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 |
|---|---|---|---|---|---|---|
When fitting a set of data with the model $y=ce^{kx}$, in order to find the regression equation, let $z=\ln y$ and transform it to get the linear equation $z=0.3x+4$. Then, the values of $c$ and $k$ are respectively \_\_\_\_\_\_ and \_\_\_\_\_\_. | 0.3 | 1 | 1,184.5625 | 1,184.5625 | -1 | |
An inverted cone with base radius $12 \mathrm{cm}$ and height $18 \mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \mathrm{cm}$. What is the height in centimeters of the water in the cylinder? | 1.5 | 1. **Calculate the volume of the water in the cone**:
The formula for the volume $V$ of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where $r$ is the radius of the base and $h$ is the height of the cone. For the given cone, $r = 12 \text{ cm}$ and $h = 18 \text{ cm}$. Plugging in these values, w... | 0.25 | 783.375 | 825.25 | 769.416667 |
Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse. | \frac{2\sqrt{5}}{5} | 0 | 7,371.375 | -1 | 7,371.375 | |
Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.) Find the probability that only the... | \frac{1}{3} | Let $a_{n}$ denote the answer when there are $2n-1$ regular townies, one Jester, and one goon. It is not hard to see that $a_{1}=\frac{1}{3}$. Moreover, we have a recursion $$a_{n}=\frac{1}{2n+1} \cdot 1+\frac{1}{2n+1} \cdot 0+\frac{2n-1}{2n+1}\left(\frac{1}{2n-1} \cdot 0+\frac{2n-2}{2n-1} \cdot a_{n-1}\right)$$ The re... | 0 | 7,505 | -1 | 7,505 |
In obtuse triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $a=7$, $b=3$, and $\cos C= \frac{ 11}{14}$.
1. Find the values of $c$ and angle $A$.
2. Find the value of $\sin (2C- \frac{ \pi }{6})$. | \frac{ 71}{98} | 1 | 4,376.5625 | 4,376.5625 | -1 | |
A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2002 is a palindrome. What is the smallest number which can be added to 2002 to produce a larger palindrome? | 110 | 0.5 | 6,262.8125 | 4,333.625 | 8,192 | |
A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit? | 23/90 | 0.6875 | 6,602.5 | 5,880 | 8,192 | |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 0.6875 | 6,064.75 | 5,362.727273 | 7,609.2 | |
Find $\log_{10} 40 +\log_{10} 25$. | 3 | 1 | 2,264.625 | 2,264.625 | -1 | |
The extension of the altitude \( BH \) of triangle \( ABC \) intersects the circumcircle at point \( D \) (points \( B \) and \( D \) lie on opposite sides of line \( AC \)). The measures of arcs \( AD \) and \( CD \) that do not contain point \( B \) are \( 120^\circ \) and \( 90^\circ \) respectively. Determine the r... | 1: \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number o... | 7 | 1. **Calculate the total volume of snow**: The driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep. Therefore, the total volume of snow is:
\[
4 \text{ yards} \times 10 \text{ yards} \times 3 \text{ yards} = 120 \text{ cubic yards}
\]
2. **Calculate the snow removed each hour**: ... | 0.25 | 7,622.1875 | 6,664 | 7,941.583333 |
Compute $\dbinom{6}{3}$. | 20 | 1 | 2,056.875 | 2,056.875 | -1 | |
Given the following three statements are true:
I. All freshmen are human.
II. All graduate students are human.
III. Some graduate students are pondering.
Considering the following four statements:
(1) All freshmen are graduate students.
(2) Some humans are pondering.
(3) No freshmen are pondering.
(4) Some of the pond... | (2). | 0 | 2,535.125 | -1 | 2,535.125 | |
What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only? | 11111111100 | 0.5625 | 7,033.375 | 6,132.222222 | 8,192 | |
Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$ ? | -2 | 0.875 | 3,851.375 | 3,577.857143 | 5,766 | |
A certain type of rice must comply with the normal distribution of weight $(kg)$, denoted as $\xi ～N(10,{σ}^{2})$, according to national regulations. Based on inspection results, $P(9.9≤\xi≤10.1)=0.96$. A company purchases a bag of this packaged rice as a welfare gift for each employee. If the company has $2000$ employ... | 40 | 0.8125 | 3,932.1875 | 3,112.923077 | 7,482.333333 | |
At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?
\begin{tabular}{ |c | c | c | }
\hline... | 14 | 0 | 1,243.6875 | -1 | 1,243.6875 | |
Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An $\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)
$\text{(A) }\{4,5,6\} \quad \text{(B) } \{4,5,7\} \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text... | \{4,5,7\} | 0 | 5,134.0625 | -1 | 5,134.0625 | |
How many positive integers $n$ satisfy $200 < n^2 < 900$? | 15 | 1 | 2,116.6875 | 2,116.6875 | -1 | |
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the ur... | \frac{1}{5} | To solve this problem, we need to calculate the probability that after four operations, the urn contains exactly three red balls and three blue balls. We start with one red ball and one blue ball in the urn.
#### Step 1: Understanding the possible sequences
To have three balls of each color, George must pick two red b... | 0.0625 | 7,845.1875 | 8,192 | 7,822.066667 |
Four planes divide space into $n$ parts at most. Calculate $n$. | 15 | 0.9375 | 5,392.625 | 5,206 | 8,192 | |
Let $a$ and $b$ be positive real numbers. Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\] | \sqrt{2} | 0.9375 | 5,618.5625 | 5,447 | 8,192 | |
A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 6x + \sin x + 3\]
Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S$. | 445 | 0 | 8,192 | -1 | 8,192 | |
An infinite geometric series has first term $328$ and a sum of $2009$. What is its common ratio? | \frac{41}{49} | 1 | 2,034.75 | 2,034.75 | -1 | |
Given two complex numbers ${z_1}$ and ${z_2}$ that correspond to points in the complex plane that are symmetric about the origin, and ${z_1 = 2 - i}$, determine the value of the complex number $\frac{z_1}{z_2}$. | -1 | 1 | 1,942.375 | 1,942.375 | -1 | |
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*unitcircle; pair[] A; pen grey1 = rgb(100/25... | 37 | To solve this problem, we need to understand the pattern of how the dots are arranged in each hexagon and how the number of dots increases as the hexagons grow in size.
1. **Observing the pattern:**
- The first hexagon has 1 dot.
- The second hexagon has 7 dots: 1 central dot and 6 surrounding dots.
- The thi... | 0.625 | 4,665.875 | 4,020.7 | 5,741.166667 |
How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem? | 160 | 0 | 7,052.125 | -1 | 7,052.125 | |
Represent the number 36 as the product of three whole number factors, the sum of which is equal to 4. What is the smallest of these factors? | -4 | 0.0625 | 7,945.5 | 4,248 | 8,192 | |
Calculate the volume of an octahedron which has an inscribed sphere of radius 1. | 4\sqrt{3} | 0.6875 | 6,961 | 6,401.454545 | 8,192 | |
How many numbers between $1$ and $3010$ are integers multiples of $4$ or $5$ but not of $20$? | 1204 | 0.0625 | 3,906.1875 | 2,136 | 4,024.2 | |
How many positive integers $n$ between 1 and 2011 make the equation $(1+i)^{2n} = 2^n i$ hold true, where $i$ is the imaginary unit? | 503 | 0.625 | 5,959.625 | 4,620.2 | 8,192 | |
As shown in the picture, the knight can move to any of the indicated squares of the $8 \times 8$ chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves. | 32 | 0.375 | 6,986.5 | 5,958.666667 | 7,603.2 | |
Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points... | 11/5 | 0.625 | 6,552.5625 | 5,845.1 | 7,731.666667 | |
Let $a_1,$ $a_2,$ $\dots$ be a sequence of positive real numbers such that
\[a_n = 11a_{n - 1} - n\]for all $n > 1.$ Find the smallest possible value of $a_1.$ | \frac{21}{100} | 0.5625 | 7,161.375 | 6,359.777778 | 8,192 | |
How many perfect squares are there between 20 and 150? | 8 | 0.9375 | 2,093.625 | 2,180.466667 | 791 | |
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\). | \frac{24}{7} | 0.125 | 8,010.25 | 6,738 | 8,192 | |
Let $ABC$ be a triangle (right in $B$ ) inscribed in a semi-circumference of diameter $AC=10$ . Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle. | 5/2 | 0.6875 | 4,533.6875 | 4,137.909091 | 5,404.4 | |
Determine the area enclosed by the curve of $y = \arccos(\cos x)$ and the $x$-axis over the interval $\frac{\pi}{4} \le x \le \frac{9\pi}{4}.$ | \frac{3\pi^2}{2} | 0 | 7,902.5625 | -1 | 7,902.5625 | |
As shown in the diagram, plane $ABDE$ is perpendicular to plane $ABC$. Triangle $ABC$ is an isosceles right triangle with $AC=BC=4$. Quadrilateral $ABDE$ is a right trapezoid with $BD \parallel AE$, $BD \perp AB$, $BD=2$, and $AE=4$. Points $O$ and $M$ are the midpoints of $CE$ and $AB$ respectively. Find the sine of t... | \frac{\sqrt{30}}{10} | 0 | 7,432.3125 | -1 | 7,432.3125 | |
When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle? | 26 | Let $a$ and $b$ be the two legs of the equation. We can find $\frac{a}{b}$ by doing $\frac{1920\pi}{800\pi}$. This simplified is $\frac{12}{5}$. We can represent the two legs as $12x$ and $5x$ for $a$ and $b$ respectively.
Since the volume of the first cone is $800\pi$, we use the formula for the volume of a cone and ... | 1 | 2,196.25 | 2,196.25 | -1 |
In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic is ahead of Dash when S... | 19.2 | 0.0625 | 7,225.125 | 8,192 | 7,160.666667 | |
Given the function $f(x)=\vec{a}\cdot \vec{b}$, where $\vec{a}=(2\cos x,\sqrt{3}\sin 2x)$, $\vec{b}=(\cos x,1)$, and $x\in \mathbb{R}$.
(Ⅰ) Find the period and the intervals of monotonic increase for the function $y=f(x)$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$,... | \frac{7\sqrt{3}}{6} | 0 | 6,399.9375 | -1 | 6,399.9375 | |
Choose one of the three conditions: ①$ac=\sqrt{3}$, ②$c\sin A=3$, ③$c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Does there exist a $\triangle ABC$ where the int... | 2\sqrt{3} | 0.0625 | 5,988.625 | 8,192 | 5,841.733333 | |
The parabola \(C_{1}: x^{2}=2 p y\) has a focus at \(F\). The hyperbola \(C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) has foci \(F_{1}\) and \(F_{2}\). Point \(P\) is a common point of the two curves in the first quadrant. If the points \(P\), \(F\), and \(F_{1}\) are collinear and there is a common tangent to \... | \sqrt{2} | 0.125 | 7,915.625 | 5,981 | 8,192 | |
Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1 (a > b > 0)$ with vertex $B$ at the top, vertex $A$ on the right, and right focus $F$. Let $E$ be a point on the lower half of the ellipse such that the tangent at $E$ is parallel to $AB$. If the eccentricity of the ellipse is $\frac {\sqrt{2}}{2}$, then... | \frac {\sqrt {2}}{4} | 0 | 4,700.5 | -1 | 4,700.5 | |
The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from the sun, in AU?
[asy]
unitsize(1 cm);
path... | 7 | 0.25 | 7,772.8125 | 7,989.25 | 7,700.666667 | |
In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have? | 13 | 0.0625 | 7,611.75 | 4,988 | 7,786.666667 | |
Let \(a_n\) be the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{k}\), where \(k = a_n\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Which values \(b_n\) occur for infinitely many \(n\)? | 87 | 0.6875 | 7,102.1875 | 6,606.818182 | 8,192 | |
In the diagram, $R$ is on $QS$ and $QR=8$.
Also, $PR=12$, $\angle PRQ=120^{\circ}$, and $\angle RPS=90^{\circ}$.
What is the area of $\triangle QPS$? | $96 \sqrt{3}$ | 0 | 4,379.8125 | -1 | 4,379.8125 | |
An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions. | 2000 | 0 | 947.9375 | -1 | 947.9375 | |
The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$ , where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$ \dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{... | 14 | 0 | 7,965.9375 | -1 | 7,965.9375 | |
The six-digit number $M=\overline{abc321}$, where $a, b, c$ are three different numbers, and all are greater than 3. If $M$ is a multiple of 7, what is the smallest value of $M$? | 468321 | 0.125 | 8,158.6875 | 7,925.5 | 8,192 | |
A high school's second-year students are studying the relationship between students' math and Chinese scores. They conducted a simple random sampling with replacement and obtained a sample of size $200$ from the second-year students. The sample observation data of math scores and Chinese scores are organized as follows... | \frac{15}{7} | 0 | 4,255.6875 | -1 | 4,255.6875 | |
Given the function $f(x)= \sqrt {3}\sin (\omega x+\varphi)-\cos (\omega x+\varphi)$ $(\omega > 0,0 < \varphi < \pi)$ is an even function, and the distance between two adjacent axes of symmetry of its graph is $\dfrac {\pi}{2}$, then the value of $f(- \dfrac {\pi}{8})$ is \_\_\_\_\_\_. | \sqrt {2} | 0 | 5,686.625 | -1 | 5,686.625 | |
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 | 0.0625 | 7,956.6875 | 8,192 | 7,941 | |
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be... | 458 | 0 | 8,192 | -1 | 8,192 | |
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
| 581 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest number that uses only the digits 2 and 3 in equal quantity and is divisible by both 2 and 3. | 223332 | 0.1875 | 7,327.25 | 5,376.333333 | 7,777.461538 | |
Given a random variable $\xi \sim N(1,4)$, and $P(\xi < 3)=0.84$, then $P(-1 < \xi < 1)=$ \_\_\_\_\_\_. | 0.34 | 0 | 5,204.375 | -1 | 5,204.375 | |
Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same? | \dfrac{1}{2} | 0.8125 | 4,501.4375 | 3,649.769231 | 8,192 | |
Determine the greatest common divisor (gcd) of all the numbers of the form
$$
(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)
$$
where $a, b, c, d$ are integers. | 12 | 0.3125 | 7,839.0625 | 7,062.6 | 8,192 | |
What is the product of $\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2006}{2005}$? | 1003 | 1. **Identify the Sequence**: The product given is a sequence of fractions where each fraction's numerator is one more than the previous fraction's denominator:
\[
\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2006}{2005}
\]
2. **Cancellation of Terms**: Notice that in the produ... | 1 | 2,313.5625 | 2,313.5625 | -1 |
Find the value of: \(\frac{\left(\sqrt{3} \cdot \tan 12^{\circ} - 3\right) \cdot \csc 12^{\circ}}{4 \cos ^{2} 12^{\circ} - 2}\). | -4 \sqrt{3} | 0.125 | 7,674.1875 | 4,049.5 | 8,192 | |
There are a few integers \( n \) such that \( n^{2}+n+1 \) divides \( n^{2013}+61 \). Find the sum of the squares of these integers. | 62 | 1 | 4,445.9375 | 4,445.9375 | -1 | |
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$ , he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-wor... | 16 | 0.875 | 5,521.1875 | 5,139.642857 | 8,192 | |
How many groups of integer solutions are there for the equation $xyz = 2009$? | 72 | 0.5 | 6,920 | 5,648 | 8,192 | |
Given an ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > 0)$ with one of its foci at $F(-1, 0)$. Points $A$ and $B$ are the left and right vertices of the ellipse's major axis, respectively. A line $l$ passes through $F$ and intersects the ellipse at distinct points $C$ and $D$.
1. Find the equation of the ellips... | \sqrt{3} | 0.25 | 7,858.3125 | 7,053 | 8,126.75 | |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 | 0.3125 | 7,604.9375 | 6,313.4 | 8,192 | |
Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:
[asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5); pair G = intersectionpoints(B--F, D--C)[0]; pair E = i... | 16 | 0 | 7,757 | -1 | 7,757 | |
Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\frac{n}{180}$ terminates. | 60 | 0 | 4,926.5625 | -1 | 4,926.5625 | |
How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not? | 41 | 0.25 | 7,952.0625 | 7,232.25 | 8,192 | |
A gambler starts with $100. During a series of 4 rounds, they bet each time one-third of the amount they have. They win twice and lose twice, but now the wins are twice the staked amount, and losses mean losing the staked amount. Determine the final amount of money the gambler has, assuming the wins and losses happen i... | \frac{8000}{81} | 0 | 8,192 | -1 | 8,192 | |
Betty used a calculator to find the product $0.075 \times 2.56$. She forgot to enter the decimal points. The calculator showed $19200$. If Betty had entered the decimal points correctly, the answer would have been | .192 | 1. **Identify the error in decimal placement**: Betty entered the numbers without the decimal points, so she effectively multiplied $75$ by $256$ instead of $0.075$ by $2.56$.
2. **Calculate the product without decimal error**:
\[
75 \times 256 = 19200
\]
This is the result shown by the calculator.
3. **... | 0.75 | 1,903.125 | 1,755.416667 | 2,346.25 |
Calculate the number of different rectangles with sides parallel to the grid that can be formed by connecting four of the dots in a $5\times 5$ square array of dots. (Two rectangles are different if they do not share all four vertices.) | 100 | 0.25 | 6,815.75 | 5,985.5 | 7,092.5 | |
There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$ | \frac{1}{2001} | The problem involves calculating the probabilities $P_s$ and $P_d$ and finding their absolute difference.
1. **Total number of marbles**:
There are $1001$ red marbles and $1001$ black marbles, making a total of $1001 + 1001 = 2002$ marbles.
2. **Calculating $P_s$ (probability that two marbles drawn are the same co... | 1 | 4,552.125 | 4,552.125 | -1 |
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|+1?$
A) $\frac{\pi}{2} + 2$
B) $\frac{3\pi}{2}$
C) $\frac{3\pi}{2} + 2$
D) $2\pi + 2$ | \frac{3\pi}{2} + 2 | 0 | 8,192 | -1 | 8,192 | |
A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is: | \frac{81}{256} | 1. **Initial Setup**: The radiator starts with 16 quarts of water.
2. **First Replacement**:
- 4 quarts of water are removed, leaving $16 - 4 = 12$ quarts of water.
- 4 quarts of antifreeze are added, making the total still 16 quarts.
- The fraction of water remaining is $\frac{12}{16} = \frac{3}{4}$.
3. **... | 0.8125 | 4,664.875 | 3,850.923077 | 8,192 |
Let $g$ be the function defined by $g(x) = -3 \sin(2\pi x)$. How many values of $x$ such that $-3 \le x \le 3$ satisfy the equation $g(g(g(x))) = g(x)$? | 48 | 0 | 8,192 | -1 | 8,192 | |
A high school math preparation group consists of six science teachers and two liberal arts teachers. During a three-day period of smog-related class suspensions, they need to arrange teachers to be on duty for question-answering sessions. The requirement is that each day, there must be one liberal arts teacher and two ... | 540 | 0.1875 | 7,959.9375 | 6,954.333333 | 8,192 | |
The length of a chord intercepted on the circle $x^2+y^2-2x+4y-20=0$ by the line $5x-12y+c=0$ is 8. Find the value(s) of $c$. | -68 | 0.8125 | 3,797.25 | 2,783.076923 | 8,192 | |
The numbers from 1 to 9 are placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
x+3y+3=0$ is parallel to the line $x+(2m-1)y+m=0$, then the real number $m=$ \_\_\_\_\_\_. | -\frac{5}{2} | 0.3125 | 7,238.3125 | 6,981 | 7,355.272727 | |
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$ . Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$ ? | 1/6 | 0 | 5,283.8125 | -1 | 5,283.8125 | |
If 2006 integers $a_1, a_2, \ldots a_{2006}$ satisfy the following conditions: $a_1=0$, $|a_2|=|a_1+2|$, $|a_3|=|a_2+2|$, $\ldots$, $|a_{2006}|=|a_{2005}+2|$, then the minimum value of $a_1+a_2+\ldots+a_{2005}$ is. | -2004 | 0.125 | 8,124.4375 | 7,651.5 | 8,192 | |
Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respectively. Line $\ell$ divides ... | 69 | 0 | 8,192 | -1 | 8,192 | |
The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)
Height of the Players on the Basketball Team (inches)
$4|8$
$5|0\;1\;4\;6\;7\;7\;9$
$6|0\;3\;4\;5\... | 61.44 | 0.125 | 880.6875 | 1,169.5 | 839.428571 | |
If $f(x) = 8x^3 - 6x^2 - 4x + 5$, find the value of $f( -2)$. | -75 | 0.9375 | 3,655 | 3,352.533333 | 8,192 | |
Harriet ran a 1000 m course in 380 seconds. She ran the first 720 m of the course at a constant speed of 3 m/s. What was her speed for the remaining part of the course? | 2 | Since Harriet ran 720 m at 3 m/s, then this segment took her 720 m / 3 m/s = 240 s. In total, Harriet ran 1000 m in 380 s, so the remaining part of the course was a distance of 1000 m - 720 m = 280 m which she ran in 380 s - 240 s = 140 s. Since she ran this section at a constant speed of v m/s, then 280 m / 140 s = v ... | 0.5625 | 1,048.25 | 1,393.777778 | 604 |
One dimension of a cube is increased by $2$, another is decreased by $2$, and the third is increased by $3$. The volume of the new rectangular solid is $7$ less than the volume of the cube. Find the original volume of the cube. | 27 | 0 | 7,727 | -1 | 7,727 | |
In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(... | 39 | 1. **Assign Variables to Circles**: Let the numbers in the circles be $a$, $b$, $c$, $d$, $e$, and $f$ starting from the top circle and moving clockwise.
2. **Set Up Equations for Each Side of the Triangle**:
- The sum of the numbers on the first side is $S = a + b + c$.
- The sum of the numbers on the second si... | 0.5625 | 6,160.125 | 4,579.777778 | 8,192 |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -5 \\ 2 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | \frac{11}{2} | 1 | 2,001.6875 | 2,001.6875 | -1 | |
A certain school holds a men's table tennis team competition. The final match adopts a points system. The two teams in the final play three matches in sequence, with the first two matches being men's singles matches and the third match being a men's doubles match. Each participating player can only play in one match in... | 36 | 0.0625 | 7,329.125 | 3,553 | 7,580.866667 | |
Given a moving point $A$ on the curve $y=x^{2}$, let $m$ be the tangent line to the curve at point $A$. Let $n$ be a line passing through point $A$, perpendicular to line $m$, and intersecting the curve at another point $B$. Determine the minimum length of the line segment $AB$. | \frac{3\sqrt{3}}{2} | 0 | 6,927.75 | -1 | 6,927.75 | |
In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the tot... | 50\pi | 0.125 | 7,808.125 | 7,071.5 | 7,913.357143 | |
Factor the following expression: $145b^2 +29b$. | 29b(5b+1) | 1 | 1,333.875 | 1,333.875 | -1 | |
\frac{3 \times 5}{9 \times 11} \times \frac{7 \times 9 \times 11}{3 \times 5 \times 7}= | 1 | 1. **Identify the expression to simplify**:
\[
\frac{3 \times 5}{9 \times 11} \times \frac{7 \times 9 \times 11}{3 \times 5 \times 7}
\]
2. **Rewrite the expression using the associative property**:
The associative property allows us to regroup the factors without changing the product. We can rearrange th... | 0.875 | 526.5625 | 509.642857 | 645 |
Given positive numbers $a$ and $b$ satisfying $a+b=1$, $c\in R$, find the minimum value of $\frac{3a}{b{c}^{2}+b}+\frac{1}{ab{c}^{2}+ab}+3c^{2}$. | 6\sqrt{2} - 3 | 0.125 | 7,559.0625 | 6,216 | 7,750.928571 | |
Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-2 - i \sqrt{5}$ as a root. | x^2 + 4x + 9 | 1 | 2,098.3125 | 2,098.3125 | -1 | |
How many different positive three-digit integers can be formed using only the digits in the set $\{3, 3, 4, 4, 4, 7, 8\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 | 0.0625 | 7,683 | 5,945 | 7,798.866667 |
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