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 points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \_\_\_\_\_\_. | 5\sqrt{13} | 0.8125 | 5,710.125 | 5,137.384615 | 8,192 | |
In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can tak... | 234 | 0.125 | 7,693.625 | 5,451 | 8,014 | |
In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces, there are a total of ▲ different parking ... | 840 | 0 | 8,165.5 | -1 | 8,165.5 | |
A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, if two bulbs are lit, whether they are in t... | 44 | 0 | 8,026.5 | -1 | 8,026.5 | |
I ponder some numbers in bed, all products of three primes I've said, apply $\phi$ they're still fun: $$n=37^{2} \cdot 3 \ldots \phi(n)= 11^{3}+1 ?$$ now Elev'n cubed plus one. What numbers could be in my head? | 2007, 2738, 3122 | The numbers expressible as a product of three primes are each of the form $p^{3}, p^{2} q$, or $p q r$, where $p, q$, and $r$ are distinct primes. Now, $\phi\left(p^{3}\right)=p^{2}(p-1), \phi\left(p^{2} q\right)=$ $p(p-1)(q-1)$, and $\phi(p q r)=(p-1)(q-1)(r-1)$. We require $11^{3}+1=12 \cdot 111=2^{2} 3^{2} 37$. The ... | 0 | 8,114.0625 | -1 | 8,114.0625 |
Given that $\cos α=-\dfrac{4}{5}\left(\dfrac{π}{2}<α<π\right)$, find $\cos\left(\dfrac{π}{6}-α\right)$ and $\cos\left(\dfrac{π}{6}+α\right)$. | -\dfrac{3+4\sqrt{3}}{10} | 0 | 5,053.9375 | -1 | 5,053.9375 | |
Let $x$, $y$, $z$, $u$, and $v$ be positive integers with $x+y+z+u+v=2505$. Let $N$ be the largest of the sums $x+y$, $y+z$, $z+u$, and $u+v$. Determine the smallest possible value of $N$. | 1253 | 0 | 7,554.5625 | -1 | 7,554.5625 | |
Determine the number of four-digit integers $n$ such that $n$ and $2n$ are both palindromes. | 20 | Let $n=\underline{a} \underline{b} \underline{b} \underline{a}$. If $a, b \leq 4$ then there are no carries in the multiplication $n \times 2$, and $2n=(2a)(2b)(2b)(2a)$ is a palindrome. We shall show conversely that if $n$ and $2n$ are palindromes, then necessarily $a, b \leq 4$. Hence the answer to the problem is $4 ... | 0.125 | 7,845.1875 | 5,957.5 | 8,114.857143 |
Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute $$\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)}$$ | 45 | Notice that if $n<8,(-1)^{s(n)}=(-1) \cdot(-1)^{s(n+8)}$ so the sum becomes $\frac{1}{255}\left(1-2^{8}\right) \sum_{0 \leq n<8} 2^{n}(-1)^{s(n)}=$ 45 . | 0.25 | 7,677.8125 | 6,135.25 | 8,192 |
Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$? | 4\pi | 0.6875 | 6,046.875 | 5,071.818182 | 8,192 | |
In a triangle with sides of lengths $a,$ $b,$ and $c,$
\[(a + b + c)(a + b - c) = 3ab.\]Find the angle opposite the side of length $c,$ in degrees. | 60^\circ | 1 | 1,522.6875 | 1,522.6875 | -1 | |
Given the sequence $\{a_n\}$ is a non-zero arithmetic sequence, $S_n$ denotes the sum of the first $n$ terms, and $S_{2n-1} = a_n^2$ for any $n \in \mathbb{N^*}$. If the inequality $\dfrac{1}{a_1a_2} + \dfrac{1}{a_2a_3} + \ldots + \dfrac{1}{a_na_{n+1}} \leqslant n\log_{\frac{1}{8}}\lambda$ holds for any $n \in \mathbb{... | \frac{1}{2} | 0.6875 | 5,867.5 | 5,549.545455 | 6,567 | |
Given the function $g(x) = \frac{6x^2 + 11x + 17}{7(2 + x)}$, find the minimum value of $g(x)$ for $x \ge 0$. | \frac{127}{24} | 0 | 6,839.0625 | -1 | 6,839.0625 | |
A grocery store manager decides to design a more compact pyramid-like stack of apples with a rectangular base of 4 apples by 6 apples. Each apple above the first level still rests in a pocket formed by four apples below, and the stack is completed with a double row of apples on top. Determine the total number of apples... | 53 | 0 | 6,303.625 | -1 | 6,303.625 | |
Let point P be the intersection point in the first quadrant of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. F\1 and F\2 are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Find the eccentricity of the hyperbola. | \frac{\sqrt{10}}{2} | 0 | 6,495 | -1 | 6,495 | |
Randomly select a number $x$ in the interval $[-1, 1]$. The probability that the value of $\cos \frac{\pi x}{2}$ falls between 0 and $\frac{1}{2}$ is ______. | \frac{1}{3} | 1 | 5,014 | 5,014 | -1 | |
Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance. | 44 | 0 | 6,225.3125 | -1 | 6,225.3125 | |
On the base \(AC\) of an isosceles triangle \(ABC\), a point \(E\) is taken, and on the sides \(AB\) and \(BC\), points \(K\) and \(M\) are taken such that \(KE \parallel BC\) and \(EM \parallel AB\). What fraction of the area of triangle \(\mathrm{ABC}\) is occupied by the area of triangle \(KEM\) if \(BM:EM = 2:3\)? | 6/25 | 0.375 | 7,241.25 | 6,189 | 7,872.6 | |
Point \( M \) belongs to the edge \( CD \) of the parallelepiped \( ABCDA_1B_1C_1D_1 \), where \( CM: MD = 1:2 \). Construct the section of the parallelepiped with a plane passing through point \( M \) parallel to the lines \( DB \) and \( AC_1 \). In what ratio does this plane divide the diagonal \( A_1C \) of the par... | 1 : 11 | 0 | 7,612.9375 | -1 | 7,612.9375 | |
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \). | 2/3 | 0 | 8,066.875 | -1 | 8,066.875 | |
An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? | 192 | 0 | 6,101.125 | -1 | 6,101.125 | |
Find all the values of $m$ for which the zeros of $2 x^{2}-m x-8$ differ by $m-1$. | 6,-\frac{10}{3} | 6,-\frac{10}{3}. | 0 | 3,900 | -1 | 3,900 |
There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even? | \frac{5}{9} | 0.1875 | 1,595.6875 | 634 | 1,817.615385 | |
Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:
\[
\frac{1}{b^2 + c^2 - 3a^2} + \frac{1}{a^2 + c^2 - 3b^2} + \frac{1}{a^2 + b^2 - 3c^2}.
\] | -3 | 0 | 8,192 | -1 | 8,192 | |
Given the equation $x^{2}+4ax+3a+1=0 (a > 1)$, whose two roots are $\tan \alpha$ and $\tan \beta$, with $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$, find $\tan \frac{\alpha + \beta}{2}$. | -2 | 0.3125 | 7,285.0625 | 5,289.8 | 8,192 | |
The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \_\_\_\_\_\_. | \frac{128\pi}{3} | 0 | 7,550.5 | -1 | 7,550.5 | |
In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$. | \frac{15}{17} | 1 | 1,853.625 | 1,853.625 | -1 | |
Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\sin \angle FAP$. | \frac{\sqrt{2}}{2} | 0 | 5,934.4375 | -1 | 5,934.4375 | |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$. | 480 | $y\le2x\le60$
Multiplying both sides by 2 yields:
$2y\le4x\le120$
Then the two inequalities can be merged to form the following inequality:
$x\le2y\le4x\le120$
Additionally, we must ensure that $2y<60$
Therefore we must find pairs $(x,y)$ that satisfy the inequality above. A bit of trial and error and observing pat... | 0.0625 | 8,078 | 7,429 | 8,121.266667 |
The product of $7d^2-3d+g$ and $3d^2+hd-8$ is $21d^4-44d^3-35d^2+14d-16$. What is $g+h$? | -3 | 0.9375 | 3,017.1875 | 2,672.2 | 8,192 | |
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$? | 6 | 1. **Divisibility by 15**: A number is divisible by 15 if and only if it is divisible by both 3 and 5.
2. **Divisibility by 5**: A number is divisible by 5 if it ends in 0 or 5. Since we are considering positive integers and uphill integers cannot end in 0 (as 0 cannot be preceded by any positive digit in an uphill i... | 0.3125 | 7,158.125 | 5,129.2 | 8,080.363636 |
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited. | \frac{1}{3} | 0 | 8,192 | -1 | 8,192 | |
The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a "wealth number." Among the numbers from 1 to 100, how many "wealth numbers" are there? | 88 | 0 | 7,765.625 | -1 | 7,765.625 | |
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
How many different paths can the particle take from $(0,0)$ to $(5,5)$?
| 83 | 0 | 8,192 | -1 | 8,192 | |
The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving: | 4x^2-17x+4 = 0 | 1. **Substitute and Simplify**: Let $t = \sqrt{x}$. Then, $t^2 = x$ and $x^{-\frac{1}{2}} = \frac{1}{t}$. Substitute these into the given equation:
\[
2\sqrt{x} + 2x^{-\frac{1}{2}} = 5 \implies 2t + \frac{2}{t} = 5.
\]
2. **Form a Quadratic Equation**: Multiply through by $t$ to clear the fraction:
\[
2... | 0 | 2,146.5 | -1 | 2,146.5 |
Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}}.$$ | -\frac{9}{10} | 0.8125 | 4,938.25 | 4,497.230769 | 6,849.333333 | |
Consider the function $f(x) = x^2 +2\sqrt{x}$. Evaluate $2f(2) - f(8)$. | -56 | 1 | 2,025.8125 | 2,025.8125 | -1 | |
A hiker is exploring a trail. The trail has three sections: the first $25 \%$ of the trail is along a river, the next $\frac{5}{8}$ of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail? | 24 \text{ km} | Since $25 \%$ is equivalent to $\frac{1}{4}$, then the fraction of the trail covered by the section along the river and the section through the forest is $\frac{1}{4}+\frac{5}{8}=\frac{2}{8}+\frac{5}{8}=\frac{7}{8}$. This means that the final section up a hill represents $1-\frac{7}{8}=\frac{1}{8}$ of the trail. Since ... | 0.8125 | 735.25 | 741.692308 | 707.333333 |
Sara conducted a survey among a group to find out how many people were aware that bats can transmit diseases. She found out that $75.3\%$ believed bats could transmit diseases. Among those who believed this, $60.2\%$ incorrectly thought that all bats transmit Zika virus, which amounted to 37 people. Determine how many ... | 81 | 0.0625 | 5,058.0625 | 7,489 | 4,896 | |
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. | 27 | 0.1875 | 6,846.75 | 6,443.666667 | 6,939.769231 | |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs? | 46 | 0 | 8,192 | -1 | 8,192 | |
The projection of $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$ onto a certain vector $\mathbf{w}$ is $\begin{pmatrix} -9/10 \\ 3/10 \end{pmatrix}.$ Find the projection of $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ onto $\mathbf{w}.$ | \begin{pmatrix} 33/10 \\ -11/10 \end{pmatrix} | 0 | 5,351.375 | -1 | 5,351.375 | |
How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers? | 49 | 0.0625 | 7,927.6875 | 7,399 | 7,962.933333 | |
Given the function $f(x) = \log_{m}(m - x)$, if the maximum value in the interval $[3, 5]$ is 1 greater than the minimum value, determine the real number $m$. | 3 + \sqrt{6} | 0.8125 | 4,986.125 | 4,246.307692 | 8,192 | |
Calculate \(\int_{0}^{1} e^{-x^{2}} \, dx\) to an accuracy of 0.001. | 0.747 | 0.25 | 7,397.25 | 6,412.75 | 7,725.416667 | |
If 8 is added to the square of 5, the result is divisible by: | 11 | 0.25 | 5,276.1875 | 3,241 | 5,954.583333 | |
Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \perp x$-axis, find the eccentricity of the hyperbola. | \sqrt{5} | 0.5 | 6,237.9375 | 4,428 | 8,047.875 | |
What is the remainder when $5^{137}$ is divided by 8? | 5 | 1 | 2,248.1875 | 2,248.1875 | -1 | |
Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of 5 or 7 or both? Express your answer as a common fraction. | \frac{8}{25} | 1 | 1,839.9375 | 1,839.9375 | -1 | |
Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.
A) $54 + 18\sqrt{2} - 9\pi$
B) $54 + 36\sqrt{2} - 9\pi$... | 54 + 36\sqrt{2} - 9\pi | 0 | 8,192 | -1 | 8,192 | |
When every vector on the line $y = \frac{5}{2} x + 4$ is projected onto a certain vector $\mathbf{w},$ the result is always the vector $\mathbf{p}.$ Find the vector $\mathbf{p}.$ | \begin{pmatrix} -40/29 \\ 16/29 \end{pmatrix} | 0 | 7,281.625 | -1 | 7,281.625 | |
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way? | 1 | 1. **Identify the Requirement**: Danica wants to arrange her model cars in rows with exactly 6 cars in each row. This means that the total number of cars must be a multiple of 6.
2. **Current Number of Cars**: She currently has 23 model cars.
3. **Finding the Next Multiple of 6**: We need to find the smallest multipl... | 1 | 3,306.6875 | 3,306.6875 | -1 |
If the product $\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{a}{b} = 9$, what is the sum of $a$ and $b$? | 37 | 1. **Identify the pattern in the product sequence**:
The given product sequence is $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b}$. Notice that each fraction $\dfrac{n+1}{n}$ cancels out the $n$ in the denominator of the previous fraction, leaving only the numerato... | 0 | 2,406.375 | -1 | 2,406.375 |
Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? | 075 | 0.1875 | 8,040.875 | 7,802.333333 | 8,095.923077 | |
Given regular hexagon $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$ , where $P$ is the midpoint of $AB$ , $Q$ is the midpoint of $CD$ , and $R$ is the midpoint of $EF$ . | \frac{3}{8} | 0.9375 | 5,385.375 | 5,198.266667 | 8,192 | |
A research study group is investigating the traffic volume at a certain intersection near the school during the rush hour from 8:00 to 10:00. After long-term observation and statistics, they have established a simple function model between traffic volume and average vehicle speed. The model is as follows: Let the traff... | \frac{25}{3} | 0.875 | 3,811 | 3,840.571429 | 3,604 | |
How many ways are there for Nick to travel from $(0,0)$ to $(16,16)$ in the coordinate plane by moving one unit in the positive $x$ or $y$ direction at a time, such that Nick changes direction an odd number of times? | 2 \cdot\binom{30}{15} = 310235040 | This condition is equivalent to the first and last step being in different directions, as if you switch directions an odd number of times, you must end in a different direction than you started. If the first step is in the $x$ direction and the last step is in the $y$ direction, it suffices to count the number of paths... | 0 | 8,192 | -1 | 8,192 |
Let $f(x) = 4x + c$ and $g(x) = cx + 2.$ If $f(g(x)) = 12x + d,$ then find $d.$ | 11 | 1 | 1,547.625 | 1,547.625 | -1 | |
Evaluate: $6 - 8\left(9 - 4^2\right) \div 2 - 3.$ | 31 | 1 | 2,557.5 | 2,557.5 | -1 | |
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$? | $2$ | Given the determinant formula for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as:
\[ \text{det} = ad - bc \]
For the specific matrix $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$, we identify the elements as:
- $a = 3$
- $b = 4$
- $c = 1$
- $d = 2$
Using the de... | 0 | 664.75 | -1 | 664.75 |
In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.
(1) Find the equation of the ellipse $(C)$;
(2) If points $A$ and $B$ both... | \frac{3 \sqrt{2}}{2} | 0 | 7,486.0625 | -1 | 7,486.0625 | |
Solve in integers the system of equations: $$x^2-y^2=z$$
$$3xy+(x-y)z=z^2$$ | {(2, 1, 3), (1, 2, -3), (1, 0, 1), (0, 1, -1), (0, 0, 0)} |
To solve the system of equations in integers:
\[
x^2 - y^2 = z
\]
\[
3xy + (x-y)z = z^2
\]
we begin by analyzing the first equation. The expression \( x^2 - y^2 = z \) can be factored as:
\[
(x+y)(x-y) = z
\]
Substituting this into the second equation, we have:
\[
3xy + (x-y)z = z^2
\]
Given the expression for \( z \... | 0 | 7,143.75 | -1 | 7,143.75 |
If the odds for pulling a prize out of the box are $3:4$, what is the probability of not pulling the prize out of the box? Express your answer as a common fraction. | \frac{4}{7} | 1 | 1,321.875 | 1,321.875 | -1 | |
A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \(\frac{1}{4}\) inch. | 2\% | 0.3125 | 7,297.125 | 6,450.2 | 7,682.090909 | |
In a store, we paid with a 1000 forint bill. On the receipt, the amount to be paid and the change were composed of the same digits but in a different order. What is the sum of the digits? | 14 | 0 | 8,050.8125 | -1 | 8,050.8125 | |
Given that $\{a\_n\}$ is a geometric sequence, $a\_2=2$, $a\_6=162$, find $a\_{10}$ = $\_\_\_\_\_\_$ . | 13122 | 1 | 2,389.125 | 2,389.125 | -1 | |
In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD. | \frac{3}{32} | 0 | 6,365.25 | -1 | 6,365.25 | |
How many cubic centimeters are in the volume of a cone having a diameter of 10cm and a height of 6cm? Express your answer in terms of $\pi$. | 50 \pi | 1 | 1,206.5 | 1,206.5 | -1 | |
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$. | 17.28125 | 0.4375 | 439.875 | 512.142857 | 383.666667 | |
A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from... | 927 | To simplify matters, we want a power of $2$. Hence, we will add $48$ 'fake' cards which we must discard in our actual count. Using similar logic as Solution 1, we find that 1999 has position $1024$ in a $2048$ card stack, where the fake cards towards the front.
Let the fake cards have positions $1, 3, 5, \cdots, 95$. ... | 0 | 8,192 | -1 | 8,192 |
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 vot... | \frac{1513}{2017} | Let $P_{n}(m)$ be the probability that after $n$ voters have voted, Alice gets $m$ votes. We show by induction that for $n \geq 3$, the ratio $P_{n}(2): P_{n}(3): \cdots: P_{n}(n-1)$ is equal to $1: 2: \cdots:(n-2)$. We take a base case of $n=3$, for which the claim is obvious. Then suppose the claim holds for $n=k$. T... | 0 | 7,864.8125 | -1 | 7,864.8125 |
A basketball championship has been played in a round-robin format, with each pair of teams playing twice and no ties (overtime is played until one team wins). The winner of a match receives 2 points, and the loser receives 1 point. At the end of the championship, the sum of the points obtained by all the teams except t... | 39 | 0.5625 | 6,076.5 | 4,431.111111 | 8,192 | |
How many even numbers are greater than 202 and less than 405? | 101 | 1 | 3,461.9375 | 3,461.9375 | -1 | |
Let the function \( f(x) \) be defined on \( \mathbb{R} \), and for any \( x \), the condition \( f(x+2) + f(x) = x \) holds. It is also known that \( f(x) = x^3 \) on the interval \( (-2, 0] \). Find \( f(2012) \). | 1006 | 0 | 8,192 | -1 | 8,192 | |
A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe. | 1/4 | 0 | 6,871.375 | -1 | 6,871.375 | |
There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them? | 41/56 | 0 | 8,004.3125 | -1 | 8,004.3125 | |
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the int... | 1247 | 0.4375 | 7,599.5 | 6,837.714286 | 8,192 | |
Liu Yulei bought 4 packs of yogurt and 4 packs of fresh milk at the supermarket, paying a total of 14 yuan. Later, she returned 2 packs of yogurt and bought 4 more packs of fresh milk, and the cashier gave her 1 yuan back. The price of each pack of yogurt is ____ yuan. | 2.5 | 0.1875 | 822.8125 | 820.666667 | 823.307692 | |
For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$. | 3n |
For a given integer \( n \ge 2 \), let \( a_0, a_1, \ldots, a_n \) be integers satisfying \( 0 = a_0 < a_1 < \ldots < a_n = 2n-1 \). We aim to find the smallest possible number of elements in the set \( \{ a_i + a_j \mid 0 \le i \le j \le n \} \).
First, we prove that the set \( \{ a_i + a_j \mid 1 \le i \le j \le n-... | 0.125 | 8,140.3125 | 7,778.5 | 8,192 |
If a positive four-digit number's thousand digit \\(a\\), hundred digit \\(b\\), ten digit \\(c\\), and unit digit \\(d\\) satisfy the relation \\((a-b)(c-d) < 0\\), then it is called a "Rainbow Four-Digit Number", for example, \\(2012\\) is a "Rainbow Four-Digit Number". How many "Rainbow Four-Digit Numbers" are there... | 3645 | 0.125 | 7,610.0625 | 5,864.5 | 7,859.428571 | |
What is the maximum number of kings that can be placed on a chessboard such that no two of them attack each other? | 16 | 0.5 | 7,017.625 | 6,066.875 | 7,968.375 | |
In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ$ and $\angle BED = \angle BDE$, then $\angle CDE =$ | 95^\circ | 1. **Identify the properties of triangle BED**: Given that $\angle BED = \angle BDE$ and $\angle B = 90^\circ$, triangle BED is an isosceles right triangle. By the triangle angle sum property, the sum of angles in any triangle is $180^\circ$. Therefore, we can set up the equation:
\[
\angle B + \angle BED + \angl... | 0 | 8,192 | -1 | 8,192 |
The polynomial $ax^4 + bx^3 + 32x^2 - 16x + 6$ has a factor of $3x^2 - 2x + 1.$ Find the ordered pair $(a,b).$ | (18,-24) | 0.9375 | 3,678.75 | 3,377.866667 | 8,192 | |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$? | 4625 | 0.125 | 8,027.75 | 6,878 | 8,192 | |
Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[\left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1.\] | 8 | 0.75 | 4,865.0625 | 4,580.25 | 5,719.5 | |
What is the ratio of the area of the shaded triangle to the area of the square? The square is divided into a 5x5 grid of smaller, equal-sized squares. A triangle is shaded such that it covers half of a square at the center of the grid and three full squares adjacent to this half-covered square. The vertices of the tria... | \frac{7}{50} | 0.125 | 8,142.3125 | 7,794.5 | 8,192 | |
James borrows $2000$ dollars from Alice, who charges an interest of $3\%$ per month (which compounds monthly). What is the least integer number of months after which James will owe more than three times as much as he borrowed? | 37 | 0.0625 | 6,261.0625 | 6,760 | 6,227.8 | |
A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: | 245 yd. | 1. **Define the problem in terms of equations:**
- Let $p(t)$ be the distance in feet that the powderman is from the starting point at time $t$ seconds after the fuse is lit. Since he runs at $8$ yards per second, and there are $3$ feet in a yard, he runs $8 \times 3 = 24$ feet per second. Therefore, $p(t) = 24t$.
... | 0 | 7,362.0625 | -1 | 7,362.0625 |
Find all real values of $x$ that satisfy $x + \frac{45}{x-4} = -10.$
Enter all the solutions, separated by commas. | -1, \; -5 | 0 | 1,373.25 | -1 | 1,373.25 | |
The probability of getting rain on any given day in June in Capital City is $\frac{1}{10}$. What is the probability that it rains on at most 2 days in June? Express your answer as a decimal to the nearest thousandth. | 0.411 | 0.25 | 7,636.6875 | 7,277.25 | 7,756.5 | |
If $e^{i \alpha} = \frac{3}{5} +\frac{4}{5} i$ and $e^{i \beta} = -\frac{12}{13} + \frac{5}{13} i,$ then find $\sin (\alpha + \beta).$ | -\frac{33}{65} | 0.9375 | 3,631.4375 | 3,327.4 | 8,192 | |
If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: | 4 | 1. **Identify the original setup**: Let the angle be $\theta$ and the sides enclosing it be $a$ and $b$. The area of the triangle can be expressed using the formula for the area of a triangle given two sides and the included angle:
\[
A = \frac{a \cdot b \cdot \sin(\theta)}{2}
\]
2. **Modify the sides**: Each... | 0.8125 | 3,670.6875 | 2,627.307692 | 8,192 |
For a certain complex number $c$, the polynomial
\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]has exactly 4 distinct roots. What is $|c|$? | \sqrt{10} | 1. **Factorize the given polynomial**:
The polynomial \( P(x) \) is given as:
\[
P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)
\]
We start by factorizing each quadratic term using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
2. **Roots of \( x^2 - 2x + 2 \)**:
\[
x =... | 0.8125 | 4,639 | 4,016.923077 | 7,334.666667 |
In the isosceles right triangle \(ABC\) with \(\angle A = 90^\circ\) and \(AB = AC = 1\), a rectangle \(EHGF\) is inscribed such that \(G\) and \(H\) lie on the side \(BC\). Find the maximum area of the rectangle \(EHGF\). | 1/4 | 0.1875 | 7,873.375 | 6,492.666667 | 8,192 | |
What is the largest four-digit number whose digits add up to 23? | 9950 | 0.4375 | 7,558.5625 | 6,744.142857 | 8,192 | |
Compute $(4+8)^2+(4^2+8^2)$. | 224 | 1 | 2,301.625 | 2,301.625 | -1 | |
Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same. | 10 | 0 | 7,970.5 | -1 | 7,970.5 | |
Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______. | 4\sqrt{3}\pi | 0.375 | 6,851.0625 | 5,657.833333 | 7,567 | |
A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.
| 59 | 0.4375 | 6,907.75 | 5,256.571429 | 8,192 | |
Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is 2, compute the product $(AB)(CD).$ | 65 | 1 | 2,981.25 | 2,981.25 | -1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.