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 |
|---|---|---|---|---|---|---|
Consider the matrix
\[\mathbf{N} = \begin{pmatrix} 2x & -y & z \\ y & x & -2z \\ y & -x & z \end{pmatrix}\]
and it is known that $\mathbf{N}^T \mathbf{N} = \mathbf{I}$. Find $x^2 + y^2 + z^2$. | \frac{2}{3} | 0 | 8,192 | -1 | 8,192 | |
A non-zero digit is chosen in such a way that the probability of choosing digit $d$ is $\log_{10}{(d+1)}-\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\frac{1}{2}$ the probability that the digit chosen is in the set | {4, 5, 6, 7, 8} | 1. **Understanding the Probability Function**: The probability of choosing a digit $d$ is given by $\log_{10}(d+1) - \log_{10}(d)$. Using the properties of logarithms, this can be rewritten as:
\[
\log_{10}\left(\frac{d+1}{d}\right)
\]
2. **Calculating the Probability for Digit 2**: Substituting $d = 2$ into ... | 0 | 7,502.125 | -1 | 7,502.125 |
Given the relationship between shelf life and storage temperature is an exponential function $y = ka^x$, where milk has a shelf life of about $100$ hours in a refrigerator at $0°C$, and about $80$ hours in a refrigerator at $5°C$, determine the approximate shelf life of milk in a refrigerator at $10°C$. | 64 | 0.625 | 5,944.6875 | 4,596.3 | 8,192 | |
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are unit vectors, then find the largest possible value of
\[\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2.\]Note: A unit vector is a vector of magnitude 1. | 9 | 0.625 | 7,032.8125 | 6,337.3 | 8,192 | |
What is the sum of all numbers in the following square matrix?
$$
\begin{array}{l}
1, 2, 3, \ldots, 98, 99, 100 \\
2, 3, 4, \ldots, 99, 100, 101 \\
3, 4, 5, \ldots, 100, 101, 102 \\
\ldots \\
100, 101, 102, \ldots, 197, 198, 199
\end{array}
$$ | 1000000 | 0.125 | 7,759.1875 | 4,729.5 | 8,192 | |
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$. | 2, 128, 2000 |
To solve the problem, we need to find all positive integers \( n \) such that the cube of the number of divisors of \( n \), denoted \( d(n)^3 \), is equal to \( 4n \). The equation we need to solve is:
\[
d(n)^3 = 4n.
\]
First, recall that for a number \( n \) with the prime factorization \( n = p_1^{a_1} p_2^{a_2} ... | 0 | 8,192 | -1 | 8,192 |
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Be... | \frac{37}{56} | To solve this problem, we need to consider the probability that Bernardo's number is larger than Silvia's number under different scenarios.
#### Case 1: Bernardo picks 9.
If Bernardo picks a 9, then his number will definitely be larger than Silvia's, as Silvia can only pick numbers from 1 to 8. We calculate the probab... | 0.125 | 7,784.4375 | 6,531 | 7,963.5 |
Solve for $2d$ if $10d + 8 = 528$. | 104 | Since $10d + 8 = 528$, then $10d = 520$ and so $\frac{10d}{5} = \frac{520}{5}$ which gives $2d = 104$. | 1 | 1,619.4375 | 1,619.4375 | -1 |
Given the equation in terms of $x$, $\frac{2x-a}{3}- \frac{x-a}{2}=x-1$, has the same solution as the equation $3(x-2)-4(x- \frac{5}{4})=0$, find the value of $a$. | -11 | 1 | 2,227.9375 | 2,227.9375 | -1 | |
It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? | 3 | 1 | 1,492.1875 | 1,492.1875 | -1 | |
If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$ , what is the smallest possible value of $k$ greater than $100$ ? | 111 | 0.625 | 6,208.875 | 5,019 | 8,192 | |
Find the largest real number $\lambda$ such that
\[a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd\]for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$ | \frac{3}{2} | 0.25 | 8,020.3125 | 7,505.25 | 8,192 | |
What is the greatest possible value of $n$ if Juliana chooses three different numbers from the set $\{-6,-4,-2,0,1,3,5,7\}$ and multiplies them together to obtain the integer $n$? | 168 | Since $3 \times 5 \times 7=105$, then the greatest possible value of $n$ is at least 105. For the product of three numbers to be positive, either all three numbers are positive or one number is positive and two numbers are negative. If all three numbers are positive, the greatest possible value of $n$ is $3 \times 5 \t... | 0.625 | 7,025.25 | 6,325.2 | 8,192 |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$ .
*Proposed by David Tang* | 10 | 0.1875 | 7,579.5 | 4,962.666667 | 8,183.384615 | |
Given $\theta \in (0, \frac{\pi}{2})$, and $\sin\theta = \frac{4}{5}$, find the value of $\cos\theta$ and $\sin(\theta + \frac{\pi}{3})$. | \frac{4 + 3\sqrt{3}}{10} | 0 | 2,490.375 | -1 | 2,490.375 | |
What is the value of \(\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}\)? | 2 | We start with the given expression:
\[
\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}
\]
First, we use the change of base formula for logarithms, which states that $\log_b a = \frac{1}{\log_a b}$. Applying this, we have:
\[
\log_{40}2 = \frac{1}{\log_2 40} \quad \text{and} \quad \log_{20}2 = \frac{1}{\log_... | 1 | 4,350.6875 | 4,350.6875 | -1 |
On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road cou... | 400 | 0.125 | 8,072.75 | 7,872 | 8,101.428571 | |
Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. What is the probability that at the end of t... | \frac{1}{4} | 0 | 7,316.1875 | -1 | 7,316.1875 | |
In polar coordinates, the point $\left( -2, \frac{3 \pi}{8} \right)$ is equivalent to what other point, in the standard polar coordinate representation? Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{11 \pi}{8} \right) | 1 | 1,782.125 | 1,782.125 | -1 | |
The net change in the population over these four years is a 20% increase, then a 30% decrease, then a 20% increase, and finally a 30% decrease. Calculate the net change in the population over these four years. | -29 | 0 | 2,982.0625 | -1 | 2,982.0625 | |
Let $p(n)$ denote the product of decimal digits of a positive integer $n$ . Computer the sum $p(1)+p(2)+\ldots+p(2001)$ . | 184320 | 0.3125 | 8,031.125 | 7,677.2 | 8,192 | |
The device consists of three independently operating elements. The probabilities of failure-free operation of the elements (over time $t$) are respectively: $p_{1}=0.7$, $p_{2}=0.8$, $p_{3}=0.9$. Find the probabilities that over time $t$, the following will occur:
a) All elements operate without failure;
b) Two ele... | 0.006 | 0.3125 | 6,431.4375 | 3,599.4 | 7,718.727273 | |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$. | \frac{\sqrt{2}}{2} | 0 | 7,029.125 | -1 | 7,029.125 | |
Mary typed a six-digit number, but the two $1$ s she typed didn't show. What appeared was $2002$ . How many different six-digit numbers could she have typed? | 15 | 0.25 | 4,264.0625 | 2,781 | 4,758.416667 | |
Given that the center of an ellipse is at the origin, the focus is on the $x$-axis, and the eccentricity $e= \frac { \sqrt {2}}{2}$, the area of the quadrilateral formed by connecting the four vertices of the ellipse in order is $2 \sqrt {2}$.
(1) Find the standard equation of the ellipse;
(2) Given that line $l$ inter... | \frac{\sqrt{6}}{3} | 0 | 6,497.3125 | -1 | 6,497.3125 | |
Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \neq 0 \), \( \omega > 0 \), \( 0 < \varphi < \frac{\pi}{2} \), if \( f\left(\frac{5\pi}{6}\right) + f(0) = 0 \), find the minimum value of \( \omega \). | \frac{6}{5} | 0.625 | 6,746.9375 | 5,879.9 | 8,192 | |
On the lateral side \( CD \) of trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is drawn to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \). | 18 | 0.0625 | 7,993.875 | 5,022 | 8,192 | |
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible. | 25 | 0.125 | 8,054.625 | 7,093 | 8,192 | |
A regular triangle $EFG$ with a side length of $a$ covers a square $ABCD$ with a side length of 1. Find the minimum value of $a$. | 1 + \frac{2}{\sqrt{3}} | 0 | 7,958.4375 | -1 | 7,958.4375 | |
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ ... | n^2 - n + 1 |
To solve the problem involving cable car companies \( A \) and \( B \), we must determine the smallest integer \( k \) such that there are always two stations linked by both companies.
### Understanding the Problem
1. **Stations and Connections:**
- We have \( n^2 \) stations on a mountain, each at different alti... | 0 | 8,167.3125 | -1 | 8,167.3125 |
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is
[asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1... | 24 |
Let's denote the five squares as follows, where $a$, $b$, $c$, $d$, and $e$ represent the numbers 1, 4, 7, 10, and 13 placed in the squares:
\[
\begin{array}{|c|c|}
\hline
a & b \\
\hline
c & d \\
\hline
\end{array}
\quad
\begin{array}{|c|}
\hline
e \\
\hline
\end{array}
\]
The problem states that the sum of the thr... | 0.75 | 4,848.375 | 3,733.833333 | 8,192 |
A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days? | 185 | 1. **Identify the initial amount of water**: The pool initially contains 200 gallons of water.
2. **Determine the rate of evaporation**: Water evaporates at a rate of 0.5 gallons per day.
3. **Calculate the total evaporation over 30 days**: Multiply the daily evaporation rate by the number of days:
\[
30 \text{... | 1 | 762.0625 | 762.0625 | -1 |
Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), satisfying
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\). | \frac{3 \sqrt{3}}{2} | 0 | 8,142.5625 | -1 | 8,142.5625 | |
Find $ 8^8 \cdot 4^4 \div 2^{28}$. | 16 | 0.9375 | 3,279.875 | 2,952.4 | 8,192 | |
In the rectangular coordinate system $(xOy)$, the curve $C\_1$: $ \begin{cases} x=a\cos φ \ y=b\sin φ\end{cases}(φ)$ is a parameter, where $(a > b > 0)$, and in the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the curve $C\_2$: $ρ=2\cos θ$, the ray $l$: $θ=α(ρ≥0)$, i... | \dfrac {3 \sqrt {30}}{20} | 0 | 7,509.375 | -1 | 7,509.375 | |
Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16? | \frac{103}{108} | 0.125 | 7,859.1875 | 5,529.5 | 8,192 | |
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \] | f(x) = x |
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function satisfying the functional equation:
\[
f(x^2 + f(y)) = y + (f(x))^2 \quad \text{for all } x, y \in \mathbb{R}.
\]
**Step 1**: Evaluate the functional equation at specific points.
First, let's substitute \( y = 0 \) into the functional equation:
\[
f(x^2 ... | 0.1875 | 7,882.625 | 6,542 | 8,192 |
How many three-digit multiples of 9 consist only of odd digits? | 11 | 0.4375 | 7,649 | 6,950.857143 | 8,192 | |
Given the function \( f(x) = ax^2 + 8x + 3 \) where \( a < 0 \). For a given negative \( a \), there exists a largest positive real number \( L(a) \) such that the inequality \( |f(x)| \leq 5 \) holds for the entire interval \([0, L(a)]\). Find the value of \( a \) that maximizes \( L(a) \) and determine this maximum \... | \frac{\sqrt{5} + 1}{2} | 0 | 8,192 | -1 | 8,192 | |
Let points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) be located on the unit sphere. Find the maximum value of \( \min \left\{A_{i} A_{j} \mid 1 \leq i < j \leq 5 \right\} \) and determine all cases where this maximum value is achieved. | \sqrt{2} | 0.25 | 7,598.9375 | 6,495.75 | 7,966.666667 | |
Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most 5 such that $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)^{2}=(a+b+c+d)(a-b+c-d)\left((a-c)^{2}+(b-d)^{2}\right)$ | 49 | Let $x=a+c, y=a-c, w=b+d$, and $z=b-d$. Then $$\left(w^{2}+x^{2}+y^{2}+z^{2}\right)^{2}=4\left(x^{2}-w^{2}\right)\left(y^{2}+z^{2}\right)$$ and since $\left|x^{2}+w^{2}\right| \geq\left|x^{2}-w^{2}\right|$ it follows that $w=0$ or $y=z=0$. Now $y=z=0$ implies $a=b=c=d=0$. Now $w=0$ gives $b=-d$. Then for equality to ho... | 0 | 8,192 | -1 | 8,192 |
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedro... | 9 |
To determine the maximum number of edges of a regular octahedron that can be seen from a point outside the octahedron, we start by considering the geometric properties of the octahedron and the visibility conditions.
A regular octahedron has 12 edges. The visibility of an edge from an external point depends on whethe... | 0 | 7,343.1875 | -1 | 7,343.1875 |
Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes. | 50 | 0 | 7,827.5625 | -1 | 7,827.5625 | |
In how many ways can four married couples sit around a circular table such that no man sits next to his wife? | 1488 | 0.0625 | 8,174.25 | 7,908 | 8,192 | |
There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time.
Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$ . | 1/30 | 0 | 7,978.4375 | -1 | 7,978.4375 | |
If $\mathbf{a}$ and $\mathbf{b}$ are vectors such that $\|\mathbf{a}\| = 7$ and $\|\mathbf{b}\| = 11$, then find all possible values of $\mathbf{a} \cdot \mathbf{b}$.
Submit your answer in interval notation. | [-77,77] | 1 | 1,096.625 | 1,096.625 | -1 | |
Calculate the limit of the function:
$$\lim _{x \rightarrow \pi}\left(\operatorname{ctg}\left(\frac{x}{4}\right)\right)^{1 / \cos \left(\frac{x}{2}\right)}$$ | e | 0.3125 | 7,562.4375 | 6,177.4 | 8,192 | |
Find the positive value of $x$ which satisfies
\[\log_5 (x + 2) + \log_{\sqrt{5}} (x^2 + 2) + \log_{\frac{1}{5}} (x + 2) = 3.\] | \sqrt{\sqrt{125} - 2} | 0 | 5,763.25 | -1 | 5,763.25 | |
Calculate the definite integral
$$
\int_{0}^{\pi / 2} \frac{\sin x}{2+\sin x} \, dx
$$ | \frac{\pi}{2} - \frac{2 \pi}{3 \sqrt{3}} | 0 | 6,347.375 | -1 | 6,347.375 | |
Find $s$ given that \[3 \log_2 s = \log_2 (3s).\] | \sqrt{3} | 1 | 1,581.5625 | 1,581.5625 | -1 | |
Given the function $f(x)=2x^{2}-3x-\ln x+e^{x-a}+4e^{a-x}$, where $e$ is the base of the natural logarithm, if there exists a real number $x_{0}$ such that $f(x_{0})=3$ holds, then the value of the real number $a$ is \_\_\_\_\_\_. | 1-\ln 2 | 0.9375 | 4,048.8125 | 3,772.6 | 8,192 | |
In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? | 3 | 1. **Define Variables:**
Let $n$ be the total number of people in the room. Let $x$ be the number of people wearing both a hat and a glove.
2. **Fractional Constraints:**
Given that $\frac{2}{5}$ of the people are wearing gloves and $\frac{3}{4}$ of the people are wearing hats, we need $n$ to be a common multipl... | 1 | 3,467.9375 | 3,467.9375 | -1 |
An \( n \)-digit number \( x \) has the following property: if the last digit of \( x \) is moved to the front, the result is \( 2x \). Find the smallest possible value of \( n \). | 18 | 0.25 | 7,677.625 | 6,134.5 | 8,192 | |
Find the area of the circle described by the equation $x^2 - 4x + y^2 - 8y + 12 = 0$ that lies above the line $y = 3$. | 4\pi | 0 | 8,192 | -1 | 8,192 | |
What is the maximum number of possible points of intersection of a circle and a triangle? | 6 | To determine the maximum number of possible points of intersection between a circle and a triangle, we need to consider the interaction between the circle and each side of the triangle.
1. **Intersection of a Circle and a Line Segment**:
A circle can intersect a line segment at most at two points. This occurs whe... | 0.9375 | 3,597 | 3,290.666667 | 8,192 |
Given the functions $f(x)=(x-2)e^{x}$ and $g(x)=kx^{3}-x-2$,
(1) Find the range of $k$ such that the function $g(x)$ is not monotonic in the interval $(1,2)$;
(2) Find the maximum value of $k$ such that the inequality $f(x)\geqslant g(x)$ always holds when $x\in[0,+\infty)$. | \frac{1}{6} | 0 | 8,192 | -1 | 8,192 | |
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows a rule: at each vertex, it randomly chooses one of the three edges with equal probability and crawls to the opposite vertex. Define $q = \frac m{2187}$ as the probability that the... | 547 | 0.75 | 6,523.8125 | 5,967.75 | 8,192 | |
Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. | 100800 | Since there are 9 cow and pigs combined and 7 horses, there must be a pair with 1 cow and 1 pig, and all the other pairs must contain a horse. There are $4 \times 5$ ways of selecting the cow-pig pair, and 7 ! ways to select the partners for the horses. It follows that the answer is $4 \times 5 \times 7!=100800$. | 0 | 8,100.4375 | -1 | 8,100.4375 |
Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$.
Point $X$ is the midpoint of $AD$ and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are jo... | 12 | 0 | 8,150.875 | -1 | 8,150.875 | |
There exist integers $a,$ $b,$ and $c$ such that
\[(x - a)(x - 10) + 1 = (x + b)(x + c).\]Enter all possible values of $a,$ separated by commas. | 8,12 | 0 | 5,597.5625 | -1 | 5,597.5625 | |
One corner of a cube is cut off, creating a new triangular face. How many edges does this new solid have? | 15 | 0.25 | 7,390.375 | 6,384.5 | 7,725.666667 | |
Penta chooses 5 of the vertices of a unit cube. What is the maximum possible volume of the figure whose vertices are the 5 chosen points? | \frac{1}{2} | Label the vertices of the cube $A, B, C, D, E, F, G, H$, such that $A B C D$ is the top face of the cube, $E$ is directly below $A, F$ is directly below $B, G$ is directly below $C$, and $H$ is directly below $D$. We can obtain a volume of $\frac{1}{2}$ by taking the vertices $A, B, C, F$, and $H$. To compute the volum... | 0 | 8,191.6875 | -1 | 8,191.6875 |
Given that $\sqrt{51.11}\approx 7.149$ and $\sqrt{511.1}\approx 22.608$, determine the value of $\sqrt{511100}$. | 714.9 | 0.25 | 7,990.6875 | 7,532 | 8,143.583333 | |
What is the remainder when $13^{51}$ is divided by 5? | 2 | 1 | 2,343.375 | 2,343.375 | -1 | |
In the tetrahedron \( ABCD \), \( AC = 8 \), \( AB = CD = 7 \), \( BC = AD = 5 \), and \( BD = 6 \). Given a point \( P \) on \( AC \), find the minimum value of \( BP + PD \). | 2\sqrt{21} | 0.125 | 8,119.9375 | 7,615.5 | 8,192 | |
Let \( ABC \) be a triangle such that \( AB = 2 \), \( CA = 3 \), and \( BC = 4 \). A semicircle with its diameter on \(\overline{BC}\) is tangent to \(\overline{AB}\) and \(\overline{AC}\). Compute the area of the semicircle. | \frac{27 \pi}{40} | 0.3125 | 5,689.25 | 4,972.8 | 6,014.909091 | |
10 students (one of whom is the captain, and 9 are team members) formed a team to participate in a math competition and won first prize. The organizing committee decided to award each team member 200 yuan as a prize. The captain received 90 yuan more than the average prize of all 10 team members. How much prize money d... | 300 | 0.3125 | 2,236.5625 | 2,715.2 | 2,019 | |
Two different numbers are randomly selected from the set $\{-3, -2, 0, 0, 5, 6, 7\}$. What is the probability that the product of these two numbers is $0$?
**A)** $\frac{1}{4}$
**B)** $\frac{1}{5}$
**C)** $\frac{5}{21}$
**D)** $\frac{1}{3}$
**E)** $\frac{1}{2}$ | \frac{5}{21} | 0 | 8,147.75 | -1 | 8,147.75 | |
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. | 3 \pi | 0.8125 | 5,231.25 | 4,548 | 8,192 | |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$? | \frac{13}{27} | 1. **Understanding the Problem**: We need to find the probability that the expression $abc + ab + a$ is divisible by $3$ when $a$, $b$, and $c$ are chosen randomly from the set $\{1, 2, 3, \dots, 2010\}$.
2. **Divisibility Analysis**: We can rewrite the expression as $a(bc + b + 1)$. For this expression to be divisibl... | 0.6875 | 5,827 | 5,067.909091 | 7,497 |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $19 | 1. **Identify the total number of days and total cost**: Al takes the pills for two weeks, which is 14 days. The total cost for these two weeks is $\$546$.
2. **Calculate daily expenditure**:
\[
\text{Daily cost} = \frac{\text{Total cost}}{\text{Number of days}} = \frac{546}{14} = 39 \text{ dollars}
\]
3. *... | 0 | 1,403.3125 | -1 | 1,403.3125 |
A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period? | 43 | 0.125 | 6,477.3125 | 4,804.5 | 6,716.285714 | |
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing a number by 3. How can one obtain the number 11 from the number 1 using this calculator? | 11 | 0.125 | 8,035.875 | 6,943 | 8,192 | |
Let \( x \) and \( y \) be positive numbers, and let \( s \) be the smallest of the numbers \( x \), \( y + \frac{1}{x} \), and \( \frac{1}{y} \). Find the maximum possible value of \( s \). For which values of \( x \) and \( y \) is it achieved? | \sqrt{2} | 1 | 5,011.75 | 5,011.75 | -1 | |
Given the rectangular coordinate system xOy, establish a polar coordinate system with O as the pole and the non-negative semi-axis of the x-axis as the polar axis. The line l passes through point P(-1, 2) with an inclination angle of $\frac{2π}{3}$, and the polar coordinate equation of circle C is $ρ = 2\cos(θ + \frac{... | 6 + 2\sqrt{3} | 0.625 | 6,768.9375 | 5,915.1 | 8,192 | |
\(ABCD\) is a square with sides \(8 \, \text{cm}\). \(M\) is a point on \(CB\) such that \(CM = 2 \, \text{cm}\). If \(N\) is a variable point on the diagonal \(DB\), find the least value of \(CN + MN\). | 10 | 0.5625 | 6,380.4375 | 5,589.555556 | 7,397.285714 | |
What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$? | 46656 | 0 | 5,625.8125 | -1 | 5,625.8125 | |
Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way? | 16 | 0 | 7,761.1875 | -1 | 7,761.1875 | |
There are four people in a room. For every two people, there is a $50 \%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the probability that every pair of people in this ro... | \frac{19}{32} | We label the four people in the room $A, B, C$, and $D$. We represent each person by a point. There are six possible pairs of friends: $AB, AC, AD, BC, BD$, and $CD$. We represent a friendship by joining the corresponding pair of points and a non-friendship by not joining the pair of points. Since each pair of points i... | 0.125 | 7,997.125 | 6,633 | 8,192 |
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? | 27,720 | 1. **Define normalized variables**: Let $A = \frac{a}{77}$, $B = \frac{b}{77}$, $C = \frac{c}{77}$, and $D = \frac{d}{77}$. This transformation ensures that $\gcd(A, B, C, D) = 1$. The problem then translates to finding the least common multiple of $A, B, C, D$ multiplied by $77$ to give $n$, i.e., $\operatorname{lcm}(... | 0 | 8,192 | -1 | 8,192 |
How many values of $x$, $-30<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1$? | 48 | 0.875 | 3,953.375 | 3,668.928571 | 5,944.5 | |
Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is | 1150 |
#### Method 1: Using the Principle of Inclusion-Exclusion
We are given:
- Total plants in Bed A, $|A| = 500$
- Total plants in Bed B, $|B| = 450$
- Total plants in Bed C, $|C| = 350$
- Plants shared between Bed A and Bed B, $|A \cap B| = 50$
- Plants shared between Bed A and Bed C, $|A \cap C| = 100$
We need to find ... | 1 | 3,075.9375 | 3,075.9375 | -1 |
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\] | (1, 1, 3, 1) |
To solve the problem of determining all quadruples \((x, y, z, t)\) of positive integers such that:
\[
20^x + 14^{2y} = (x + 2y + z)^{zt},
\]
we begin by analyzing the constraints in the given equation with small values for \(x\), \(y\), \(z\), and \(t\). Our task is to find values that satisfy the equation when sub... | 0 | 8,192 | -1 | 8,192 |
A three-digit natural number with digits in the hundreds, tens, and units places denoted as $a$, $b$, $c$ is called a "concave number" if and only if $a > b$, $b < c$, such as $213$. If $a$, $b$, $c \in \{1,2,3,4\}$, and $a$, $b$, $c$ are all different, then the probability of this three-digit number being a "concave n... | \frac{1}{3} | 0.8125 | 5,699.375 | 5,124.153846 | 8,192 | |
Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data falls in [31.5, 43.5). | \frac{1}{3} | 0.8125 | 2,808.25 | 2,656.384615 | 3,466.333333 | |
Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.
Generalize and solve the problem for a number system w... | 6210001000 | 0 | 8,192 | -1 | 8,192 | |
For the one-variable quadratic equation $x^{2}+3x+m=0$ with two real roots for $x$, determine the range of values for $m$. | \frac{9}{4} | 0 | 2,125.125 | -1 | 2,125.125 | |
If $x, y$, and $z$ are real numbers such that $2 x^{2}+y^{2}+z^{2}=2 x-4 y+2 x z-5$, find the maximum possible value of $x-y+z$. | 4 | The equation rearranges as $(x-1)^{2}+(y+2)^{2}+(x-z)^{2}=0$, so we must have $x=1$, $y=-2, z=1$, giving us 4 . | 0.1875 | 7,890.0625 | 7,345.666667 | 8,015.692308 |
Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahedron? | 15 | 0 | 7,974.625 | -1 | 7,974.625 | |
Given two plane vectors, the angle between them is $120^\circ$, and $a=1$, $|b|=2$. If the plane vector $m$ satisfies $m\cdot a=m\cdot b=1$, then $|m|=$ ______. | \frac{ \sqrt{21}}{3} | 0 | 5,380.9375 | -1 | 5,380.9375 | |
Find the smallest natural number ending in the digit 2 such that it doubles when this digit is moved to the beginning. | 105263157894736842 | 0.25 | 7,608.25 | 5,857 | 8,192 | |
Given that player A needs to win 2 more games and player B needs to win 3 more games, and the probability of winning each game for both players is $\dfrac{1}{2}$, calculate the probability of player A ultimately winning. | \dfrac{11}{16} | 0.375 | 7,203.625 | 5,556.333333 | 8,192 | |
In the rectangular coordinate system xOy, it is known that 0 < α < 2π. Point P, with coordinates $(1 - \tan{\frac{\pi}{12}}, 1 + \tan{\frac{\pi}{12}})$, lies on the terminal side of angle α. Determine the value of α. | \frac{\pi}{3} | 0.9375 | 4,497.5625 | 4,251.266667 | 8,192 | |
In triangle $DEF$, points $D'$, $E'$, and $F'$ are on the sides $EF$, $FD$, and $DE$, respectively. Given that $DD'$, $EE'$, and $FF'$ are concurrent at the point $P$, and that $\frac{DP}{PD'}+\frac{EP}{PE'}+\frac{FP}{PF'}=94$, find $\frac{DP}{PD'}\cdot \frac{EP}{PE'}\cdot \frac{FP}{PF'}$. | 92 | 0.125 | 6,152.125 | 4,189.5 | 6,432.5 | |
Four friends went fishing one day and caught a total of 11 fish. Each person caught at least one fish. Which statement must be true: (A) At least one person caught exactly one fish. (B) At least one person caught exactly three fish. (C) At least one person caught more than three fish. (D) At least one person caught few... | D | Choice (A) is not necessarily true, since the four friends could have caught 2,3, 3, and 3 fish. Choice (B) is not necessarily true, since the four friends could have caught 1, 1, 1, and 8 fish. Choice (C) is not necessarily true, since the four friends could have caught 2, 3, 3, and 3 fish. Choice (E) is not necessari... | 0.5 | 4,659.0625 | 4,508.875 | 4,809.25 |
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this?
A) 12
B) 24
C) 48
D) 96
E) 198 | 24 | 0 | 8,192 | -1 | 8,192 | |
Mark has 75% more pencils than John, and Luke has 50% more pencils than John. Find the percentage relationship between the number of pencils that Mark and Luke have. | 16.67\% | 0.125 | 2,067.4375 | 1,332.5 | 2,172.428571 | |
What is the base $2$ representation of $125_{10}$? | 1111101_2 | 0.9375 | 3,567.5 | 3,663.066667 | 2,134 | |
A regular hexagon's center and vertices together make 7 points. Calculate the number of triangles that can be formed using any 3 of these points as vertices. | 32 | 0.625 | 5,855.1875 | 4,453.1 | 8,192 | |
Given a school library with four types of books: A, B, C, and D, and a student limit of borrowing at most 3 books, determine the minimum number of students $m$ such that there must be at least two students who have borrowed the same type and number of books. | 15 | 0.4375 | 5,227.375 | 3,933.714286 | 6,233.555556 |
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