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 |
|---|---|---|---|---|---|---|
What percentage of students did not receive a muffin, given that 38\% of students received a muffin? | 62\% | Since $38\%$ of students received a muffin, then $100\% - 38\% = 62\%$ of students did not receive a muffin. | 1 | 335.875 | 335.875 | -1 |
Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$. What is the greatest integer that does not exceed $100x$? | 241 | A slight variant of the above solution, note that
\begin{eqnarray*} \sum_{n=1}^{44} \cos n + \sum_{n=1}^{44} \sin n &=& \sum_{n=1}^{44} \sin n + \sin(90-n)\\ &=& \sqrt{2}\sum_{n=1}^{44} \cos(45-n) = \sqrt{2}\sum_{n=1}^{44} \cos n\\ \sum_{n=1}^{44} \sin n &=& (\sqrt{2}-1)\sum_{n=1}^{44} \cos n \end{eqnarray*}
This is t... | 0.75 | 5,350.6875 | 4,403.583333 | 8,192 |
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Enter all possible values of $b,$ separated by commas. | -330,90 | 0 | 7,959.375 | -1 | 7,959.375 | |
The eccentricity of the hyperbola defined by the equation $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ given that a line with a slope of -1 passes through its right vertex A and intersects the two asymptotes of the hyperbola at points B and C, and if $\overrightarrow {AB}= \frac {1}{2} \overrightarrow {BC}$, deter... | \sqrt{5} | 0.9375 | 4,564.4375 | 4,322.6 | 8,192 | |
The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$? | 12 | 0 | 7,365.4375 | -1 | 7,365.4375 | |
The top section of an 8 cm by 6 cm rectangular sheet of paper is folded along a straight line so that when the top section lies flat on the bottom section, corner $P$ lies on top of corner $R$. What is the length of the crease? | 7.5 | Suppose that the crease intersects $PS$ at $X$, $QR$ at $Y$, and the line $PR$ at $Z$. We want to determine the length of $XY$. Since $P$ folds on top of $R$, then line segment $PZ$ folds on top of line segment $RZ$, since after the fold $Z$ corresponds with itself and $P$ corresponds with $R$. This means that $PZ=RZ$ ... | 0.25 | 4,737.8125 | 4,066.25 | 4,961.666667 |
A point $Q$ is randomly placed in the interior of the right triangle $XYZ$ with $XY = 10$ units and $XZ = 6$ units. What is the probability that the area of triangle $QYZ$ is less than one-third of the area of triangle $XYZ$? | \frac{1}{3} | 0 | 7,818.875 | -1 | 7,818.875 | |
If two points are randomly selected from the eight vertices of a cube, the probability that the line determined by these two points intersects each face of the cube is ______. | \frac{1}{7} | 0.3125 | 6,710.875 | 4,771.6 | 7,592.363636 | |
Evaluate $\frac{2016!^{2}}{2015!2017!}$. Here $n$ ! denotes $1 \times 2 \times \cdots \times n$. | \frac{2016}{2017} | $\frac{2016!^{2}}{2015!2017!}=\frac{2016!}{2015!} \frac{2016!}{2017!}=\frac{2016}{1} \frac{1}{2017}=\frac{2016}{2017}$ | 0.875 | 3,459.75 | 2,783.714286 | 8,192 |
Given $\sin \left(\alpha+ \frac {\pi}{3}\right)=- \frac {4}{5}$, and $- \frac {\pi}{2} < \alpha < 0$, find $\cos \alpha=$ ______. | \frac {3-4 \sqrt {3}}{10} | 0 | 8,192 | -1 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\left(a\cos B-c\right)$. Find:<br/>
$(1)$ The value of angle $A$;<br/>
$(2)$ If $a\cos C=\sqrt{3}$ and $b=1$, find the value of $c$. | 2\sqrt{3} - 2 | 0.75 | 5,148.9375 | 4,284.333333 | 7,742.75 | |
Simplify $(x+15)+(100x+15)$. | 101x+30 | 1 | 1,230.1875 | 1,230.1875 | -1 | |
What is the largest three-digit multiple of 9 whose digits' sum is 18? | 990 | 0.9375 | 3,955.75 | 3,673.333333 | 8,192 | |
A digital watch displays time in a 24-hour format, showing hours and minutes. Calculate the largest possible sum of the digits in this display. | 24 | 0.125 | 8,127.8125 | 7,820 | 8,171.785714 | |
In triangle $XYZ$, side $y = 7$, side $z = 3$, and $\cos(Y - Z) = \frac{17}{32}$. Find the length of side $x$. | \sqrt{41} | 0 | 7,919.625 | -1 | 7,919.625 | |
Given that the focus of the parabola $x^{2}=2py$ coincides with the lower focus of the ellipse $\frac{x^{2}}{3}+\frac{y^{2}}{4}=1$, find the value of $p$. | -2 | 1 | 2,579.375 | 2,579.375 | -1 | |
There exist two distinct unit vectors $\mathbf{v}$ such that the angle between $\mathbf{v}$ and $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}$ is $45^\circ,$ and the angle between $\mathbf{v}$ and $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ is $60^\circ.$ Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be these vectors. Find $\... | \sqrt{2} | 0.375 | 6,815.375 | 4,609 | 8,139.2 | |
Given that the function $f(x)=x^{3}-3x^{2}$, find the value of $f( \frac {1}{2015})+f( \frac {2}{2015})+f( \frac {3}{2015})+…+f( \frac {4028}{2015})+f( \frac {4029}{2015})$. | -8058 | 0.1875 | 7,722.625 | 5,688.666667 | 8,192 | |
Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$. | \frac{2\sqrt{2} + \sqrt{3}}{6} | 0 | 7,158.125 | -1 | 7,158.125 | |
(Ⅰ) Find the equation of the line that passes through the intersection point of the two lines $2x-3y-3=0$ and $x+y+2=0$, and is perpendicular to the line $3x+y-1=0$.
(Ⅱ) Given the equation of line $l$ in terms of $x$ and $y$ as $mx+y-2(m+1)=0$, find the maximum distance from the origin $O$ to the line $l$. | 2 \sqrt {2} | 0 | 5,723.6875 | -1 | 5,723.6875 | |
The constant term in the expansion of $(x+3)\left(2x- \frac{1}{4x\sqrt{x}}\right)^5$ is ______. | 15 | 0.6875 | 6,026.9375 | 5,139.090909 | 7,980.2 | |
The regular price for a bicycle is $\$320$. The bicycle is on sale for $20\%$ off. The regular price for a helmet is $\$80$. The helmet is on sale for $10\%$ off. If Sandra bought both items on sale, what is her percentage savings on the total purchase? | 18\% | Since the regular price for the bicycle is $\$320$ and the savings are $20\%$, then the amount of money that Sandra saves on the bicycle is $\$320 \times 20\%=\$320 \times 0.2=\$64$. Since the regular price for the helmet is $\$80$ and the savings are $10\%$, then the amount of money that Sandra saves on the helmet is ... | 1 | 1,568.1875 | 1,568.1875 | -1 |
Given that ${{a_n}}$ is an arithmetic sequence with a non-zero common difference, and ${{a_9}}$, ${{a_1}}$, ${{a_5}}$ form a geometric sequence, and ${{a_1} + 3{{a_5}} + {{a_9}} = 20}$, determine the value of ${{a_{13}}}$. | 28 | 0.9375 | 3,402.3125 | 3,083 | 8,192 | |
Given the curve $C$: $y^{2}=4x$ with a focus at point $F$, a line $l$ passes through point $F$ and intersects curve $C$ at points $P$ and $Q$. If the relationship $\overrightarrow{FP}+2\overrightarrow{FQ}=\overrightarrow{0}$ holds, calculate the area of triangle $OPQ$. | \frac{3\sqrt{2}}{2} | 0 | 5,119.25 | -1 | 5,119.25 | |
Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ there is one point where the graphs of $y=h(2x)$ and $y=2j(x)$ must intersect. What is the sum of the coordinates of that point? | 16 | 0.875 | 4,832.25 | 4,352.285714 | 8,192 | |
Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by maintaining the same ratio of ingredients. | 26 | 0.3125 | 7,016.25 | 4,905.8 | 7,975.545455 | |
Given the function $f(x) = \sin x + \cos x$.
(1) If $f(x) = 2f(-x)$, find the value of $\frac{\cos^2x - \sin x\cos x}{1 + \sin^2x}$;
(2) Find the maximum value and the intervals of monotonic increase for the function $F(x) = f(x) \cdot f(-x) + f^2(x)$. | \frac{6}{11} | 0 | 4,501.3125 | -1 | 4,501.3125 | |
A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive $x$ and $y$ axes is shown in each figure I to IV below.
The inequalities
\(|x|+|y| \leq \sqrt{2(x^{2}+y^{2})} \leq 2\mbox{Max}(|x|, |y|)\)
are represented geometrically* by the figure numbered
... | II | To solve this problem, we need to analyze the given inequalities and understand their geometric representations. We will consider each inequality separately and then find the figure that correctly represents all three inequalities simultaneously.
1. **Understanding the Inequalities:**
- The first inequality $|x| + ... | 0.25 | 7,241 | 5,241 | 7,907.666667 |
Misha made himself a homemade dartboard at the summer house. The round board is divided into sectors by circles - it can be used to throw darts. Points are awarded according to the number written in the sector, as indicated in the diagram.
Misha threw 8 darts 3 times. The second time, he scored twice as many points as... | 48 | 0 | 2,855.3125 | -1 | 2,855.3125 | |
How many four-digit positive integers are multiples of 7? | 1286 | 1 | 3,671.6875 | 3,671.6875 | -1 | |
Eva chooses a three-digit positive number and subtracts the sum of its three digits from it. She finds that the result is a three-digit number in which all three digits are the same. How many different starting numbers could Eva have chosen? | 20 | 0.375 | 7,167.9375 | 5,845.666667 | 7,961.3 | |
There exists a complex number of the form $z = x + yi,$ where $x$ and $y$ are positive integers, such that
\[z^3 = -74 + ci,\]for some integer $c.$ Find $z.$ | 1 + 5i | 1 | 4,401.3125 | 4,401.3125 | -1 | |
At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given away a hat more times than they had receiv... | 10 | 0 | 8,103.875 | -1 | 8,103.875 | |
Evaluate
\[\begin{vmatrix} y + 1 & y & y \\ y & y + 1 & y \\ y & y & y + 1 \end{vmatrix}.\] | 3y + 1 | 0.375 | 7,483.5625 | 6,302.833333 | 8,192 | |
Given the sequence \(\{a_n\}\) with the sum of its first \(n\) terms denoted by \(S_n\), let \(T_n = \frac{S_1 + S_2 + \cdots + S_n}{n}\). \(T_n\) is called the "mean" of the sequence \(a_1, a_2, \cdots, a_n\). It is known that the "mean" of the sequence \(a_1, a_2, \cdots, a_{1005}\) is 2012. Determine the "mean" of t... | 2009 | 0.3125 | 6,859.1875 | 3,927 | 8,192 | |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{4}{9} | 0 | 7,405 | -1 | 7,405 | |
Given that $-7$ is a solution to $x^2 + bx -28 = 0$, what is the value of $b$? | 3 | 1 | 1,853.4375 | 1,853.4375 | -1 | |
Suppose that $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of $g$ is $[-2,1]$. The range of $f(x) \cdot g(x)$ is $[a,b]$. What is the largest possible value of $b$? | 10 | 0.9375 | 3,580.5 | 3,542.2 | 4,155 | |
A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? | 10 | 1. **Equating Volume and Surface Area**: Given a rectangular box with dimensions $a \times b \times c$, the volume is $abc$ and the surface area is $2(ab + bc + ca)$. Setting these equal gives:
\[
2(ab + bc + ca) = abc.
\]
Dividing both sides by $2abc$:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \fr... | 0.0625 | 8,147.875 | 7,957 | 8,160.6 |
The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\). | 3/7 | 0.5 | 7,695.375 | 7,198.75 | 8,192 | |
Person A and person B start simultaneously from points A and B, respectively, and move towards each other. When person A reaches the midpoint C of A and B, person B is still 240 meters away from point C. When person B reaches point C, person A has already moved 360 meters past point C. What is the distance between poin... | 144 | 0 | 7,576 | -1 | 7,576 | |
Find the distance from the point $(1,-1,2)$ to the line passing through $(-2,2,1)$ and $(-1,-1,3).$ | \sqrt{5} | 0.9375 | 3,189.25 | 2,855.733333 | 8,192 | |
In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \sqrt{2}$. Find the value of $x$. | -6 | 0.6875 | 6,813.0625 | 6,473.545455 | 7,560 | |
How many whole numbers between $100$ and $400$ contain the digit $2$? | 138 | 1. **Define the range and the problem**: We need to find how many whole numbers between $100$ and $400$ contain the digit $2$. This range includes numbers from $100$ to $399$.
2. **Use the Principle of Inclusion-Exclusion (PIE)**: Define events:
- $A$: the first digit is $2$.
- $B$: the second digit is $2$.
-... | 0.3125 | 7,294.5625 | 5,320.2 | 8,192 |
Given a linear function \( f(x) \), it is known that the distance between the intersection points of the graphs \( y=x^{2}+2 \) and \( y=f(x) \) is \(\sqrt{10}\), and the distance between the intersection points of the graphs \( y=x^{2}-1 \) and \( y=f(x)+1 \) is \(\sqrt{42}\). Find the distance between the intersectio... | \sqrt{34} | 0.6875 | 6,486.875 | 6,181.272727 | 7,159.2 | |
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah has to pull out to guarantee that he has a set of shirts that includes either 3 of the same colour or 3 of different colou... | 5 | Zebadiah must remove at least 3 shirts. If he removes 3 shirts, he might remove 2 red shirts and 1 blue shirt. If he removes 4 shirts, he might remove 2 red shirts and 2 blue shirts. Therefore, if he removes fewer than 5 shirts, it is not guaranteed that he removes either 3 of the same colour or 3 of different colours.... | 0 | 5,727.0625 | -1 | 5,727.0625 |
A secret agent is trying to decipher a passcode. So far, he has obtained the following information:
- It is a four-digit number.
- It is not divisible by seven.
- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.
- The number formed by the first two digits of t... | 4583 | 0.375 | 5,258.1875 | 2,788.666667 | 6,739.9 | |
A circle is inscribed in a circular sector with an arc that measures $60^{\circ}$. Find the ratio of the area of this circle to the area of the sector. | 2/3 | 0.8125 | 5,204.3125 | 4,514.846154 | 8,192 | |
While walking on a plane surface, a traveler first headed 18 miles north, then 11 miles west, then 6 miles south and finally 6 miles east. How many miles from the starting point was the traveler after these four legs of the journey? | 13 | 1 | 1,393.125 | 1,393.125 | -1 | |
Let $f(n)=1 \times 3 \times 5 \times \cdots \times (2n-1)$ . Compute the remainder when $f(1)+f(2)+f(3)+\cdots +f(2016)$ is divided by $100.$ *Proposed by James Lin* | 74 | 0.0625 | 7,443.8125 | 6,975 | 7,475.066667 | |
In $\triangle DEF$ with sides $5$, $12$, and $13$, a circle with center $Q$ and radius $2$ rolls around inside the triangle, always keeping tangency to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled? | 18 | 0 | 8,192 | -1 | 8,192 | |
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? | 67 | 1. **Identify the total distance and total time**: Sam drove a total of $96$ miles in $90$ minutes. To convert minutes to hours, we divide by $60$:
\[
90 \text{ minutes} = \frac{90}{60} \text{ hours} = 1.5 \text{ hours}
\]
2. **Calculate the overall average speed**: The average speed for the entire trip is t... | 0.9375 | 2,015.5625 | 1,603.8 | 8,192 |
Find the sum $m + n$ where $m$ and $n$ are integers, such that the positive difference between the two roots of the quadratic equation $2x^2 - 5x - 12 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, and $m$ is not divisible by the square of any prime number. | 123 | 0.625 | 7,197.0625 | 6,600.1 | 8,192 | |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{3^{5 x}-2^{x}}{x-\sin 9 x}$$ | \frac{1}{8} \ln \frac{2}{243} | 0 | 4,027.6875 | -1 | 4,027.6875 | |
For a finite sequence \( B = (b_1, b_2, \dots, b_{50}) \) of numbers, the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_{50}}{50},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 50 \).
If the Cesaro sum of the 50-term sequence \( (b_1, \dots, b_{50}) \) is 500, what is the Cesaro sum of the 51-term ... | 492 | 0 | 7,377.6875 | -1 | 7,377.6875 | |
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square. | 3 |
Let \( n \) be an integer such that \( n \geq 1 \), and we want to find all values of \( n \) for which \( n \cdot 2^{n+1} + 1 \) is a perfect square. We set:
\[
n \cdot 2^{n+1} + 1 = k^2
\]
for some integer \( k \). Rearranging gives:
\[
n \cdot 2^{n+1} = k^2 - 1
\]
which can be factored as:
\[
(k - 1)(k + 1) = ... | 0.125 | 8,103.5 | 7,484 | 8,192 |
A five-digit positive integer is created using each of the odd digits $1, 3, 5, 7, 9$ once so that the thousands digit is larger than the hundreds digit, the thousands digit is larger than the ten thousands digit, the tens digit is larger than the hundreds digit, and the tens digit is larger than the units digit. How m... | 16 | We write such a five-digit positive integer with digits $V W X Y Z$. We want to count the number of ways of assigning $1, 3, 5, 7, 9$ to the digits $V, W, X, Y, Z$ in such a way that the given properties are obeyed. From the given conditions, $W > X, W > V, Y > X$, and $Y > Z$. The digits 1 and 3 cannot be placed as $W... | 0 | 7,833 | -1 | 7,833 |
Compute the sum of all positive real numbers \(x \leq 5\) satisfying \(x=\frac{\left\lceil x^{2}\right\rceil+\lceil x\rceil \cdot\lfloor x\rfloor}{\lceil x\rceil+\lfloor x\rfloor}\). | 85 | Note that all integer \(x\) work. If \(x\) is not an integer then suppose \(n<x<n+1\). Then \(x=n+\frac{k}{2n+1}\), where \(n\) is an integer and \(1 \leq k \leq 2n\) is also an integer, since the denominator of the fraction on the right hand side is \(2n+1\). We now show that all \(x\) of this form work. Note that \(x... | 0 | 7,900.5625 | -1 | 7,900.5625 |
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \_\_\_\_\_\_ (answer in numbers). | 600 | 0 | 7,653.1875 | -1 | 7,653.1875 | |
There are 15 girls in a class of 27 students. The ratio of boys to girls in this class is: | 4:5 | 1 | 1,548.25 | 1,548.25 | -1 | |
Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether? | 3932 | To determine the total number of toothpicks used in constructing a grid that is $60$ toothpicks long and $32$ toothpicks wide, we need to consider both the vertical and horizontal toothpicks.
1. **Vertical Toothpicks:**
- Each vertical line extends from the top to the bottom of the grid.
- Since the grid is $60$... | 0.3125 | 7,250.8125 | 5,180.2 | 8,192 |
For any positive integer \( n \), let \( f(n) \) represent the last digit of \( 1 + 2 + 3 + \cdots + n \). For example, \( f(1) = 1 \), \( f(2) = 3 \), \( f(5) = 5 \), and so on. Find the value of \( f(2) + f(4) + f(6) + \cdots + f(2012) \). | 3523 | 0.3125 | 7,264.25 | 6,353.4 | 7,678.272727 | |
Antal and Béla start from home on their motorcycles heading towards Cegléd. After traveling one-fifth of the way, Antal for some reason turns back. As a result, he accelerates and manages to increase his speed by one quarter. He immediately sets off again from home. Béla, continuing alone, decreases his speed by one qu... | 40 | 0 | 8,094.4375 | -1 | 8,094.4375 | |
A circle with equation $x^{2}+y^{2}=1$ passes through point $P(1, \sqrt {3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$. | \sqrt {3} | 0 | 5,086.75 | -1 | 5,086.75 | |
The coefficient of $x^3y^5$ in the expansion of $(x+y)(x-y)^7$ is __________. | 14 | 1 | 4,600.625 | 4,600.625 | -1 | |
A bug is on one exterior vertex of solid $S$, a $3 \times 3 \times 3$ cube that has its center $1 \times 1 \times 1$ cube removed, and wishes to travel to the opposite exterior vertex. Let $O$ denote the outer surface of $S$ (formed by the surface of the $3 \times 3 \times 3$ cube). Let $L(S)$ denote the length of the ... | \frac{\sqrt{29}}{3 \sqrt{5}} \text{ OR } \frac{\sqrt{145}}{15} | By $\left(^{*}\right)$, the shortest route in $O$ has length $2 \sqrt{1.5^{2}+3^{2}}=3 \sqrt{5}$. By $\left({ }^{* *}\right)$, the shortest route overall (in $S$ ) has length $2 \sqrt{1.5^{2}+1^{2}+2^{2}}=\sqrt{3^{2}+2^{2}+4^{2}}=\sqrt{29}$. Therefore the desired ratio is $\frac{\sqrt{29}}{3 \sqrt{5}}=\frac{\sqrt{145}}... | 0 | 8,097.3125 | -1 | 8,097.3125 |
How many points $(x, y)$, with $x$ and $y$ both integers, are on the line with equation $y=4x+3$ and inside the region bounded by $x=25, x=75, y=120$, and $y=250$? | 32 | We determine the number of integers $x$ with $25 \leq x \leq 75$ for which $120 \leq 4x+3 \leq 250$. When $x=30$, $4x+3=123$ and when $x=61$, $4x+3=247$. Therefore, $4x+3$ is between 120 and 250 exactly when $30 \leq x \leq 61$. There are $61-30+1=32$ such values of $x$. | 0.9375 | 2,861.625 | 2,791.6 | 3,912 |
Solve for $X$ if $\sqrt[4]{X^5} = 32\sqrt[16]{32}$. | 16\sqrt[4]{2} | 0.5 | 4,739.125 | 4,612.125 | 4,866.125 | |
Find the area of a trapezoid with diagonals of 7 cm and 8 cm, and bases of 3 cm and 6 cm. | 12 \sqrt{5} | 0.8125 | 4,660.5 | 4,445.153846 | 5,593.666667 | |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{6}}{3}$, the distance from the origin to the line passing through points $A(0, -b)$ and $B(a, 0)$ is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Given a fixed point $E(-1, 0)$, if the line $... | \frac{7}{6} | 0.75 | 6,414.8125 | 5,822.416667 | 8,192 | |
Find all the triples of positive integers $(a,b,c)$ for which the number
\[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\]
is an integer and $a+b+c$ is a prime. | (1, 1, 1), (2, 2, 1), (6, 3, 2) |
To solve this problem, we are tasked with finding all triples of positive integers \((a, b, c)\) such that the expression
\[
\frac{(a+b)^4}{c} + \frac{(b+c)^4}{a} + \frac{(c+a)^4}{b}
\]
is an integer and the sum \(a + b + c\) is a prime number.
### Step-by-step Solution
1. **Initial Constraints**:
Each term ... | 0 | 8,192 | -1 | 8,192 |
The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is: | 36.5 | 1. **Calculate the total sum of the original set of numbers**:
Given that the arithmetic mean of $50$ numbers is $38$, the total sum of these numbers can be calculated using the formula for the arithmetic mean:
\[
\text{Sum} = \text{Mean} \times \text{Number of elements} = 38 \times 50 = 1900
\]
2. **Adju... | 0 | 3,492.125 | -1 | 3,492.125 |
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$. | t < \frac{1}{2} |
To find all positive real numbers \( t \) with the property that there exists an infinite set \( X \) of real numbers such that the inequality
\[
\max\{|x-(a-d)|,|y-a|,|z-(a+d)|\} > td
\]
holds for all \( x, y, z \in X \), all real numbers \( a \), and all positive real numbers \( d \), we proceed as follows:
Firs... | 0 | 8,192 | -1 | 8,192 |
In the rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinates of point $A$ are $\left( 4\sqrt{2}, \frac{\pi}{4} \right)$, and the polar equation of line $l$ is $\rho \cos \... | \frac{32}{7} | 0.3125 | 7,507.5 | 6,577.8 | 7,930.090909 | |
Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be... | 719 | Denote $r = \frac{a}{b}$, where $\left( a, b \right) = 1$. We have $55 r = \frac{55a}{b}$. Suppose $\left( 55, b \right) = 1$, then the sum of the numerator and the denominator of $55r$ is $55a + b$. This cannot be equal to the sum of the numerator and the denominator of $r$, $a + b$. Therefore, $\left( 55, b \right) \... | 0.4375 | 6,960.3125 | 5,376.714286 | 8,192 |
Bob is writing a sequence of letters of the alphabet, each of which can be either uppercase or lowercase, according to the following two rules: If he had just written an uppercase letter, he can either write the same letter in lowercase after it, or the next letter of the alphabet in uppercase. If he had just written a... | 376 | The smallest possible sequence from $a$ to $z$ is $a A B C D \ldots Z z$, which has 28 letters. To insert 4 more letters, we can either switch two (not necessarily distinct) letters to lowercase and back again (as in $a A B C c C D E F f F G H \ldots Z z$ ), or we can insert a lowercase letter after its corresponding u... | 0 | 7,915.4375 | -1 | 7,915.4375 |
Given the function $f(x)= \sqrt {2}\cos (x+ \frac {\pi}{4})$, after translating the graph of $f(x)$ by the vector $\overrightarrow{v}=(m,0)(m > 0)$, the resulting graph exactly matches the function $y=f′(x)$. The minimum value of $m$ is \_\_\_\_\_\_. | \frac {3\pi}{2} | 0.625 | 6,664.25 | 6,235.9 | 7,378.166667 | |
We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle. | 60^\circ, 60^\circ, 60^\circ |
We are given that the lines \((\ell_1)\) and \((\ell_2)\) pass through the orthocenter \(H\) of triangle \(ABC\) and each line divides the triangle into two figures of equal area and equal perimeters. We need to determine the angles of the triangle.
The orthocenter \(H\) of a triangle is the intersection of its altit... | 0 | 7,290.8125 | -1 | 7,290.8125 |
Arrange the numbers 3, 4, 5, 6 into two natural numbers A and B, so that the product A×B is maximized. Find the value of A×B. | \left( 3402 \right) | 0 | 8,019.4375 | -1 | 8,019.4375 | |
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. | 1 - \frac{35}{12 \pi^2} | \textbf{First solution:} (by Daniel Kane) The probability is $1 - \frac{35}{12\pi^2}$. We start with some notation and simplifications. For simplicity, we assume without loss of generality that the circle has radius 1. Let $E$ denote the expected value of a random variable over all choices of $P,Q,R$. Write $[XYZ]$ for... | 0 | 7,120.5625 | -1 | 7,120.5625 |
Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____. | 1012 | 0.75 | 6,279.1875 | 5,882 | 7,470.75 | |
Find $\left(\frac{2}{3}\right)^{6} \cdot \left(\frac{5}{6}\right)^{-4}$. | \frac{82944}{456375} | 0 | 5,444 | -1 | 5,444 | |
Find minimum of $x+y+z$ where $x$ , $y$ and $z$ are real numbers such that $x \geq 4$ , $y \geq 5$ , $z \geq 6$ and $x^2+y^2+z^2 \geq 90$ | 16 | 0.375 | 7,575.5625 | 6,548.166667 | 8,192 | |
In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$. | \frac{15}{17} | 1 | 2,298.4375 | 2,298.4375 | -1 | |
Let $R$ be the region in the Cartesian plane of points $(x, y)$ satisfying $x \geq 0, y \geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor \leq 5$. Determine the area of $R$. | \frac{9}{2} | We claim that a point in the first quadrant satisfies the desired property if the point is below the line $x+y=3$ and does not satisfy the desired property if it is above the line. To see this, for a point inside the region, $x+y<3$ and $\lfloor x\rfloor+\lfloor y\rfloor \leq x+y<3$ However, $\lfloor x\rfloor+\lfloor y... | 0.25 | 7,519 | 6,786.75 | 7,763.083333 |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}[t]{r} ABBCB \\ + \\ BCADA \\ \hline DBDDD \end{array}$ | 7 | 1. **Analyze the last column of the addition:**
- The equation from the last column is $B + A = D$ or $B + A = D + 10$. Since the sum $B + A$ must be a single digit (as carrying over would make the sum a six-digit number), we conclude that $B + A = D$.
2. **Analyze the first column of the addition:**
- The equat... | 0.0625 | 8,071.375 | 6,262 | 8,192 |
For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons? | $k \ne 1$ |
Given a \(1 \times k\) rectangle, we want to determine for which real values of \(k > 0\) it is possible to dissect the rectangle into two similar, but noncongruent, polygons.
First, let's understand the requirements: two polygons are similar if their corresponding angles are equal and their corresponding sides are i... | 0 | 8,192 | -1 | 8,192 |
Given that $0\le x_3 \le x_2 \le x_1\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\frac{1}{4},$ find $x_1.$ | \frac{3}{4} | 0.875 | 5,632.0625 | 5,266.357143 | 8,192 | |
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. | 720 | 0.3125 | 7,125.625 | 6,482 | 7,418.181818 | |
(The full score of this question is 12 points) In a box, there are three cards labeled 1, 2, and 3, respectively. Now, two cards are drawn from this box with replacement in succession, and their labels are denoted as $x$ and $y$, respectively. Let $\xi = |x-2| + |y-x|$.
(1) Find the range of the random variable $\xi$... | \frac{2}{9} | 0.75 | 4,775.75 | 4,424.666667 | 5,829 | |
For all positive integers $m>10^{2022}$ , determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$ . | 10 | 0 | 8,192 | -1 | 8,192 | |
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.
| 486 | 0.125 | 8,096.9375 | 8,145 | 8,090.071429 | |
Given the function $f(x)= \sqrt {3}\sin x+\cos x$ $(x\in R)$
(1) Find the value of $f( \frac {5π}{6})$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\[- \frac {π}{2}, \frac {π}{2}\]$ and their respective $x$ values. | -\sqrt {3} | 0 | 4,097 | -1 | 4,097 | |
There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$ | 330 | We plug -20 into the equation obtaining $(-20)^3-20a+b$, likewise, plugging -21 into the second equation gets $(-21)^3+441c+d$.
Both equations must have 3 solutions exactly, so the other two solutions must be $m + \sqrt{n} \cdot i$ and $m - \sqrt{n} \cdot i$.
By Vieta's, the sum of the roots in the first equation is ... | 0.4375 | 6,618.125 | 5,095.571429 | 7,802.333333 |
The polynomial sequence is defined as follows: \( f_{0}(x)=1 \) and \( f_{n+1}(x)=\left(x^{2}-1\right) f_{n}(x)-2x \) for \( n=0,1,2, \ldots \). Find the sum of the absolute values of the coefficients of \( f_{6}(x) \). | 190 | 0 | 7,717.6875 | -1 | 7,717.6875 | |
Let $S$ be the set of lattice points inside the circle $x^{2}+y^{2}=11$. Let $M$ be the greatest area of any triangle with vertices in $S$. How many triangles with vertices in $S$ have area $M$? | 16 | The boundary of the convex hull of $S$ consists of points with $(x, y)$ or $(y, x)=(0, \pm 3)$, $( \pm 1, \pm 3)$, and $( \pm 2, \pm 2)$. For any triangle $T$ with vertices in $S$, we can increase its area by moving a vertex not on the boundary to some point on the boundary. Thus, if $T$ has area $M$, its vertices are ... | 0 | 8,192 | -1 | 8,192 |
A "double-single" number is a three-digit number made up of two identical digits followed by a different digit. For example, 553 is a double-single number. How many double-single numbers are there between 100 and 1000? | 81 | 0.8125 | 4,741.4375 | 3,945.153846 | 8,192 | |
Mr. Morgan G. Bloomgarten wants to distribute 1,000,000 dollars among his friends. He has two specific rules for distributing the money:
1. Each gift must be either 1 dollar or a power of 7 (7, 49, 343, 2401, etc.).
2. No more than six people can receive the same amount.
How can he distribute the 1,000,000 dollars und... | 1,000,000 | 0 | 8,192 | -1 | 8,192 | |
Given that $a > 0$, $b > 0$, and $4a - b \geq 2$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$. | \frac{1}{2} | 0.875 | 5,418.4375 | 5,022.214286 | 8,192 | |
Given the function $f(x)= \dfrac {x+3}{x+1}$, let $f(1)+f(2)+f(4)+f(8)+f(16)=m$ and $f( \dfrac {1}{2})+f( \dfrac {1}{4})+f( \dfrac {1}{8})+f( \dfrac {1}{16})=n$, then $m+n=$ \_\_\_\_\_\_. | 18 | 0.5625 | 7,242.5 | 6,504 | 8,192 |
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