_{1}+a_{10}=a_{9}$, find $\frac{{a}_{1}+{a}_{2}+…+{a}_{9}}{{a}_{10}}$. | \frac{27}{8} | 1 | 2,482.5625 | 2,482.5625 | -1 | |
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000. | 7 | If we write out the remainders when powers of $2$ are divided by $1000$, we must eventually write a number we have already written down. After this happens, we will fall into a cycle, and thus, nothing new will be written down.
The answer extraction of the problem is equivalent to asking for $1+2+4 + \dots + 2^n \pmod... | 0 | 8,192 | -1 | 8,192 |
If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$. | \frac{25}{9} | 0.1875 | 7,781.625 | 6,003.333333 | 8,192 | |
In the polar coordinate system, circle $C$ is centered at point $C\left(2, -\frac{\pi}{6}\right)$ with a radius of $2$.
$(1)$ Find the polar equation of circle $C$;
$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\theta = -\frac{5\pi}{12} (\rho \in \mathbb{R})$. | 2\sqrt{2} | 0.1875 | 7,535.4375 | 5,223 | 8,069.076923 | |
Let \(ABCD\) be a convex trapezoid such that \(\angle BAD = \angle ADC = 90^{\circ}\), \(AB = 20\), \(AD = 21\), and \(CD = 28\). Point \(P \neq A\) is chosen on segment \(AC\) such that \(\angle BPD = 90^{\circ}\). Compute \(AP\). | 143/5 | 0.75 | 6,152.625 | 5,472.833333 | 8,192 | |
How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.) | 15 | To find how many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order, we start by analyzing the possible structures of such numbers.
1. **Determine the range of the first two digits:**
- The integers must be between $2020$ and $2400$. Therefore, the first two digits can only be... | 0.625 | 6,674.8125 | 5,764.5 | 8,192 |
Define $E(n)$ as the sum of the even digits of $n$ and $O(n)$ as the sum of the odd digits of $n$. Find the value of $E(1) + O(1) + E(2) + O(2) + \dots + E(150) + O(150)$.
A) 1200
B) 1300
C) 1350
D) 1400
E) 1450 | 1350 | 0 | 8,192 | -1 | 8,192 | |
The number $839$ can be written as $19q+r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$? | 41 | 1 | 4,745.4375 | 4,745.4375 | -1 | |
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \] | 1 |
To solve the problem, we need to determine all positive integers that are relatively prime to every term of the sequence defined by:
\[
a_n = 2^n + 3^n + 6^n - 1, \quad n \geq 1.
\]
**Step 1: Understanding the sequence properties**
To determine an integer relatively prime to all \( a_n \), we first investigate the ... | 0.125 | 8,110.5625 | 7,540.5 | 8,192 |
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their rout... | 279 | When they meet at the milepost, Sparky has been ridden for $n$ miles total. Assume Butch rides Sparky for $a$ miles, and Sundance rides for $n-a$ miles. Thus, we can set up an equation, given that Sparky takes $\frac{1}{6}$ hours per mile, Butch takes $\frac{1}{4}$ hours per mile, and Sundance takes $\frac{2}{5}$ hours... | 0 | 8,082.3125 | -1 | 8,082.3125 |
Find the length of side $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair X,Y,Z;
X = (0,0);
Y= (2,0);
Z = (0,sqrt(3));
draw (X--Y--Z--X,linewidth(0.9));
draw(rightanglemark(Y,X,Z,3));
label("$X$",X,S);
label("$Y$",Y,S);
label("$Z$",Z,N);
label("$12$",Z/2,W);
label("$60^\circ$",(1.2,0),N);
[/asy] | 24 | 0 | 5,789.5625 | -1 | 5,789.5625 | |
A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest \( x \) for which \( f(x) = f(2022) \). | 2022 | 0 | 8,192 | -1 | 8,192 | |
Nine points are evenly spaced at intervals of one unit around a $3 \times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{5}$
D) $\f... | \frac{1}{3} | 0 | 7,655 | -1 | 7,655 | |
Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five? | 72000 | 0.3125 | 6,632.875 | 5,801.8 | 7,010.636364 | |
Find the product of the values of $x$ that satisfy the equation $|5x| + 7 = 47$. | -64 | 1 | 1,105 | 1,105 | -1 | |
In rectangle \(ABCD\), \(E\) and \(F\) are chosen on \(\overline{AB}\) and \(\overline{CD}\), respectively, so that \(AEFD\) is a square. If \(\frac{AB}{BE} = \frac{BE}{BC}\), determine the value of \(\frac{AB}{BC}\). | \frac{3 + \sqrt{5}}{2} | 0 | 4,994.5625 | -1 | 4,994.5625 | |
The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$? | 41 | 0.0625 | 8,090.625 | 8,192 | 8,083.866667 | |
Calculate the expression $16 \times 0.5 - (4.5 - 0.125 \times 8)$.
A) $4.5$
B) $4$
C) $4 \frac{1}{2}$
D) $6$
E) $7$ | 4 \frac{1}{2} | 0 | 448.375 | -1 | 448.375 | |
Given that $\dfrac{\pi}{2} < \alpha < \beta < \dfrac{3\pi}{4}, \cos(\alpha - \beta) = \dfrac{12}{13}, \sin(\alpha + \beta) = -\dfrac{3}{5}$, find the value of $\sin 2\alpha$. | -\dfrac{56}{65} | 0.25 | 6,530 | 5,469.75 | 6,883.416667 | |
In a party with $1982$ people, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else? | \[ 1979 \] | We induct on $n$ to prove that in a party with $n$ people, there must be at least $(n-3)$ people who know everyone else. (Clearly this is achievable by having everyone know everyone else except three people $A, B, C$ , who do not know each other.)
Base case: $n = 4$ is obvious.
Inductive step: Suppose in a party with $... | 0 | 6,837 | -1 | 6,837 |
Let $a$ and $b$ be acute angles such that
\begin{align*}
3 \sin^2 a + 2 \sin^2 b &= 1, \\
3 \sin 2a - 2 \sin 2b &= 0.
\end{align*}Find $a + 2b,$ as measured in radians. | \frac{\pi}{2} | 0.6875 | 6,379.3125 | 5,555.363636 | 8,192 | |
Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step. | 10 | 0 | 5,359.875 | -1 | 5,359.875 | |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\frac{c}{b\sin B}$$. | \frac{2\sqrt{3}}{3} | 0 | 6,125.4375 | -1 | 6,125.4375 | |
The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day... | 5491 | 0.9375 | 2,733.6875 | 2,848.133333 | 1,017 | |
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to $n$ and let $a_{i}$ be the number of friends of the $i$-th resident. Suppose that $\sum_{i=1}^{n} a_{i}^{2}=n^{2}-n$. Let $k$ be the smallest number of residents (at ... | \[
k = 5
\] | Let us define the simple, undirected graph $G$ so that the vertices of $G$ are the town's residents and the edges of $G$ are the friendships between the residents. Let $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$ denote the vertices of $G ; a_{i}$ is degree of $v_{i}$ for every $i$. Let $E(G)$ denote the edges of $G$. In thi... | 0 | 8,192 | -1 | 8,192 |
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 $c$ denote the number of positive integers $n \leq 1000$ with $S(n)$ odd, and let $d$ denote the number of positive integers $n \leq 1000... | 33 | 0 | 7,846.75 | -1 | 7,846.75 | |
In this subtraction problem, \( P, Q, R, S, T \) represent single digits. What is the value of \( P + Q + R + S + T \)?
\[
\begin{array}{rrrrr}
7 & Q & 2 & S & T \\
-P & 3 & R & 9 & 6 \\
\hline
2 & 2 & 2 & 2 & 2
\end{array}
\] | 29 | 0.0625 | 6,225.625 | 4,616 | 6,332.933333 | |
If the average of a set of sample data 4, 5, 7, 9, $a$ is 6, then the variance $s^2$ of this set of data is \_\_\_\_\_\_. | \frac{16}{5} | 0.625 | 3,104.125 | 3,140.8 | 3,043 | |
Inside an isosceles triangle $\mathrm{ABC}$ with equal sides $\mathrm{AB} = \mathrm{BC}$ and an angle of 80 degrees at vertex $\mathrm{B}$, a point $\mathrm{M}$ is taken such that the angle $\mathrm{MAC}$ is 10 degrees and the angle $\mathrm{MCA}$ is 30 degrees. Find the measure of the angle $\mathrm{AMB}$. | 70 | 0.375 | 7,379.3125 | 6,024.833333 | 8,192 | |
During the Spring Festival, a supermarket holds a promotional lottery event. When the amount spent by a customer reaches a certain threshold, they can participate in a lottery. The rules of the event are: from a box containing 3 black balls, 2 red balls, and 1 white ball (identical except for color), customers can draw... | \frac{8}{9} | 0.3125 | 6,297.1875 | 3,445 | 7,593.636364 | |
Find an ordered pair $(u,v)$ that solves the system: \begin{align*} 5u &= -7 - 2v,\\ 3u &= 4v - 25 \end{align*} | (-3,4) | 1 | 1,938.875 | 1,938.875 | -1 | |
At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize and the winner bowls #2.... | 32 | 0.5 | 7,015.375 | 5,838.75 | 8,192 | |
A $1 \times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
A) $\frac{9\pi}{8}$
B) $\frac{12\pi}{8}$
C) $\frac{13\pi}{8}$
D) $\frac{15\pi}{8}$
E) $\frac{16\pi}{8}$ | \frac{13\pi}{8} | 0 | 3,449.125 | -1 | 3,449.125 | |
Given \(\triangle DEF\), where \(DE=28\), \(EF=30\), and \(FD=16\), calculate the area of \(\triangle DEF\). | 221.25 | 0 | 6,618.5 | -1 | 6,618.5 | |
The integers \( r \) and \( k \) are randomly selected, where \(-5 < r < 10\) and \(0 < k < 10\). What is the probability that the division \( r \div k \) results in \( r \) being a square number? Express your answer as a common fraction. | \frac{8}{63} | 0 | 7,476.75 | -1 | 7,476.75 |
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