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 |
|---|---|---|---|---|---|---|
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? | 36 | 1. **Understanding the Problem Setup:**
- We have two squares, $ABCD$ and $EFGH$, where $EFGH$ is inside $ABCD$.
- Each side of $EFGH$ can be extended to pass through a vertex of $ABCD$.
- The side length of $ABCD$ is given as $\sqrt{50}$.
- The distance from vertex $B$ of square $ABCD$ to the nearest point... | 0 | 8,192 | -1 | 8,192 |
Given the sets \( M=\{x, xy, \lg(xy)\} \) and \( N=\{0, |x|, y\} \), and that \( M=N \), find the value of \( \left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right) \). | -2 | 0.125 | 6,577.875 | 4,798.5 | 6,832.071429 | |
A straight line $l$ passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to $l$ is one quarter of its minor axis length, calculate the eccentricity of the ellipse. | \dfrac{1}{2} | 0.0625 | 7,860.1875 | 6,547 | 7,947.733333 | |
Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | 36 | 0.375 | 6,696.375 | 5,817 | 7,224 | |
Given the function $f(x)=a\sin x - \sqrt{3}\cos x$, one of its graphs has an axis of symmetry at $x=-\frac{\pi}{6}$, and $f(x_1) - f(x_2) = -4$, calculate the minimum value of $|x_1+x_2|$. | \frac{2\pi}{3} | 0.5 | 7,074.3125 | 6,294.125 | 7,854.5 | |
In plane $\alpha$, there are four points, and in plane $\beta$, there are five points. From these nine points, the maximum number of planes that can be determined by any three points is ; the maximum number of tetrahedrons that can be determined by any four points is . (Answer with numbers) | 120 | 0.625 | 5,849.125 | 4,443.4 | 8,192 | |
There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways are distinguished by the sizes of the square and the triangle). | 16 | 1 | 4,115.375 | 4,115.375 | -1 | |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 0 | 8,083.5 | -1 | 8,083.5 | |
Consider the sequence defined by $a_k =\dfrac{1}{k^2+k}$ for $k\geq 1$. Given that $a_m+a_{m+1}+\cdots+a_{n-1}=\dfrac{1}{29}$, for positive integers $m$ and $n$ with $m<n$, find $m+n$. | 840 | Note that $a_1 + a_2 + \cdots + a_i = \dfrac{i}{i+1}$. This can be proven by induction. Thus, $\sum\limits_{i=m}^{n-1} a_i = \sum\limits_{i=1}^{n-1} a_i - \sum\limits_{i=1}^{m-1} a_i = \dfrac{n-1}{n} - \dfrac{m-1}{m} = \dfrac{n-m}{mn} = 1/29$. Cross-multiplying yields $29n - 29m - mn = 0$, and adding $29^2$ to both sid... | 0.9375 | 4,216.75 | 3,951.733333 | 8,192 |
The term containing \(x^7\) in the expansion of \((1 + 2x - x^2)^4\) arises when \(x\) is raised to the power of 3 in three factors and \(-x^2\) is raised to the power of 1 in one factor. | -8 | 0.0625 | 5,722.6875 | 2,203 | 5,957.333333 | |
The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ has the property that
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_6 = 60,$ then find $b_7.$ | 97 | 1 | 3,059.4375 | 3,059.4375 | -1 | |
Convert the binary number $10110100$ to a decimal number. | 180 | 0.875 | 1,319.6875 | 868.642857 | 4,477 | |
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are p... | 87 | 0.9375 | 3,801.0625 | 3,508.333333 | 8,192 | |
In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) ... | 1:2 | 0 | 8,190.6875 | -1 | 8,190.6875 | |
Let $O$ be the origin, the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ and the hyperbola $C_{2}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ have a common focus $F$. The line passing through $F$ and perpendicular to the $x$-axis intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$... | \frac{\sqrt{6} + \sqrt{2}}{2} | 0 | 8,111.5 | -1 | 8,111.5 | |
Given that $\theta$ is an angle in the second quadrant, and $\tan 2\theta = -2\sqrt{2}$.
(1) Find the value of $\tan \theta$.
(2) Calculate the value of $\frac {2\cos^{2} \frac {\theta}{2}-\sin\theta-\tan \frac {5\pi}{4}}{\sqrt {2}\sin(\theta + \frac {\pi}{4})}$. | 3 + 2\sqrt{2} | 0.9375 | 5,089.5625 | 4,882.733333 | 8,192 | |
Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\mathrm{th}}$ power. | 2^n - 1 |
Let \( n \geq 2 \) be a given integer. We are tasked with finding the greatest value of \( N \) such that there are infinitely many ways to select \( N \) consecutive integers where none of them has a divisor greater than 1 that is a perfect \( n^{\text{th}} \) power.
To solve this, consider the properties of divisor... | 0 | 7,631.4375 | -1 | 7,631.4375 |
$\triangle PQR$ is similar to $\triangle XYZ$. What is the number of centimeters in the length of $\overline{YZ}$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(10,-2)--(8,6)--cycle);
label("10cm",(6,3),NW);
label("7cm",(10.2,2.5),NE);
draw((15,0)--(23,-1.8)--(22,4.5)--cycle);
label("$P$",(... | 5.7 | 0.0625 | 7,218.375 | 3,985 | 7,433.933333 | |
Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 3 & 4 \\ 6 & 3 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\] | 3 - 2\sqrt{6} | 0.9375 | 2,814.875 | 2,456.4 | 8,192 | |
What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) are on the same line (regardless of the angle of inclination)? | 12 | 0.375 | 7,828.875 | 7,346.666667 | 8,118.2 | |
Given $f(x) = \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2) \end{cases}$, determine the value of $x$ if $f(x)=3$. | \sqrt{3} | 1 | 2,475.4375 | 2,475.4375 | -1 | |
Find the fifth term of the geometric sequence with first term $2$ and second term $\frac{1}{4}$. | \frac{1}{2048} | 1 | 1,747.3125 | 1,747.3125 | -1 | |
Determine $\sqrt[6]{1061520150601}$ without a calculator. | 101 | 0.6875 | 5,252.9375 | 4,042.363636 | 7,916.2 | |
What is the average of five-digit palindromic numbers? (A whole number is a palindrome if it reads the same backward as forward.) | 55000 | 0.375 | 6,961.1875 | 5,672.833333 | 7,734.2 | |
If six people decide to come to a basketball game, but three of them are only 2/5 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 5 people stayed the entire time? | \frac{44}{125} | 1 | 2,853.125 | 2,853.125 | -1 | |
Six students taking a test sit in a row of seats with aisles only on the two sides of the row. If they finish the test at random times, what is the probability that some student will have to pass by another student to get to an aisle? | \frac{43}{45} | The probability $p$ that no student will have to pass by another student to get to an aisle is the probability that the first student to leave is one of the students on the end, the next student to leave is on one of the ends of the remaining students, etc.: $p=\frac{2}{6} \cdot \frac{2}{5} \cdot \frac{2}{4} \cdot \fra... | 0 | 7,781.25 | -1 | 7,781.25 |
Given the function g(n) = log<sub>27</sub>n if log<sub>27</sub>n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n). | 12 | 0.8125 | 4,811.6875 | 4,031.615385 | 8,192 | |
A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$. | -6 | 0 | 4,932.25 | -1 | 4,932.25 | |
The sum of the positive divisors of a positive integer of the form $2^i3^j$ is equal to $600$. What is $i + j$? | 6 | 0.9375 | 4,767.9375 | 4,539.666667 | 8,192 | |
Find $53\cdot\left(3\frac{1}{5} - 4\frac{1}{2}\right) \div \left(2\frac{3}{4} + 1\frac{2}{3} \right)$. Express your answer as a mixed number. | -15\frac{3}{5} | 1 | 4,121.625 | 4,121.625 | -1 | |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - 2xy\]for all $x,$ $y.$ Find the sum of all possible values of $g(1).$ | -\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope. | 10 | 0.3125 | 3,947.875 | 3,120.2 | 4,324.090909 | |
The Euler family consists of four girls, each aged 8 years, and three boys, one aged 10 and twins both aged 12. Calculate the mean age of the children after one year passes. | 10.43 | 0.3125 | 607.125 | 637.2 | 593.454545 | |
Zhang Bing was born in 1953. In a certain year before this year, his age was a multiple of 9 and equal to the sum of the digits of that year. How old was he that year? | 18 | 0.6875 | 5,634.1875 | 4,471.545455 | 8,192 | |
Given \( a > b \), the quadratic inequality \( ax^{2}+2x+b \geqslant 0 \) holds for all real numbers \( x \), and there exists \( x_{0} \in \mathbb{R} \) such that \( ax_{0}^{2}+2x_{0}+b=0 \) is satisfied. Find the minimum value of \( 2a^{2}+b^{2} \). | 2\sqrt{2} | 0.1875 | 7,446.1875 | 6,159.333333 | 7,743.153846 | |
Someone bought 5 consecutive train ticket numbers, and the sum of these 5 ticket numbers is 120. What is the product of these 5 ticket numbers? | 7893600 | 0 | 1,643.875 | -1 | 1,643.875 | |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\begin{pmatrix} 1 \\ 3 \\ 8 \end{pmatrix},$ and the vector on the line at $t = 2$ is $\begin{pmatrix} 0 \\ -2 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = 3.$ | \begin{pmatrix} -1/3 \\ -11/3 \\ -8 \end{pmatrix} | 0 | 5,253.5625 | -1 | 5,253.5625 | |
In the triangle \( \triangle ABC \), the maximum value of \( \sin A + \sin B + 2 \sqrt{7} \sin C \) is ______. | \frac{27}{4} | 0.6875 | 6,834.4375 | 6,217.363636 | 8,192 | |
Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar. | -1 | 1 | 3,419.5 | 3,419.5 | -1 | |
Find
\[\min_{y \in \mathbb{R}} \max_{0 \le x \le 1} |x^2 - xy|.\] | 3 - 2 \sqrt{2} | 0.6875 | 6,617.5625 | 5,901.909091 | 8,192 | |
Pick a random integer between 0 and 4095, inclusive. Write it in base 2 (without any leading zeroes). What is the expected number of consecutive digits that are not the same (that is, the expected number of occurrences of either 01 or 10 in the base 2 representation)? | \frac{20481}{4096} | Note that every number in the range can be written as a 12-digit binary string. For $i=1,2, \ldots 11$, let $R_{i}$ be a random variable which is 1 if the $i$ th and $(i+1)$ st digits differ in a randomly chosen number in the range. By linearity of expectation, $E\left(\sum_{i} R_{i}\right)=\sum E\left(R_{i}\right)$. S... | 0.0625 | 7,176.875 | 8,192 | 7,109.2 |
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\). | 768 | 0.25 | 7,656.5 | 6,050 | 8,192 | |
In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. Consider the curve $C\_1$: $ρ^{2}-4ρ\cos θ+3=0$, $θ∈[0,2π]$, and the curve $C\_2$: $ρ= \frac {3}{4\sin ( \frac {π}{6}-θ)}$, $θ∈[0,2π]$.
(I) F... | \frac { \sqrt {15}}{2} | 0 | 5,798.25 | -1 | 5,798.25 | |
Given point P(-2,0) and line l: (1+3λ)x + (1+2λ)y = 2+5λ (λ ∈ ℝ), find the maximum value of the distance from point P to line l. | \sqrt{10} | 0.5625 | 6,764.8125 | 5,654.777778 | 8,192 | |
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of t... | 465 | Using bricks of dimensions $4''\times10''\times19''$ is comparable to using bricks of dimensions $0''\times6''\times15''$ which is comparable to using bricks of dimensions $0''\times2''\times5''$. Using 5 bricks of height $2''$ can be replaced by using 2 bricks of height $5''$ and 3 bricks of height $0''$.
It follows ... | 0 | 8,192 | -1 | 8,192 |
Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$. | 300 | 0.5 | 6,080.3125 | 4,441.875 | 7,718.75 | |
Given a parallelogram \\(ABCD\\) where \\(AD=2\\), \\(∠BAD=120^{\\circ}\\), and point \\(E\\) is the midpoint of \\(CD\\), if \\( \overrightarrow{AE} \cdot \overrightarrow{BD}=1\\), then \\( \overrightarrow{BD} \cdot \overrightarrow{BE}=\\) \_\_\_\_\_\_. | 13 | 0.625 | 6,082.3125 | 4,816.5 | 8,192 | |
A circle with center at $O$ has radius 1. Points $P$ and $Q$ outside the circle are placed such that $P Q$ passes through $O$. Tangent lines to the circle through $P$ hit the circle at $P_{1}$ and $P_{2}$, and tangent lines to the circle through $Q$ hit the circle at $Q_{1}$ and $Q_{2}$. If $\angle P_{1} P P_{2}=45^{\c... | \frac{\pi}{12} | $(45-30)^{\circ}=\frac{\pi}{12}$. | 0 | 8,192 | -1 | 8,192 |
Dean is playing a game with calculators. The 42 participants (including Dean) sit in a circle, and Dean holds 3 calculators. One calculator reads 1, another 0, and the last one -1. Dean starts by pressing the cube button on the calculator that shows 1, pressing the square button on the one that shows 0, and on the calc... | 0 | 0.5 | 6,126.5625 | 4,666.875 | 7,586.25 | |
If $\left|\frac{12}{x}+3\right|=2$, find the product of all possible values of $x$. Express your answer as an improper fraction. | \frac{144}{5} | 1 | 1,653.5625 | 1,653.5625 | -1 | |
Given that there is an extra $32.13 on the books due to a misplaced decimal point, determine the original amount of the sum of money. | 3.57 | 0.625 | 1,049.3125 | 1,416 | 438.166667 | |
Two vertical towers, \( AB \) and \( CD \), are located \( 16 \mathrm{~m} \) apart on flat ground. Tower \( AB \) is \( 18 \mathrm{~m} \) tall and tower \( CD \) is \( 30 \mathrm{~m} \) tall. Ropes are tied from \( A \) to \( C \) and from \( B \) to \( C \). Assuming the ropes are taut, calculate the total length of r... | 54 | 0.4375 | 4,650.8125 | 2,901.571429 | 6,011.333333 | |
How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive? | 6788 | 0 | 8,089.625 | -1 | 8,089.625 | |
How many pairs of real numbers $(x, y)$ satisfy the equation $y^{4}-y^{2}=x y^{3}-x y=x^{3} y-x y=x^{4}-x^{2}=0$? | 9 | We can see that if they solve the first and fourth equations, they are automatically solutions to the second and third equations. Hence, the solutions are just the $3^{2}=9$ points where $x, y$ can be any of $-1,0,1$. | 0.875 | 5,770.125 | 5,424.142857 | 8,192 |
My three friends and I have dinner together every weekend. Each weekend, two of us cook and the other two clean up afterwards. How many different ways are there for us to choose who cooks and who cleans? | 6 | 1 | 3,619 | 3,619 | -1 | |
Find $7463_{8} - 3254_{8}$. Express your answer first in base $8$, then convert it to base $10$. | 2183_{10} | 0.1875 | 5,222.3125 | 3,343.666667 | 5,655.846154 | |
A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the fewest relatively prime dates? | 10 | 0.4375 | 5,660 | 5,246.571429 | 5,981.555556 | |
Let $S$ be a set of intervals defined recursively as follows: Initially, $[1,1000]$ is the only interval in $S$. If $l \neq r$ and $[l, r] \in S$, then both $\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right],\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right] \in S$. An integer $i$ is chosen uniformly at rando... | 10.976 | The answer is given by computing the sum of the lengths of all intervals in $S$ and dividing this value by 1000, where the length of an interval $[i, j]$ is given by $j-i+1$. An interval may be categorized based on how many times $[1,1000]$ must be split to attain it. An interval that is derived from splitting $[1,1000... | 0 | 8,192 | -1 | 8,192 |
Find the number of natural numbers \( k \) that do not exceed 291000 and such that \( k^{2} - 1 \) is divisible by 291. | 4000 | 0.6875 | 6,530.625 | 5,820.090909 | 8,093.8 | |
The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$. | 2020 | 0.625 | 5,814.875 | 4,607.8 | 7,826.666667 | |
Given two circles intersecting at points A(1, 3) and B(m, -1), where the centers of both circles lie on the line $x - y + c = 0$, find the value of $m + c$. | -1 | 0.125 | 4,210.0625 | 4,949 | 4,104.5 | |
What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 37\cdot 2x + 37^2$ is a multiple of 47? | 5 | 1 | 2,331.5625 | 2,331.5625 | -1 | |
In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? | 5720 | 0.125 | 7,996.8125 | 6,630.5 | 8,192 | |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40, m] = 120$ and $\mathop{\text{lcm}}[m, 45] = 180$, what is $m$? | m = 36 | 0 | 7,465.25 | -1 | 7,465.25 | |
How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$ ? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$ ; for example, $\left\lfloor\pi\right\rfloor=3$ .
*2018 CCA Math Bonanza Lightning Round #3.2* | 86 | 0 | 7,979.8125 | -1 | 7,979.8125 | |
In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900? | 44 | 0 | 8,179.5625 | -1 | 8,179.5625 | |
Find the area of the shape enclosed by the curve $y=x^2$ (where $x>0$), the tangent line at point A(2, 4), and the x-axis. | \frac{2}{3} | 0.5 | 6,496.25 | 5,131.875 | 7,860.625 | |
What is the 100th digit to the right of the decimal point in the decimal representation of $\frac{13}{90}$? | 4 | 0.9375 | 2,650.375 | 2,280.933333 | 8,192 | |
In a particular state, the design of vehicle license plates was changed from an old format to a new one. Under the old scheme, each license plate consisted of two letters followed by three digits (e.g., AB123). The new scheme is made up of four letters followed by two digits (e.g., ABCD12). Calculate by how many times ... | \frac{26^2}{10} | 0.625 | 1,166.25 | 643.6 | 2,037.333333 | |
The distance between every two utility poles along the road is 50 meters. Xiao Wang travels at a constant speed in a car, and sees 41 utility poles in 2 minutes after seeing the first pole. How many meters does the car travel per hour? | 60000 | 0.6875 | 859.3125 | 987.636364 | 577 | |
Given that $a > 0, b > 0$ and $a+b=1$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$. | 3 + 2\sqrt{2} | 0.875 | 4,367.4375 | 3,821.071429 | 8,192 | |
Call a $7$-digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0, 1, 2, \ldots, 9$, the number of different memorable telepho... | 19990 | To solve this problem, we need to count the number of memorable telephone numbers based on the given conditions. A memorable telephone number is defined as $d_1d_2d_3-d_4d_5d_6d_7$ where the sequence $d_1d_2d_3$ matches either $d_4d_5d_6$ or $d_5d_6d_7$.
1. **Counting the total possibilities for $d_4d_5d_6d_7$:**
E... | 0.3125 | 7,053.0625 | 5,557.8 | 7,732.727273 |
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? | \frac{8a^2}{a^2+1} |
#### Step 1: Analyze the given equations
The equations given are:
1. \((x+ay)^2 = 4a^2\)
2. \((ax-y)^2 = a^2\)
These can be rewritten as:
- For equation 1: \(x+ay = \pm 2a\)
- For equation 2: \(ax-y = \pm a\)
#### Step 2: Convert to standard line equations
From the rewritten forms:
- \(x + ay = 2a\) and \(x + ay = -... | 0 | 8,089.9375 | -1 | 8,089.9375 |
The expansion of (ax- \frac {3}{4x}+ \frac {2}{3})(x- \frac {3}{x})^{6} is given, and the sum of its coefficients is 16. Determine the coefficient of the x^{3} term in this expansion. | \frac{117}{2} | 0.5 | 7,007.5625 | 6,745.75 | 7,269.375 | |
Given the real numbers $a, x, y$ that satisfy the equation:
$$
x \sqrt{a(x-a)}+y \sqrt{a(y-a)}=\sqrt{|\lg (x-a)-\lg (a-y)|},
$$
find the value of the algebraic expression $\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$. | \frac{1}{3} | 0.5625 | 5,740.5625 | 3,833.888889 | 8,192 | |
Let $A,$ $B,$ and $C$ be constants such that the equation \[\frac{(x+B)(Ax+28)}{(x+C)(x+7)} = 2\]has infinitely many solutions for $x.$ For these values of $A,$ $B,$ and $C,$ it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the sum of these values of $x.$ | -21 | 1 | 2,531.375 | 2,531.375 | -1 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively. The radius of the circumcircle of $\triangle ABC$ is $1$, and $b = acosC - \frac{{\sqrt{3}}}{6}ac$.
$(Ⅰ)$ Find the value of $a$;
$(Ⅱ)$ If $b = 1$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{4} | 0 | 6,121.4375 | -1 | 6,121.4375 | |
Given that the graph of the function $f(x)=ax^3+bx^2+c$ passes through the point $(0,1)$, and the equation of the tangent line at $x=1$ is $y=x$.
(1) Find the analytic expression of $y=f(x)$;
(2) Find the extreme values of $y=f(x)$. | \frac{23}{27} | 0.8125 | 2,804.8125 | 2,876.846154 | 2,492.666667 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, then the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is ______. | \dfrac {\pi}{4} | 0.6875 | 3,444.8125 | 3,166.454545 | 4,057.2 | |
An obtuse triangle with positive area has side lengths 10, 17, and \(x\). If \(x\) is an integer, what is the sum of all possible values of \(x\)? | 224 | 0.8125 | 4,802.1875 | 4,300.307692 | 6,977 | |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 12$ and $X$ is an integer, what is the smallest possible value of $X$? | 925 | 0.5 | 7,383.625 | 6,575.25 | 8,192 | |
When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______. | 2618 | 0.25 | 5,251.9375 | 4,051 | 5,652.25 | |
Place four different balls - red, black, blue, and yellow - into three different boxes, with at least one ball in each box. The red and blue balls cannot be in the same box. How many different arrangements are there? | 30 | 0.375 | 7,566.9375 | 6,563.666667 | 8,168.9 | |
The sum of the digits of the result of the expression $\underbrace{99 \cdots 99}_{2021 \text{ digits}} \times \underbrace{99 \cdots 99}_{2020 \text{ digits}}$ is $\qquad$ | 18189 | 0.125 | 8,041.3125 | 6,986.5 | 8,192 | |
Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \[
n^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6
\] | 8 | 0.75 | 6,456.375 | 5,877.833333 | 8,192 | |
How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$ | 180 | 0 | 7,915.5 | -1 | 7,915.5 | |
If $a,b,c$ are non-negative integers less than $7$ such that \begin{align*}
a+2b+3c&\equiv 0\pmod 7,\\
2a+3b+c&\equiv 4\pmod 7,\\
3a+b+2c&\equiv 4\pmod 7,
\end{align*}then determine the remainder when $abc$ is divided by $7$. | 6 | 1 | 3,329.25 | 3,329.25 | -1 | |
Given that the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, points $P$ and $Q$ are on planes $\alpha$ and $\beta$ respectively. The distance from $P$ to plane $\beta$ is $\sqrt{3}$, and the distance from $Q$ to plane $\alpha$ is $2 \sqrt{3}$. What is the minimum distance between points $P$ and $Q$? | 2\sqrt{3} | 0.0625 | 8,071.5625 | 7,814 | 8,088.733333 | |
Simplify $\frac{36}{54}$. | \frac{2}{3} | 1 | 2,183.1875 | 2,183.1875 | -1 | |
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spen... | 110 | 0 | 751 | -1 | 751 | |
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.
| 283 | 1 | 2,077.375 | 2,077.375 | -1 | |
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | 17 | 1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.
2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January, $P_1$, is calculated as:
\[
P_1 = P_0 + 0.20 \times P_0 = 1... | 0.9375 | 3,354 | 3,031.466667 | 8,192 |
There are 10 different televisions, including 3 type A, 3 type B, and 4 type C. Now, 3 televisions are randomly selected from them. If at least two different types are included, calculate the total number of different ways to select them. | 114 | 0.6875 | 5,336.0625 | 4,037.909091 | 8,192 | |
Compute
\[\begin{vmatrix} -5 & 3 \\ 4 & -4 \end{vmatrix}.\] | 8 | 1 | 2,181.6875 | 2,181.6875 | -1 | |
A store received a large container of milk. The salesperson has a balance scale that lacks weights (milk bottles can be placed on the scale), and there are 3 identical milk bottles, two of which are empty, and one has 1 liter of milk. How can exactly 85 liters of milk be measured into one bottle using the balance scale... | 85 | 0 | 8,192 | -1 | 8,192 | |
If \(\tan \gamma = 5\) and \(\tan \beta = 3,\) find \(\tan(\gamma + \beta)\) and \(\tan(\gamma - \beta)\). | \frac{1}{8} | 1 | 2,634 | 2,634 | -1 | |
The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value? | 45 | 0 | 7,405.6875 | -1 | 7,405.6875 | |
Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \). | \frac{9 \sqrt[4]{3}}{8} | 0 | 7,867.4375 | -1 | 7,867.4375 | |
Find the remainder when $1^3 + 2^3 + 3^3 + \dots + 100^3$ is divided by 6. | 4 | 1 | 3,293.25 | 3,293.25 | -1 | |
In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\angle SAC = 60^\circ$, $\angle SCA = 45^\circ$, and the ratio of the area of triangle $AOB$ to the area of triangle $ABC$ is $\frac{1}{2 + \sqrt{3}}$. Find... | 75 | 0 | 8,192 | -1 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.