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 the function $f(x)=4\cos x\sin \left(x- \frac{\pi}{3}\right)+a$ has a maximum value of $2$.
$(1)$ Find the value of $a$ and the smallest positive period of the function $f(x)$;
$(2)$ In $\triangle ABC$, if $A < B$, and $f(A)=f(B)=1$, find the value of $\frac{BC}{AB}$. | \sqrt{2} | 0.9375 | 4,416.125 | 4,164.4 | 8,192 | |
A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ | 4 | 1. **Identify the conditions for a number to be flippy and divisible by 15**:
- A flippy number alternates between two distinct digits.
- A number is divisible by 15 if it is divisible by both 3 and 5.
2. **Condition for divisibility by 5**:
- The last digit must be either 0 or 5.
3. **Eliminate the possibil... | 0.4375 | 7,672.4375 | 7,154.428571 | 8,075.333333 |
Find the value of $n$ that satisfies $\frac{1}{n+1} + \frac{2}{n+1} + \frac{n}{n+1} = 3$. | 0 | 1 | 1,859.5 | 1,859.5 | -1 | |
What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$? | 6048 | 1 | 3,532.25 | 3,532.25 | -1 | |
What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$? | -280 | To find the minimum possible product of three different numbers from the set $\{-8,-6,-4,0,3,5,7\}$, we need to consider the sign of the product and the magnitude of the numbers involved.
1. **Sign of the Product**:
- A product of three numbers is negative if and only if exactly one or all three of the numbers are ... | 0.9375 | 5,215.3125 | 5,016.866667 | 8,192 |
Given that point $P(-15a, 8a)$ is on the terminal side of angle $\alpha$, where $a \in \mathbb{R}$ and $a \neq 0$, find the values of the six trigonometric functions of $\alpha$. | -\frac{15}{8} | 0.125 | 6,792.125 | 5,721 | 6,945.142857 | |
The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is | \frac{5}{12} | 1. **Identify the probability of even and odd outcomes for each wheel:**
- For the first wheel, the probability of landing on an even number is $\frac{1}{4}$ (since one out of four sections is even), and the probability of landing on an odd number is $\frac{3}{4}$ (since three out of four sections are odd).
- For... | 0 | 5,449.6875 | -1 | 5,449.6875 |
The digits from 1 to 9 are to be written in the nine cells of a $3 \times 3$ grid, one digit in each cell.
- The product of the three digits in the first row is 12.
- The product of the three digits in the second row is 112.
- The product of the three digits in the first column is 216.
- The product of the three digits... | 30 | 0 | 8,190.5625 | -1 | 8,190.5625 | |
The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$. | \frac{1}{1998^2} | 0 | 8,192 | -1 | 8,192 | |
A fair six-sided die with faces numbered 1 to 6 is rolled twice. Let $a$ and $b$ denote the numbers obtained in the two rolls.
1. Find the probability that $a + b \geq 9$.
2. Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
3. Find the probability that the lengths $a$, $b$... | \frac{7}{18} | 0.0625 | 7,613.375 | 7,660 | 7,610.266667 | |
Some middle school students in a city participated in a mathematics invitational competition, which consisted of 6 problems. It is known that each problem was solved by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. What is the minimum number of middle school ... | 1000 | 0.0625 | 8,151.0625 | 7,763 | 8,176.933333 | |
A number contains only two kinds of digits: 3 or 4, and both 3 and 4 appear at least once. The number is a multiple of both 3 and 4. What is the smallest such number? | 3444 | 0.3125 | 7,082.75 | 5,755.8 | 7,685.909091 | |
Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly o... | \[
383
\] | We claim that Paul wins if and only if there are exactly 1 or 2 odd integers on the board at the start. Assuming this, the answer is $\frac{2021+\left(\frac{2021}{202}\right)}{2^{2021}}$. Since the numerator is odd, this fraction is reduced. Now, $m+n \equiv 2^{2021}+21+2021 \cdot 1010 \equiv 231+2^{2021} \equiv 231+2^... | 0 | 7,992.4375 | -1 | 7,992.4375 |
Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between 1 and $n$, inclusive: if $n=1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k=1$, he will sto... | \frac{197}{60} | If we let $a_{n}$ be the expected number of rolls starting with an $n$-sided die, we see immediately that $a_{1}=0$, and $a_{n}=1+\frac{1}{n} \sum_{i=1}^{n} a_{i}$ for $n>1$. Thus $a_{2}=2$, and for $n \geq 3$, $a_{n}=1+\frac{1}{n} a_{n}+\frac{n-1}{n}\left(a_{n-1}-1\right)$, or $a_{n}=a_{n-1}+\frac{1}{n-1}$. Thus $a_{n... | 0.0625 | 7,272.625 | 5,726 | 7,375.733333 |
Let \( F_1 = (0,2) \) and \( F_2 = (6,2) \). Find the set of points \( P \) such that
\[
PF_1 + PF_2 = 10
\]
forms an ellipse. The equation of this ellipse can be written as
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
\]
Find \( h + k + a + b \). | 14 | 1 | 2,008.4375 | 2,008.4375 | -1 | |
The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction. | \frac{60}{7} | 0.4375 | 4,546.875 | 3,873.428571 | 5,070.666667 | |
Simplify $\left(\sqrt[6]{27} - \sqrt{6 \frac{3}{4} }\right)^2$ | \frac{3}{4} | 1. **Simplify $\sqrt[6]{27}$:**
\[
\sqrt[6]{27} = (3^3)^{\frac{1}{6}} = 3^{\frac{3}{6}} = 3^{\frac{1}{2}} = \sqrt{3}
\]
2. **Simplify $\sqrt{6 \frac{3}{4}}$:**
\[
6 \frac{3}{4} = 6 + \frac{3}{4} = \frac{24}{4} + \frac{3}{4} = \frac{27}{4}
\]
\[
\sqrt{6 \frac{3}{4}} = \sqrt{\frac{27}{4}} = \frac... | 1 | 2,002.3125 | 2,002.3125 | -1 |
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | 3 | To solve this problem, we need to consider the constraints given and use casework based on the position of the yellow house (Y). The constraints are:
1. Ralph passed the orange house (O) before the red house (R).
2. Ralph passed the blue house (B) before the yellow house (Y).
3. The blue house (B) was not next to the y... | 0.1875 | 7,807.1875 | 6,139.666667 | 8,192 |
In Nevada, 580 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the "Soda" sector of the graph is $198^\circ$, to the nearest whole degree. How many of the people surveyed chose "Soda"? Express your answer as a whole number. | 321 | 0 | 3,300.3125 | -1 | 3,300.3125 | |
The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares? | 77 | 1. **Define the integers**: Let the three consecutive positive integers be $a-1$, $a$, and $a+1$.
2. **Express their sum**: The sum of these integers is $(a-1) + a + (a+1) = 3a$.
3. **Set up the equation**: According to the problem, the product of these integers is $8$ times their sum. Therefore, we have:
\[
(... | 1 | 1,999.375 | 1,999.375 | -1 |
Find the result of $46_8 - 63_8$ and express your answer in base 10. | -13 | 0.875 | 5,677.3125 | 5,385.857143 | 7,717.5 | |
As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot Y moves clockwise, and robots J and B move... | 56 | 0 | 7,491.9375 | -1 | 7,491.9375 | |
Compute the smallest positive integer $n$ for which $\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$ is an integer. | 6156 | The number $\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$ is a positive integer if and only if its square is a perfect square. We have $$(\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}})^{2} =(100+\sqrt{n})+(100-\sqrt{n})+2 \sqrt{(100+\sqrt{n})(100-\sqrt{n})} =200+2 \sqrt{10000-n}$$ To minimize $n$, we should maximize the value of ... | 0.25 | 7,612.375 | 5,873.5 | 8,192 |
Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola. | \frac{\sqrt{17}}{2} | 0 | 8,168.5 | -1 | 8,168.5 | |
There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, calculate the number of different arrangements. | 72 | 0 | 8,192 | -1 | 8,192 | |
For finite sets $A$ and $B$ , call a function $f: A \rightarrow B$ an \emph{antibijection} if there does not exist a set $S \subseteq A \cap B$ such that $S$ has at least two elements and, for all $s \in S$ , there exists exactly one element $s'$ of $S$ such that $f(s')=s$ . Let $N$ be the number of an... | 1363641 | 0 | 7,902.6875 | -1 | 7,902.6875 | |
The area of a circle is $49\pi$ square units. What is the radius of the circle, in units? | 7 | 1 | 1,003.8125 | 1,003.8125 | -1 | |
To $m$ ounces of a $m\%$ solution of acid, $x$ ounces of water are added to yield a $(m-10)\%$ solution. If $m>25$, then $x$ is | \frac{10m}{m-10} | To solve this problem, we start by analyzing the initial and final conditions of the solution.
1. **Initial Condition:**
- The initial solution has $m$ ounces of a $m\%$ solution of acid.
- This means that the amount of pure acid in the solution is $\frac{m}{100} \times m = \frac{m^2}{100}$ ounces.
2. **Adding ... | 1 | 2,409.3125 | 2,409.3125 | -1 |
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there w... | 240 | Let $x$ be the total number of pages in the book.
1. **Reading on the first day:**
Hui reads $\frac{1}{5}x + 12$ pages. The remaining pages are:
\[
x - \left(\frac{1}{5}x + 12\right) = \frac{4}{5}x - 12
\]
2. **Reading on the second day:**
From the remaining pages, Hui reads $\frac{1}{4}(\frac{4}{5}x -... | 1 | 2,904.0625 | 2,904.0625 | -1 |
Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$, uniformly and at random with replacement. Set $M=a+2 b+4 c+8 d+16 e+32 f$. What is the expected value of the remainder when $M$ is divided by 64? | \frac{63}{2} | Consider $M$ in binary. Assume we start with $M=0$, then add $a$ to $M$, then add $2 b$ to $M$, then add $4 c$ to $M$, and so on. After the first addition, the first bit (defined as the rightmost bit) of $M$ is toggled with probability $\frac{1}{2}$. After the second addition, the second bit of $M$ is toggled with prob... | 0.0625 | 8,174.75 | 7,916 | 8,192 |
The field shown has been planted uniformly with wheat. [asy]
draw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));
label("$60^\circ$",(0.06,0.1),E);
label("$120^\circ$",(1/2-0.05,sqrt(3)/2-0.1),E);
label("$120^\circ$",(3/2+0.05,sqrt(3)/2-0.1),W);
label("$60^\circ$",(2-0.05,0.1),W);
label("100 m",(... | \frac{5}{12} | 0 | 8,151.625 | -1 | 8,151.625 | |
A math teacher requires Noelle to do one homework assignment for each of the first four homework points she wants to earn; for each of the next four homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, she has to do $\lceil n\div4 \rceil$ homework ass... | 40 | 0.5625 | 7,496.5 | 6,955.555556 | 8,192 | |
A reader mentioned that his friend's house in location $A$, where he was invited for lunch at 1 PM, is located 1 km from his own house in location $B$. At 12 PM, he left $B$ in his wheelchair heading towards location $C$ for a stroll. His friend, intending to join him and help him reach on time for lunch, left $A$ at 1... | 2/3 | 0 | 7,842.4375 | -1 | 7,842.4375 | |
A candy company makes 5 colors of jellybeans, which come in equal proportions. If I grab a random sample of 5 jellybeans, what is the probability that I get exactly 2 distinct colors? | \frac{12}{125} | There are $\binom{5}{2}=10$ possible pairs of colors. Each pair of colors contributes $2^{5}-2=30$ sequences of beans that use both colors. Thus, the answer is $10 \cdot 30 / 5^{5}=12 / 125$. | 0.875 | 4,909.375 | 4,440.428571 | 8,192 |
If $x,y>0$, $\log_y(x)+\log_x(y)=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ | 13\sqrt{3} | 1. **Rewrite the logarithmic equation**: Given $\log_y(x) + \log_x(y) = \frac{10}{3}$, we can use the change of base formula to rewrite this as:
\[
\frac{\log x}{\log y} + \frac{\log y}{\log x} = \frac{10}{3}
\]
Let $u = \frac{\log x}{\log y}$. Then the equation becomes:
\[
u + \frac{1}{u} = \frac{10}... | 1 | 4,440.4375 | 4,440.4375 | -1 |
A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$ . Find $100\thet... | 1.101 | 0 | 8,183.6875 | -1 | 8,183.6875 | |
Suppose that a real number $x$ satisfies \[\sqrt{49-x^2}-\sqrt{25-x^2}=3.\]What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$? | 8 | 1 | 2,102.1875 | 2,102.1875 | -1 | |
The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that
$\sum_{n=0}^{7}\log_{3}(x_{n}) = 308$ and $56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,$
find $\log_{3}(x_{14}).$ | 91 | All these integral powers of $3$ are all different, thus in base $3$ the sum of these powers would consist of $1$s and $0$s. Thus the largest value $x_7$ must be $3^{56}$ in order to preserve the givens. Then we find by the given that $x_7x_6x_5\dots x_0 = 3^{308}$, and we know that the exponents of $x_i$ are in an ari... | 0.125 | 7,852.875 | 5,679 | 8,163.428571 |
In the acute triangle \( \triangle ABC \),
\[
\sin(A+B) = \frac{3}{5}, \quad \sin(A-B) = \frac{1}{5}, \quad AB = 3.
\]
Find the area of \( \triangle ABC \). | \frac{3(\sqrt{6} + 2)}{2} | 0 | 8,192 | -1 | 8,192 | |
Find the value of $k$ so that the line $3x + 5y + k = 0$ is tangent to the parabola $y^2 = 24x.$ | 50 | 1 | 2,725.1875 | 2,725.1875 | -1 | |
A subset \( S \) of the set of integers \{ 0, 1, 2, ..., 99 \} is said to have property \( A \) if it is impossible to fill a 2x2 crossword puzzle with the numbers in \( S \) such that each number appears only once. Determine the maximal number of elements in sets \( S \) with property \( A \). | 25 | 0 | 7,832.375 | -1 | 7,832.375 | |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$ .
*Proposed by Aaron Lin* | 2083 | 0.0625 | 7,346.75 | 4,122 | 7,561.733333 | |
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A particular solid right cylinder of radius 2 feet has a total surface area of $12\pi$ square feet. What is the height of this cylinder? | 1 | 0.9375 | 1,835.375 | 1,411.6 | 8,192 | |
Calculate $\frac{1}{6} \cdot \frac{2}{7} \cdot \frac{3}{8} \cdot \frac{4}{9} \cdots \frac{94}{99} \cdot \frac{95}{100}$. Express your answer as a common fraction. | \frac{1}{75287520} | 0.3125 | 7,806.625 | 6,958.8 | 8,192 | |
A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is: | 490 | To find the total area to be painted, we need to calculate the areas of the walls and the ceiling that will be painted. The barn is a rectangular prism with dimensions:
- Width = $10$ yd
- Length = $13$ yd
- Height = $5$ yd
#### Step 1: Calculate the area of each wall
There are four walls in the barn, two pairs of opp... | 0 | 7,096.4375 | -1 | 7,096.4375 |
Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \). | 16 | 0.625 | 6,136.5625 | 4,903.3 | 8,192 | |
How many lattice points lie on the hyperbola \(x^2 - y^2 = 999^2\)? | 21 | 0 | 7,961.375 | -1 | 7,961.375 | |
Fill the numbers $1,2,\cdots,36$ into a $6 \times 6$ grid, placing one number in each cell, such that the numbers in each row are in increasing order from left to right. What is the minimum possible sum of the six numbers in the third column? | 108 | 0 | 8,192 | -1 | 8,192 | |
Let set $A=\{(x,y) \,|\, |x|+|y| \leq 2\}$, and $B=\{(x,y) \in A \,|\, y \leq x^2\}$. Calculate the probability that a randomly selected element $P(x,y)$ from set $A$ belongs to set $B$. | \frac {17}{24} | 0 | 6,475.4375 | -1 | 6,475.4375 | |
How many natural numbers from 1 to 700, inclusive, contain the digit 6 at least once? | 133 | 0.0625 | 7,968.8125 | 7,594 | 7,993.8 | |
Let $a$, $b$, and $c$ be positive real numbers such that $abc = 4$. Find the minimum value of
\[(3a + b)(2b + 3c)(ac + 4).\] | 384 | 0 | 8,192 | -1 | 8,192 | |
The Fibonacci numbers are defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$ for $n \geq 1$. The Lucas numbers are defined by $L_{1}=1, L_{2}=2$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n \geq 1$. Calculate $\frac{\prod_{n=1}^{15} \frac{F_{2 n}}{F_{n}}}{\prod_{n=1}^{13} L_{n}}$. | 1149852 | It is easy to show that $L_{n}=\frac{F_{2 n}}{F_{n}}$, so the product above is $L_{1} 4 L_{1} 5=843$. $1364=1149852$. | 0 | 6,922.25 | -1 | 6,922.25 |
Natural numbers \( a, b, c \) are such that \( 1 \leqslant a < b < c \leqslant 3000 \). Find the largest possible value of the quantity
$$
\gcd(a, b) + \gcd(b, c) + \gcd(c, a)
$$ | 3000 | 0.1875 | 8,122.8125 | 7,823 | 8,192 | |
A manager schedules a consultation session at a café with two assistant managers but forgets to set a specific time. All three aim to arrive randomly between 2:00 and 5:00 p.m. The manager, upon arriving, will leave if both assistant managers are not present. Each assistant manager is prepared to wait for 1.5 hours for... | \frac{1}{4} | 0 | 7,706.3125 | -1 | 7,706.3125 | |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $\begin{cases} x=t^{2} \\ y=2t \end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho=5\cos \... | 20 | 0.6875 | 5,814.8125 | 5,088.545455 | 7,412.6 | |
Find the maximum value of
\[\frac{x + 2y + 3}{\sqrt{x^2 + y^2 + 1}}\]over all real numbers $x$ and $y.$ | \sqrt{14} | 0.25 | 7,387.625 | 4,974.5 | 8,192 | |
Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$. | 2001 | 0.0625 | 7,808.5 | 6,358 | 7,905.2 | |
The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwic... | 111 | 0.125 | 6,621.875 | 6,426 | 6,649.857143 | |
There exists a scalar $c$ so that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) = c \mathbf{v}\]for all vectors $\mathbf{v}.$ Find $c.$ | 2 | 0.1875 | 7,298.5625 | 3,427 | 8,192 | |
A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: | 12 | 1. **Understanding the Problem**: We are given a regular polygon with $n$ sides inscribed in a circle of radius $R$. The area of the polygon is given as $3R^2$. We need to find the value of $n$.
2. **Area of a Regular Polygon**: The area $A$ of a regular polygon inscribed in a circle can be calculated using the formul... | 1 | 3,366.875 | 3,366.875 | -1 |
Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$. | \frac{\sqrt{3}}{3} | 0 | 6,790.8125 | -1 | 6,790.8125 | |
If $\log_9 (x-2)=\frac{1}{2}$, find $\log_{625} x$. | \frac14 | 1 | 1,739.5625 | 1,739.5625 | -1 | |
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\] | {f(n)=n} |
We are tasked with finding all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) that satisfy the condition:
\[
f(n+1) > f(f(n)), \quad \forall n \in \mathbb{N}.
\]
To solve this problem, let us first analyze the condition given:
\[
f(n+1) > f(f(n)).
\]
This inequality implies that the function \( f \) must order ... | 0 | 7,836.8125 | -1 | 7,836.8125 |
One yuan, two yuan, five yuan, and ten yuan RMB notes, each one piece, can form a total of \_\_\_\_\_ different denominations. (Fill in the number) | 15 | 0.5625 | 629.3125 | 669.444444 | 577.714286 | |
In terms of $\pi$, what is the area of the circle defined by the equation $2x^2+2y^2+10x-6y-18=0$? | \frac{35}{2} \pi | 1 | 2,726.4375 | 2,726.4375 | -1 | |
Let $f(x)=|2\{x\}-1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \[nf(xf(x))=x\]has at least $2012$ real solutions. What is $n$?
Note: the fractional part of $x$ is a real number $y=\{x\}$ such that $0\le y<1$ and $x-y$ is an integer. | 32 | 0 | 8,091.75 | -1 | 8,091.75 | |
In a contest with 5 participants, there were several questions. For each question, one participant gave an incorrect answer while the others answered correctly. Petya gave 10 correct answers, which is fewer than any other participant. Vasya gave 13 correct answers, which is more than any other participant. How many que... | 14 | 0.4375 | 6,018.6875 | 3,566.857143 | 7,925.666667 | |
Find the largest positive integer $n$ such that there exist $n$ distinct positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=2017.
$$ | 16 | 0 | 8,040.375 | -1 | 8,040.375 | |
Given that $S_{n}=\sin \frac{\pi}{7}+\sin \frac{2\pi}{7}+\cdots+\sin \frac{n\pi}{7}$, determine the number of positive values in the sequence $S_{1}$, $S_{2}$, …, $S_{100}$. | 86 | 0 | 8,192 | -1 | 8,192 | |
Find all solutions $x$ (real and otherwise) to the equation
\[x^4+64=0.\]Enter all the solutions, separated by commas. | 2+2i,\,-2-2i,\,-2+2i,\,2-2i | 0 | 4,619.0625 | -1 | 4,619.0625 | |
The sum of a positive number and its square is 156. What is the number? | 12 | 1 | 2,128 | 2,128 | -1 | |
In a simulation experiment, 20 groups of random numbers were generated: 6830, 3013, 7055, 7430, 7740, 4422, 7884, 2604, 3346, 0952, 6807, 9706, 5774, 5725, 6576, 5929, 9768, 6071, 9138, 6754. If the numbers 1, 2, 3, 4, 5, 6 each appear exactly three times among these, it represents hitting the target exactly three time... | 25\% | 0 | 7,288.5 | -1 | 7,288.5 | |
Find four-thirds of $\frac{9}{2}$. | 6 | 1 | 1,369.875 | 1,369.875 | -1 | |
Find the smallest whole number that is larger than the sum
\[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\] | 16 |
1. **Break down the mixed numbers**:
Each term in the sum \(2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}\) can be separated into its integer and fractional parts:
\[
2\dfrac{1}{2} = 2 + \dfrac{1}{2}, \quad 3\dfrac{1}{3} = 3 + \dfrac{1}{3}, \quad 4\dfrac{1}{4} = 4 + \dfrac{1}{4}, \quad 5\dfrac{1}{5... | 1 | 4,245.75 | 4,245.75 | -1 |
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens,... | 37 |
We will analyze the game by observing the token distribution and the net change in tokens over a set of rounds. We will use the first solution approach as it provides a clear and concise method to determine the number of rounds.
#### Step 1: Understanding the token distribution and net change per round
In each round,... | 0.0625 | 8,192 | 8,192 | 8,192 |
A bus at a certain station departs punctually at 7:00 and 7:30 in the morning. Student Xiao Ming arrives at the station to catch the bus between 6:50 and 7:30, and his arrival time is random. The probability that he waits for less than 10 minutes for the bus is ______. | \frac{1}{2} | 0.1875 | 7,789.25 | 7,209 | 7,923.153846 | |
An odd six-digit number is called "just cool" if it consists of digits that are prime numbers, and no two identical digits are adjacent. How many "just cool" numbers exist? | 729 | 0 | 7,755.375 | -1 | 7,755.375 | |
A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units? | 56 | 0.9375 | 2,123.875 | 1,719.333333 | 8,192 | |
Suppose \( x \) and \( y \) are integers such that \( xy + 3x + 2y = -4 \). Find the greatest possible value of \( y \). | -1 | 0.9375 | 4,986.5625 | 4,772.866667 | 8,192 | |
The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Find $d$ to the nearest tenth. | 0.3 | 0.25 | 7,641.0625 | 5,988.25 | 8,192 | |
Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. | \frac{250}{7} | 0 | 8,192 | -1 | 8,192 | |
The shortest distance from a point on the parabola $x^2=y$ to the line $y=2x+m$ is $\sqrt{5}$. Find the value of $m$. | -6 | 0.1875 | 7,725.875 | 5,988 | 8,126.923077 | |
Solve for $x$: $4x^{1/3}-2 \cdot \frac{x}{x^{2/3}}=7+\sqrt[3]{x}$. | 343 | 1 | 2,246.9375 | 2,246.9375 | -1 | |
Let \( x \neq y \), and the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. Then \(\frac{b_{4}-b_{3}}{a_{2}-a_{1}}\) equals $\qquad$. | 2.6666666666666665 | 0 | 3,602.25 | -1 | 3,602.25 | |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 0.3125 | 7,945.1875 | 7,402.2 | 8,192 | |
(1) Calculate the value of $(\frac{2}{3})^{0}+3\times(\frac{9}{4})^{{-\frac{1}{2}}}+(\log 4+\log 25)$.
(2) Given $\alpha \in (0,\frac{\pi }{2})$, and $2\sin^{2}\alpha - \sin \alpha \cdot \cos \alpha - 3\cos^{2}\alpha = 0$, find the value of $\frac{\sin \left( \alpha + \frac{\pi }{4} \right)}{\sin 2\alpha + \cos 2\alph... | \frac{9}{4} | 0 | 8,192 | -1 | 8,192 | |
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? | \frac{3}{10} | To find the probability that 4 is the largest value selected, we need to consider two scenarios:
1. The number 5 is not selected.
2. The numbers selected are all less than or equal to 4.
#### Step 1: Calculate the probability of not selecting 5.
The probability that the first card selected is not 5 is $\frac{4}{5}$ (s... | 0.9375 | 3,225.25 | 2,894.133333 | 8,192 |
Given that $17^{-1} \equiv 11 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 42 | 0 | 7,150.875 | -1 | 7,150.875 | |
Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$ 0.50 $, and every century thereafter has a $ 50\% $ chance of doubling his money and a $ 50\% $ chance of having his money reset to $ \ $0.50$ . What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries... | 13 | 0.375 | 3,886.5625 | 3,891.666667 | 3,883.5 | |
In the diagram, $AB$ is a line segment. What is the value of $x$?
[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ... | 38 | 0.1875 | 7,039.875 | 4,712.333333 | 7,577 | |
Let $ABC$ be a triangle with $\angle BAC=117^\circ$ . The angle bisector of $\angle ABC$ intersects side $AC$ at $D$ . Suppose $\triangle ABD\sim\triangle ACB$ . Compute the measure of $\angle ABC$ , in degrees. | 42 | 0.4375 | 7,114.25 | 5,728.571429 | 8,192 | |
Let \( M = \{1, 2, \cdots, 10\} \), and \( A_1, A_2, \cdots, A_n \) be distinct non-empty subsets of \( M \). If \(i \neq j\), then \( A_i \cap A_j \) can have at most two elements. Find the maximum value of \( n \). | 175 | 0 | 8,192 | -1 | 8,192 | |
On the board, there are \( n \) different integers, each pair of which differs by at least 10. The sum of the squares of the three largest among them is less than three million. The sum of the squares of the three smallest among them is also less than three million. What is the greatest possible \( n \)? | 202 | 0 | 8,192 | -1 | 8,192 | |
Given that $x$ and $y$ are positive numbers satisfying the equation $xy = \frac{x-y}{x+3y}$, find the maximum value of $y$. | \frac{1}{3} | 0.625 | 6,057.9375 | 4,777.5 | 8,192 | |
Kaleb defined a $\emph{clever integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 9. What fraction of all clever integers is divisible by 27? Express your answer as a common fraction. | \frac{2}{5} | 0.9375 | 4,448.9375 | 4,395.333333 | 5,253 | |
Let's call a natural number a "snail" if its representation consists of the representations of three consecutive natural numbers, concatenated in some order: for example, 312 or 121413. "Snail" numbers can sometimes be squares of natural numbers: for example, $324=18^{2}$ or $576=24^{2}$. Find a four-digit "snail" numb... | 1089 | 0 | 8,192 | -1 | 8,192 | |
If $x, y$, and $y-\frac{1}{x}$ are not $0$, then $\frac{x-\frac{1}{y}}{y-\frac{1}{x}}$ equals | \frac{x}{y} | 1. **Start with the given expression:**
\[
\frac{x-\frac{1}{y}}{y-\frac{1}{x}}
\]
2. **Multiply the numerator and the denominator by $xy$ to eliminate the fractions:**
\[
\frac{x-\frac{1}{y}}{y-\frac{1}{x}} \cdot \frac{xy}{xy} = \frac{(x-\frac{1}{y})xy}{(y-\frac{1}{x})xy}
\]
3. **Simplify the expres... | 1 | 2,408.875 | 2,408.875 | -1 |
Let $a, b$ be real numbers. If the complex number $\frac{1+2i}{a+bi} \= 1+i$, then $a=\_\_\_\_$ and $b=\_\_\_\_$. | \frac{1}{2} | 1 | 2,208.1875 | 2,208.1875 | -1 | |
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tre... | 140 | 0.1875 | 5,864.8125 | 3,976.333333 | 6,300.615385 | |
The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what? | 4\sqrt{3}\pi | 0.0625 | 8,059.3125 | 6,370 | 8,171.933333 |
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