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 |
|---|---|---|---|---|---|---|
Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$. | 734 | Note that, since $[x+n]=[x]+n$ for any integer $n$, $$f(x+3)=[x+3]+[2(x+3)]+\left[\frac{5(x+3)}{3}\right]+[3(x+3)]+[4(x+3)]=f(x)+35$$ one only needs to investigate the interval $[0,3)$. The numbers in this interval at which at least one of the real numbers $x, 2 x, \frac{5 x}{3}, 3 x, 4 x$ is an integer are - $0,1,2$ f... | 0 | 8,192 | -1 | 8,192 |
In the arithmetic sequence $\{a_n\}$, we have $a_2=4$, and $a_4+a_7=15$.
(Ⅰ) Find the general term formula for the sequence $\{a_n\}$.
(Ⅱ) Let $b_n= \frac{1}{a_n a_{n+1}}$, calculate the value of $b_1+b_2+b_3+\dots+b_{10}$. | \frac{10}{39} | 1 | 2,556.3125 | 2,556.3125 | -1 | |
Define a function \( f \) on the set of positive integers \( N \) as follows:
(i) \( f(1) = 1 \), \( f(3) = 3 \);
(ii) For \( n \in N \), the function satisfies
\[
\begin{aligned}
&f(2n) = f(n), \\
&f(4n+1) = 2f(2n+1) - f(n), \\
&f(4n+3) = 3f(2n+1) - 2f(n).
\end{aligned}
\]
Find all \( n \) such that \( n \leqslant 1... | 92 | 0 | 8,192 | -1 | 8,192 | |
In $\Delta ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin C=\sqrt{3}c\cos A$.
1. Find the size of angle $A$.
2. If $a=\sqrt{13}$ and $c=3$, find the area of $\Delta ABC$. | 3\sqrt{3} | 0.9375 | 4,129.5 | 3,858.666667 | 8,192 | |
Given a stalk of bamboo with nine sections, with three sections from the bottom holding 3.9 liters, and the four sections from the top holding three liters, determine the combined volume of the middle two sections. | 2.1 | 0.0625 | 500.4375 | 440 | 504.466667 | |
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulti... | 100 | 0 | 8,192 | -1 | 8,192 | |
If $n$ is an integer, what is the remainder when the sum of $7 - n$ and $n + 3$ is divided by $7$? | 3 | 1 | 1,604 | 1,604 | -1 | |
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$. | \boxed{987^2+1597^2} |
We are tasked with finding the maximum value of \( m^2 + n^2 \), where \( m \) and \( n \) are integers within the range \( 1, 2, \ldots, 1981 \), satisfying the equation:
\[
(n^2 - mn - m^2)^2 = 1.
\]
### Step 1: Analyze the Equation
The equation given is a Pell-like equation. Simplifying, we have:
\[
n^2 - mn - ... | 0 | 8,192 | -1 | 8,192 |
For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ a point $P$ on the ellipse makes lines connecting it to the two foci $F1$ and $F2$ perpendicular to each other. Then, the area of $\triangle PF1F2$ is ________. | 12 | 0 | 6,749.125 | -1 | 6,749.125 | |
If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\overrightarrow{a}=\left(0,-2,-1\right)$ and $\overrightarrow{b}=\left(2,0,4\right)$, calculate the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$. | \frac{2}{5} | 0.8125 | 2,487.75 | 2,685 | 1,633 | |
Conduct an experiment by throwing 2 dice, and denote the point P with the coordinates $(x, y)$, where $x$ represents the number shown on the first die, and $y$ represents the number shown on the second die.
(I) Find the probability that point P lies on the line $y = x$.
(II) Find the probability that point P does n... | \frac{7}{36} | 0.5625 | 5,704.875 | 4,464.444444 | 7,299.714286 | |
Find the least common multiple of 36 and 132. | 396 | 0.9375 | 2,463.4375 | 2,081.533333 | 8,192 | |
Given the sequence $\left\{ a_n \right\}$ such that $a_{n+1}+a_n={(-1)}^n\cdot n$ ($n\in \mathbb{N}^*$), find the sum of the first 20 terms of $\left\{ a_n \right\}$. | -100 | 0.0625 | 8,153.5625 | 7,577 | 8,192 | |
Calculate:<br/>$(1)(\sqrt{50}-\sqrt{8})÷\sqrt{2}$;<br/>$(2)\sqrt{\frac{3}{4}}×\sqrt{2\frac{2}{3}}$. | \sqrt{2} | 1 | 1,015.75 | 1,015.75 | -1 | |
A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$. | 2.7 | 0.0625 | 6,699.8125 | 5,578 | 6,774.6 | |
What is the minimum value of $y$ if $y=3x^2+6x+9?$ | 6 | 1 | 2,223.4375 | 2,223.4375 | -1 | |
Let \(a,\) \(b,\) \(c\) be distinct real numbers such that
\[\frac{a}{1 + b} = \frac{b}{1 + c} = \frac{c}{1 + a} = k.\] Find the product of all possible values of \(k.\) | -1 | 0 | 7,689.5 | -1 | 7,689.5 | |
A sign painter paints numbers for each of 150 houses, numbered consecutively from 1 to 150. How many times does the digit 9 appear in total on all the house numbers? | 25 | 0.375 | 6,211.5 | 5,606.166667 | 6,574.7 | |
Given that $a$, $b$, $c$, $d$, $e$, and $f$ are all positive numbers, and $\frac{bcdef}{a}=\frac{1}{2}$, $\frac{acdef}{b}=\frac{1}{4}$, $\frac{abdef}{c}=\frac{1}{8}$, $\frac{abcef}{d}=2$, $\frac{abcdf}{e}=4$, $\frac{abcde}{f}=8$, find $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}$. | \frac{119}{8} | 0.625 | 5,909.125 | 4,539.4 | 8,192 | |
Given the sets
$$
\begin{array}{c}
M=\{x, xy, \lg (xy)\} \\
N=\{0, |x|, y\},
\end{array}
$$
and that \( M = N \), determine 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 | 6,161.5 | 6,254.5 | 6,148.214286 | |
Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$ | \frac{5 \sqrt{3}}{3} | First we find the direction of a line perpendicular to both of these lines. By taking the cross product $(2,3,1) \times(-1,1,2)=(5,-5,5)$ we find that the plane $x-y+z+3=0$ contains the first line and is parallel to the second. Now we take a point on the second line, say the point $(3,0,-1)$ and find the distance betwe... | 0 | 2,929.5625 | -1 | 2,929.5625 |
Medians are drawn from point $A$ and point $B$ in this right triangle to divide segments $\overline{BC}$ and $\overline{AC}$ in half, respectively. The lengths of the medians are 6 and $2\sqrt{11}$ units, respectively. How many units are in the length of segment $\overline{AB}$?
[asy]
draw((0,0)--(7,0)--(0,4)--(0,0)--... | 8 | 0.9375 | 4,182.5 | 3,915.2 | 8,192 | |
Out of sixteen Easter eggs, three are red. Ten eggs were placed in a larger box and six in a smaller box at random. What is the probability that both boxes contain at least one red egg? | 3/4 | 0.8125 | 6,324.75 | 5,893.846154 | 8,192 | |
Stan has a stack of 100 blocks and starts with a score of 0, and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.)... | 4950 | Let $E(n)$ be the expected value of the score for an $n$-block game. It suffices to show that the score is invariant regardless of how the game is played. We proceed by induction. We have $E(1)=0$ and $E(2)=1$. We require that $E(n)=E(n-k)+E(k)+(n-k) k$ for all $k$. Setting $k=1$, we hypothesize that $E(n)=n(n-1) / 2$.... | 0.0625 | 7,707.3125 | 4,672 | 7,909.666667 |
Given that \( f(x) \) is a polynomial of degree \( n \) with non-negative integer coefficients, and that \( f(1)=6 \) and \( f(7)=3438 \), find \( f(2) \). | 43 | 0.8125 | 4,310.75 | 3,833.846154 | 6,377.333333 | |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated. | 18185 | 0.375 | 6,738.9375 | 5,179.333333 | 7,674.7 | |
In triangle $ABC$, $AB=2$, $AC=5$, and $\cos A= \frac {4}{5}$. A point $P$ is chosen randomly inside triangle $ABC$. The probability that the area of triangle $PAB$ is greater than $1$ and less than or equal to $2$ is __________. | \frac {1}{3} | 0.5 | 7,959 | 7,726 | 8,192 | |
Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 21 through 50 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 20 and the tile from box $D$ is either odd or greater than 40? Express your ans... | \frac{19}{45} | 1 | 2,400.375 | 2,400.375 | -1 | |
What is the minimum value of the expression $x^2+y^2-6x+4y+18$ for real $x$ and $y$? | 5 | 1 | 1,714.5625 | 1,714.5625 | -1 | |
Find the number of permutations \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \), such that \( a_{i+1} \) is not less than \( a_i - 1 \) for \( i = 1, 2, \ldots, 9 \). | 512 | 0 | 8,192 | -1 | 8,192 | |
If $\log_{10}{m}= b-\log_{10}{n}$, then $m=$ | \frac{10^{b}}{n} | 1. Start by expressing $b$ in terms of logarithm base 10:
\[
b = \log_{10}{10^b}
\]
This follows from the property of logarithms that $\log_b{b^x} = x$.
2. Substitute this expression for $b$ into the given equation:
\[
\log_{10}{m} = \log_{10}{10^b} - \log_{10}{n}
\]
3. Apply the logarithmic prop... | 0 | 1,834.4375 | -1 | 1,834.4375 |
What is the perimeter of the figure shown if $x=3$? | 23 | Since $x=3$, the side lengths of the figure are $4,3,6$, and 10. Thus, the perimeter of the figure is $4+3+6+10=23$. (Alternatively, the perimeter is $x+6+10+(x+1)=2x+17$. When $x=3$, this equals $2(3)+17$ or 23.) | 0 | 5,756.875 | -1 | 5,756.875 |
The numbers \(2, 3, 4, 5, 6, 7, 8\) are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done? | 12 | 0 | 7,752.125 | -1 | 7,752.125 | |
There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum? | 2005 | 0 | 8,099.5625 | -1 | 8,099.5625 | |
Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle? | $\left[\frac{\sqrt{3}}{3}, 1\right]$ | Say we have four points $(a, f(a)),(b, f(b)),(c, f(c)),(d, f(d))$ on the curve which form a rectangle. If we interpolate a cubic through these points, that cubic will be symmetric around the center of the rectangle. But the unique cubic through the four points is $f(x)$, and $f(x)$ has only one point of symmetry, the p... | 0 | 8,184.375 | -1 | 8,184.375 |
Determine the value of the following expression:
$$
\left\lfloor\frac{11}{2010}\right\rfloor+\left\lfloor\frac{11 \times 2}{2010}\right\rfloor+\left\lfloor\frac{11 \times 3}{2010}\right\rfloor+\\left\lfloor\frac{11 \times 4}{2010}\right\rfloor+\cdots+\left\lfloor\frac{11 \times 2009}{2010}\right\rfloor,
$$
where \(\lfl... | 10045 | 0.6875 | 5,659.5625 | 4,508.454545 | 8,192 | |
In a round-robin chess tournament, only grandmasters and masters participated. The number of masters was three times the number of grandmasters, and the total points scored by the masters was 1.2 times the total points scored by the grandmasters.
How many people participated in the tournament? What can be said about t... | 12 | 0.0625 | 8,141.875 | 7,390 | 8,192 | |
What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$? | 112 | Notice that in order for $k$ to be unique, $\frac{n}{n + k+ 1} \le \frac{8}{15}$ and $\frac{n}{n+ k-1} \ge \frac{7}{13}$ must be true. Solving these inequalities for $k$ yields $\frac{7}{6}(k-1) \le n \le \frac{8}{7}(k+1)$.
Thus, we want to find $k$ such that $\frac{7}{6}(k-1)\le \frac{8}{7}(k+1)$. Solving this inequa... | 0.0625 | 8,005.875 | 5,879 | 8,147.666667 |
A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it was found that for every $1$ yuan reducti... | 20 | 0.8125 | 4,820 | 4,567.615385 | 5,913.666667 | |
The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. Now, there is a wire 144cm long, and it needs to be cut into $n$ small segments ($n>2$), each segment being no less than 1cm in length. If any three of these... | 10 | 0.0625 | 7,748.9375 | 6,722 | 7,817.4 | |
Let $f(x)=x^{3}+3 x-1$ have roots $a, b, c$. Given that $$\frac{1}{a^{3}+b^{3}}+\frac{1}{b^{3}+c^{3}}+\frac{1}{c^{3}+a^{3}}$$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$, find $100 m+n$. | 3989 | We know that $a^{3}=-3 a+1$ and similarly for $b, c$, so $$\frac{1}{a^{3}+b^{3}}=\frac{1}{2-3 a-3 b}=\frac{1}{2+3 c}=\frac{1}{3(2 / 3+c)}$$ Now, $$f(x-2 / 3)=x^{3}-2 x^{2}+\frac{13}{3} x-\frac{89}{27}$$ has roots $a+2 / 3, b+2 / 3$, and $c+2 / 3$. Thus the answer is, by Vieta's formulas, $$\frac{1}{3} \frac{(a+2 / 3)(b... | 0.625 | 6,421.875 | 5,359.8 | 8,192 |
Some students are required to stand in lines: On June 1st, each column has 15 people; on June 2nd, everyone stands in one column; on June 3rd, each column has one person; on June 4th, each column has 6 people; and so on until June 12th, with a different number of people per column each day. However, from June 13th onwa... | 60 | 0 | 8,183.0625 | -1 | 8,183.0625 | |
Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group at two separate circular tables? | 1247400 | 0.125 | 7,915.125 | 6,737.5 | 8,083.357143 | |
How many ten-digit numbers exist in which there are at least two identical digits? | 8996734080 | 1 | 4,232.3125 | 4,232.3125 | -1 | |
The line \(y = M\) intersects the graph of the function \(y = x^{3} - 84x\) at points with abscissas \(a\), \(b\), and \(c\) (\(a < b < c\)). It is given that the distance between \(a\) and \(b\) is half the distance between \(b\) and \(c\). Find \(M\). | 160 | 0.875 | 4,450.625 | 3,916.142857 | 8,192 | |
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done? | 144 | 0 | 8,192 | -1 | 8,192 | |
If the inequality \( ab + b^2 + c^2 \geq \lambda(a + b)c \) holds for all positive real numbers \( a, b, c \) that satisfy \( b + c \geq a \), then the maximum value of the real number \( \lambda \) is \(\quad\) . | \sqrt{2} - \frac{1}{2} | 0.125 | 8,140.0625 | 7,776.5 | 8,192 | |
Let $C$ be the circle with equation $x^2+2y-9=-y^2+18x+9$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$? | 18 | 1 | 3,239.0625 | 3,239.0625 | -1 | |
Given points \( A(3,1) \) and \( B\left(\frac{5}{3}, 2\right) \), and the four vertices of quadrilateral \( \square ABCD \) are on the graph of the function \( f(x)=\log _{2} \frac{a x+b}{x-1} \), find the area of \( \square ABCD \). | \frac{26}{3} | 0.1875 | 8,097 | 7,685.333333 | 8,192 | |
Find the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the part of the surface \(z = x^2 + y^2\), cut off by the plane \(z=2\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid. | -2\pi | 0 | 8,038.1875 | -1 | 8,038.1875 | |
Find the sum $$\frac{2^1}{4^1 - 1} + \frac{2^2}{4^2 - 1} + \frac{2^4}{4^4 - 1} + \frac{2^8}{4^8 - 1} + \cdots.$$ | 1 | 0.3125 | 7,086.4375 | 5,445.4 | 7,832.363636 | |
The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor? | 125,875,000 | To find the fifth-largest divisor of $2,014,000,000$, we first need to factorize the number.
1. **Prime Factorization of $2,014,000,000$:**
\[
2,014,000,000 = 2014 \times 10^6 = (2 \times 19 \times 53) \times (10^6) = 2 \times 19 \times 53 \times (2 \times 5)^6 = 2^7 \times 5^6 \times 19 \times 53
\]
2. **Id... | 0 | 7,806.25 | -1 | 7,806.25 |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases}x=2+t \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases}x=-2+m \\ y= \frac{m}{k}\end{cases}$ (where $m$ is... | \sqrt{5} | 0.625 | 5,645.5 | 4,380.4 | 7,754 | |
Simplify the expression \(1.6 \frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}}\) given that \(a = 7.4\) and \(b = \frac{5}{37}\). | 1.6 | 0.75 | 5,883.875 | 5,249.25 | 7,787.75 | |
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the poin... | 25 | 0 | 7,935.9375 | -1 | 7,935.9375 | |
At the MP Donut Hole Factory, Niraek, Theo, and Akshaj are coating spherical donut holes in powdered sugar. Niraek's donut holes have radius 6 mm, Theo's donut holes have radius 8 mm, and Akshaj's donut holes have radius 10 mm. All three workers coat the surface of the donut holes at the same rate and start at the sa... | 400 | 0.9375 | 4,414.9375 | 4,163.133333 | 8,192 | |
Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the grain in Warehouse A will be six times tha... | 153 | 0.75 | 5,544.5 | 4,662 | 8,192 | |
Compute $\tan 75^\circ.$ | 2 + \sqrt{3} | 1 | 3,085.9375 | 3,085.9375 | -1 | |
For the function $f(x)=4\sin \left(2x+\frac{\pi }{3}\right)$, the following propositions are given:
$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\cos \left(2x-\frac{\pi }{6}\right)$; $(3)$ The graph of $f(x)$ ... | (2)(3) | 0 | 4,769.1875 | -1 | 4,769.1875 | |
A noodle shop offers classic specialty noodles to customers, who can either dine in at the shop (referred to as "dine-in" noodles) or purchase packaged fresh noodles with condiments (referred to as "fresh" noodles). It is known that the total price of 3 portions of "dine-in" noodles and 2 portions of "fresh" noodles is... | \frac{40}{9} | 0.1875 | 7,771.375 | 5,948.666667 | 8,192 | |
How many four-digit positive integers $y$ satisfy $5678y + 123 \equiv 890 \pmod{29}$? | 310 | 0.3125 | 4,975.3125 | 3,906.6 | 5,461.090909 | |
Given the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$$ (a > 0, b > 0), a circle with center at point (b, 0) and radius a is drawn. The circle intersects with one of the asymptotes of the hyperbola at points M and N, and ∠MPN = 90°. Calculate the eccentricity of the hyperbola. | \sqrt{2} | 0.5625 | 7,478.3125 | 6,931.888889 | 8,180.857143 | |
A certain school randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:... | 86 | 0.125 | 7,615.8125 | 7,801.5 | 7,589.285714 | |
How many even integers are there between $\frac{9}{2}$ and $\frac{24}{1}$? | 10 | 0.1875 | 4,175.6875 | 5,328 | 3,909.769231 | |
In a dark room, a drawer contains 120 red socks, 100 green socks, 70 blue socks, and 50 black socks. A person selects socks one by one from the drawer without being able to see their color. What is the minimum number of socks that must be selected to ensure that at least 15 pairs of socks are selected, with no sock bei... | 33 | 0.3125 | 7,483.6875 | 5,925.4 | 8,192 | |
A deck of 60 cards, divided into 5 suits of 12 cards each, is shuffled. In how many ways can we pick three different cards in sequence? (Order matters, so picking card A, then card B, then card C is different from picking card B, then card A, then card C.) | 205320 | 1 | 2,129.625 | 2,129.625 | -1 | |
Calculate the perimeter of a triangle whose vertices are located at points $A(2,3)$, $B(2,10)$, and $C(8,6)$ on a Cartesian coordinate plane. | 7 + 2\sqrt{13} + 3\sqrt{5} | 0.9375 | 2,863.25 | 2,957 | 1,457 | |
Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form. | 12\sqrt{3} | 0.4375 | 2,179.375 | 2,310.571429 | 2,077.333333 | |
Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$.
$(1)$ Find the equation of circle $C$.
$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$. | \frac{4}{3} | 0.9375 | 5,412.0625 | 5,246.8 | 7,891 | |
It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is: | 26 | 1. **Identify the nature of the roots:** Given that one root of the quadratic equation $2x^2 + rx + s = 0$ is $3+2i$, and knowing that the coefficients $r$ and $s$ are real numbers, the other root must be the complex conjugate of $3+2i$. Therefore, the other root is $3-2i$.
2. **Use Vieta's formulas:** Vieta's formula... | 1 | 1,595.875 | 1,595.875 | -1 |
Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\). | 2 - \sqrt{2} | 0.625 | 6,995.4375 | 6,552.4 | 7,733.833333 | |
What is the area enclosed by the graph of $|3x|+|4y|=12$? | 24 | 0.875 | 3,746.125 | 3,111 | 8,192 | |
Given $f(x)=2x^3-6x^2+a$ (where $a$ is a constant), the function has a maximum value of 3 on the interval $[-2, 2]$. Find the minimum value of $f(x)$ on the interval $[-2, 2]$. | -29 | 0.125 | 3,138.1875 | 2,295.5 | 3,258.571429 | |
Let the sequence $(a_n)$ be defined as $a_1=\frac{1}{3}$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac{1}{50}$, find the value of $k$. | 193 | 0.3125 | 7,066.125 | 4,589.2 | 8,192 | |
An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward? | 96 | 0 | 8,164.625 | -1 | 8,164.625 | |
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \). | 2006 | 0 | 8,192 | -1 | 8,192 | |
A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
... | 1720 | 0.0625 | 7,969.1875 | 5,255 | 8,150.133333 | |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2017,0),(2017,2018),$ and $(0,2018)$. What is the probability that $x > 9y$? Express your answer as a common fraction. | \frac{2017}{36324} | 0.3125 | 6,811.9375 | 5,585.6 | 7,369.363636 | |
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | 100 | 1. **Understanding the Offer**: The store offers one free window for every four purchased. This means for every five windows considered (four purchased + one free), the cost is for four windows only.
2. **Calculating Individual Savings**:
- **Dave's Purchase**: Dave needs 7 windows.
- Without the offer, Dave w... | 0.5625 | 5,669.8125 | 4,900.777778 | 6,658.571429 |
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. | 29 | Obviously, all of the terms must be odd. The common difference between the terms cannot be $2$ or $4$, since otherwise there would be a number in the sequence that is divisible by $3$. However, if the common difference is $6$, we find that $5,11,17,23$, and $29$ form an arithmetic sequence. Thus, the answer is $\boxed{... | 0.4375 | 7,528.625 | 6,675.714286 | 8,192 |
Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When ... | 114 | Just like in solution 1, we note that all colors will be used equally, except for the pink stripe. This must bring red and white down to $130$ each, so $34$ red and $58$ white are used, making for a total of $92$ for the pink stripe. Thus, the other stripes also use $92$. The answer is $130+164+188-4(92)=\boxed{114}$ | 0.8125 | 3,891.625 | 2,899.230769 | 8,192 |
Let $\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$.
| 927 | 0.25 | 7,858.9375 | 6,859.75 | 8,192 | |
If $A$ is an angle such that $\tan A + \sec A = 2,$ enter all possible values of $\cos A,$ separated by commas. | \frac{4}{5} | 1 | 3,660.25 | 3,660.25 | -1 | |
How many four-digit positive integers are multiples of 3? | 3000 | 0.9375 | 2,868.5625 | 2,513.666667 | 8,192 | |
In parallelogram \(ABCD\), point \(P\) is taken on side \(BC\) such that \(3PB = 2PC\), and point \(Q\) is taken on side \(CD\) such that \(4CQ = 5QD\). Find the ratio of the area of triangle \(APQ\) to the area of triangle \(PQC\). | 37/15 | 0.5625 | 7,217.5625 | 6,459.666667 | 8,192 | |
Find the matrix $\mathbf{M}$ that swaps the columns of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} b & a \\ d & c \end{pmatrix}.\]If no such matrix $\mathbf{M}$ exists, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} | 0.25 | 7,628.3125 | 6,479.25 | 8,011.333333 | |
If a four-digit number is called a "good number" when its unit digit is 1 and it has exactly three identical digits, then how many "good numbers" are there among the four-digit numbers formed by the digits 1, 2, 3, and 4 with repetitions? | 12 | 0.0625 | 7,731.625 | 6,623 | 7,805.533333 | |
Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally ... | 32 | 0.5 | 6,286.5 | 5,227.625 | 7,345.375 | |
There is a set of natural numbers (it is known that there are at least seven numbers) such that the sum of every seven of them is less than 15, and the sum of all numbers in the set is 100. What is the smallest number of numbers that can be in the set? | 50 | 0.1875 | 7,743.625 | 5,800.666667 | 8,192 | |
Calculate the difference between the sum of the first 2023 odd counting numbers and the sum of the first 2023 even counting numbers, where even numbers start from 2, and then multiply this difference by 3. | 9063 | 0 | 4,915.625 | -1 | 4,915.625 | |
Given the function \( f(x)=\frac{x^{3}}{1+x^{3}} \), find the value of the sum
\[
f\left(\frac{1}{1}\right)+f\left(\frac{2}{1}\right)+\ldots+f\left(\frac{2007}{1}\right) + f\left(\frac{1}{2}\right)+f\left(\frac{2}{2}\right)+\ldots+f\left(\frac{2007}{2}\right) + \ldots + f\left(\frac{1}{2007}\right)+f\left(\frac{2}{20... | \frac{2007^2}{2} | 0 | 8,144.9375 | -1 | 8,144.9375 | |
Given the quadratic function $f(x)=ax^{2}+bx+c$, where $a$, $b$, and $c$ are constants, if the solution set of the inequality $f(x) \geqslant 2ax+b$ is $\mathbb{R}$, find the maximum value of $\frac{b^{2}}{a^{2}+c^{2}}$. | 2\sqrt{2}-2 | 0.5625 | 6,704.125 | 5,962.111111 | 7,658.142857 | |
Given the function $f(x)=\sqrt{3}\sin\left(x+\frac{\pi}{4}\right)\sin\left(x-\frac{\pi}{4}\right)+\sin x\cos x$.
$(Ⅰ)$ Find the smallest positive period and the center of symmetry of $f(x)$;
$(Ⅱ)$ Given an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A... | 12 | 0.8125 | 6,996 | 6,720 | 8,192 | |
A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs. | \frac{3 \sqrt{15}}{2} | 0 | 8,192 | -1 | 8,192 | |
Find all rational roots of
\[4x^4 - 3x^3 - 13x^2 + 5x + 2 = 0\]Enter all the rational roots, separated by commas. | 2,-\frac{1}{4} | 0 | 4,710.0625 | -1 | 4,710.0625 | |
Regular decagon (10-sided polygon) \(A B C D E F G H I J\) has an area of 2017 square units. Determine the area (in square units) of the rectangle \(C D H I\). | 806.8 | 0 | 7,966.875 | -1 | 7,966.875 | |
The sum of the first $20$ positive even integers is also the sum of four consecutive even integers. What is the largest of these four integers? | 108 | 1 | 1,796.875 | 1,796.875 | -1 | |
Let $S$ be the set of all nonzero real numbers. The function $f : S \to S$ satisfies the following two properties:
(i) First,
\[f \left( \frac{1}{x} \right) = xf(x)\]for all $x \in S.$
(ii) Second,
\[f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right) = 1 + f \left( \frac{1}{x + y} \right)\]for all $x \in S... | 2 | 0.1875 | 7,881.75 | 6,537.333333 | 8,192 | |
How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.) | 120 | 1. **Identify the vowels and consonants in the word "CONTEST":**
- Vowels: O, E
- Consonants: C, N, T, S, T
2. **Calculate the number of ways to arrange the vowels:**
- There are 2 vowels, and they can be arranged in $2! = 2$ ways.
3. **Calculate the number of ways to arrange the consonants:**
- There are... | 0.9375 | 3,991.125 | 3,711.066667 | 8,192 |
Given a circle O with a radius of 6, the length of chord AB is 6.
(1) Find the size of the central angle α corresponding to chord AB;
(2) Find the arc length l and the area S of the sector where α is located. | 6\pi | 1 | 1,973.4375 | 1,973.4375 | -1 |
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