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 |
|---|---|---|---|---|---|---|
Liam has $x$ candies, Mia has three times as many candies as Liam, Noah has four times as many candies as Mia, and Olivia has six times the number of candies Noah has. If in total Liam, Mia, Noah, and Olivia have 468 candies, what is the value of $x?$ | \frac{117}{22} | 0.5 | 5,321.75 | 5,453.5 | 5,190 | |
Given a decreasing arithmetic sequence $\{a_n\}$, where $a_3 = -1$, and $a_1$, $a_4$, $-a_6$ form a geometric sequence. Find the value of $S_7$, where $S_n$ represents the sum of the first $n$ terms of $\{a_n\}$. | -14 | 0.9375 | 3,695 | 3,395.2 | 8,192 | |
Determine the periodicity of the following functions. If the function is periodic, find its smallest positive period:
(1) \( y = \tan x - \cot x \);
(2) \( y = \sin (\cos x) \);
(3) \( y = \sin x^{2} \). | 2\pi | 0.8125 | 5,045.75 | 4,396 | 7,861.333333 | |
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer. | 28 | 0 | 8,192 | -1 | 8,192 | |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity. | \sqrt{3} | 0.5625 | 2,529.8125 | 2,338.333333 | 2,776 | |
Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints. | 9\pi | 0.0625 | 8,157.8125 | 7,645 | 8,192 | |
Given that the coefficient of the second term in the expansion of $((x+2y)^{n})$ is $8$, find the sum of the coefficients of all terms in the expansion of $((1+x)+(1+x)^{2}+…+(1+x)^{n})$. | 30 | 0.9375 | 3,293.125 | 2,966.533333 | 8,192 | |
An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip? | \frac{3}{5} | Let $p$ be the probability of getting a head in one flip. There are 6 ways to get 2 heads and 2 tails, each with probability $p^{2}(1-p)^{2}$, and 4 ways to get 3 heads and 1 tail, each with probability $p^{3}(1-p)$. We are given that $6 p^{2}(1-p)^{2}=4 p^{3}(1-p)$. Clearly $p$ is not 0 or 1, so we can divide by $p^{2... | 1 | 1,829.625 | 1,829.625 | -1 |
In the trapezoid \(ABCD\), if \(AB = 8\), \(DC = 10\), the area of \(\triangle AMD\) is 10, and the area of \(\triangle BCM\) is 15, then the area of trapezoid \(ABCD\) is \(\quad\). | 45 | 0 | 8,192 | -1 | 8,192 | |
What is the value of the sum $\frac{3}{4}+\frac{3^2}{4^2}+\frac{3^3}{4^3}+ \ldots +\frac{3^{15}}{4^{15}}$? Express your answer as a common fraction. | \frac{3177884751}{1073741824} | 0 | 8,136.5 | -1 | 8,136.5 | |
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds: $\overline{ab}=3 \cdot \overline{cd} + 1$ . | 2809 | 0.0625 | 8,187.875 | 8,126 | 8,192 | |
Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15. | 46 | 0.75 | 5,266.9375 | 4,998.333333 | 6,072.75 | |
When 2007 bars of soap are packed into \( N \) boxes, where \( N \) is a positive integer, there is a remainder of 5. How many possible values of \( N \) are there? | 14 | 0.8125 | 4,848.5 | 4,076.923077 | 8,192 | |
Let \( g(x) = 10x + 5 \). Find the sum of all \( x \) that satisfy the equation \( g^{-1}(x) = g((3x)^{-2}) \). | 50 | 0 | 7,924.5625 | -1 | 7,924.5625 | |
Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch? | 1/2 | 0.0625 | 7,938.9375 | 4,427 | 8,173.066667 | |
In a class, there are 4 lessons in one morning, and each lesson needs a teacher to teach it. Now, from 6 teachers A, B, C, D, E, F, 4 teachers are to be arranged to teach one lesson each. The first lesson can only be taught by either A or B, and the fourth lesson can only be taught by either A or C. How many different ... | 36 | 0.75 | 5,909.3125 | 5,311.75 | 7,702 | |
The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \in (0, +\infty)$. The value of the real number $m$ is | -1 | 0 | 8,179.1875 | -1 | 8,179.1875 | |
Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53. | 22 | 0.0625 | 6,404.125 | 6,714 | 6,383.466667 | |
Let \(ABC\) be a triangle with \(AB=8, AC=12\), and \(BC=5\). Let \(M\) be the second intersection of the internal angle bisector of \(\angle BAC\) with the circumcircle of \(ABC\). Let \(\omega\) be the circle centered at \(M\) tangent to \(AB\) and \(AC\). The tangents to \(\omega\) from \(B\) and \(C\), other than \... | 16 | Redefine \(D\) as the reflection of \(A\) across the perpendicular bisector \(l\) of \(BC\). We prove that \(DB\) and \(DC\) are both tangent to \(\omega\), and hence the two definitions of \(D\) align. Indeed, this follows by symmetry; we have that \(\angle CBM=\angle CAM=\angle BAM=\angle BCM\), so \(BM=CM\) and so \... | 0 | 8,192 | -1 | 8,192 |
Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy $$x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5}$$ and $$x y z=\sqrt{3}+i \sqrt{5}.$$ If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then $$\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} ... | 1516 | From the conditions, it is clear that $a, b, c$ all have magnitude $\sqrt{2}$. Conjugating the first equation gives $2\left(\frac{a b+b c+c a}{a b c}\right)=-\frac{\sqrt{3}}{2}+i \sqrt{5}$, which means $a b+b c+c a=\left(-\frac{\sqrt{3}}{4}+i \frac{\sqrt{5}}{2}\right)(\sqrt{3}+i \sqrt{5})=\frac{-13+i \sqrt{15}}{4}$. Th... | 0 | 8,192 | -1 | 8,192 |
In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. ... | 8-4\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability t... | 76 | We first assume a population of $100$ to facilitate solving. Then we simply organize the statistics given into a Venn diagram.
[asy] pair A,B,C,D,E,F,G; A=(0,55); B=(60,55); C=(60,0); D=(0,0); draw(A--B--C--D--A); E=(30,35); F=(20,20); G=(40,20); draw(circle(E,15)); draw(circle(F,15)); draw(circle(G,15)); draw("$A$",(3... | 0.8125 | 3,604.3125 | 2,913.846154 | 6,596.333333 |
If the solution set of the inequality $tx^2-6x+t^2<0$ with respect to $x$ is $(-\infty, a) \cup (1, +\infty)$, then the value of $a$ is \_\_\_\_\_\_. | -3 | 0.6875 | 5,207.5625 | 3,851 | 8,192 | |
The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were randomly selected, with 5 students from eac... | \dfrac{2}{5} | 0.6875 | 3,979.4375 | 4,225.181818 | 3,438.8 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^2 + b^2 = 2017c^2$, calculate the value of $\frac{\tan C}{\tan A} + \frac{\tan C}{\tan B}$. | \frac{1}{1008} | 0.6875 | 5,754.6875 | 4,646.818182 | 8,192 | |
Let \( g(x) \) be the function defined on \(-2 \le x \le 2\) by the formula
\[ g(x) = 2 - \sqrt{4 - x^2}. \]
If a graph of \( x = g(y) \) is overlaid on the graph of \( y = g(x) \), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth? | 2.28 | 0.3125 | 7,728.9375 | 7,081.8 | 8,023.090909 | |
If $z=1+i$, then $|{iz+3\overline{z}}|=\_\_\_\_\_\_$. | 2\sqrt{2} | 0.9375 | 3,759.125 | 3,463.6 | 8,192 | |
In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and additionally ensure my bike key is next to my mailbox key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order by reflection or rota... | 24 | 0 | 8,082.375 | -1 | 8,082.375 | |
What is the tens digit of the smallest positive integer that is divisible by each of 20, 16, and 2016? | 8 | We note that $20=2^{2} \cdot 5$ and $16=2^{4}$ and $2016=16 \cdot 126=2^{5} \cdot 3^{2} \cdot 7$. For an integer to be divisible by each of $2^{2} \cdot 5$, $2^{4}$, and $2^{5} \cdot 3^{2} \cdot 7$, it must include at least 5 factors of 2, at least 2 factors of 3, at least 1 factor of 5, and at least 1 factor of 7. The... | 1 | 3,099.6875 | 3,099.6875 | -1 |
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | 1462 |
To solve this problem, we need to determine how many miles the car has actually traveled when the odometer reads 002005, given that it skips every occurrence of the digit 4.
#### Method 1: Counting Skipped Numbers
1. **Counting Numbers with Digit 4**:
- **Hundreds Place**: There are 200 numbers from 0000 to 1999 ... | 0.0625 | 7,175.875 | 6,112 | 7,246.8 |
In a hypothetical math competition, contestants are given the problem to find three distinct positive integers $X$, $Y$, and $Z$ such that their product $X \cdot Y \cdot Z = 399$. What is the largest possible value of the sum $X+Y+Z$? | 29 | 0 | 5,255.25 | -1 | 5,255.25 | |
For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$.
$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$.
$(2)$ The inequality $t\leqslant \left(m-1\right)^{2}+\left(n-1\right)^{2}+\left(m-n\right)^{2}$ always holds. Find the max... | \frac{9}{8} | 0.1875 | 7,206.625 | 6,465.666667 | 7,377.615385 | |
The graph of the equation $y = \frac{x}{x^3 + Ax^2 + Bx + C}$, where $A,B,C$ are integers, is shown below. Find $A + B + C$. [asy]
import graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.52,xmax=4.62,ymin=-3.66,ymax=3.94;
pen cqcqcq=rgb(0.75,0.75,0.75)... | -1 | 1 | 2,449.75 | 2,449.75 | -1 | |
Find $\overrightarrow{a}+2\overrightarrow{b}$, where $\overrightarrow{a}=(2,0)$ and $|\overrightarrow{b}|=1$, and then calculate the magnitude of this vector. | 2\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels? | 40 | 0.1875 | 6,181.6875 | 3,358.666667 | 6,833.153846 | |
In the sequence $00$ , $01$ , $02$ , $03$ , $\cdots$ , $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$ , $39$ , or $28$ , but not by $30$ or $20$ ). What is the maximal numb... | 50 | 0.25 | 7,898.75 | 7,316.5 | 8,092.833333 | |
Given that the length of the major axis of the ellipse is 4, the left vertex is on the parabola \( y^2 = x - 1 \), and the left directrix is the y-axis, find the maximum value of the eccentricity of such an ellipse. | \frac{2}{3} | 0.75 | 6,008.5 | 5,280.666667 | 8,192 | |
It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot. | 15 | 0.0625 | 8,066.375 | 7,204 | 8,123.866667 | |
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. The pr... | .2 | 0 | 7,222.8125 | -1 | 7,222.8125 | |
Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{1}{4}$
B) $\frac{57}{64}$
C) $\frac{199}{256}$
D) $\frac{57}{256}$
E) $\frac{63}{64}$ | \frac{199}{256} | 0 | 8,192 | -1 | 8,192 | |
Find all the solutions to
\[\sqrt[3]{15x - 1} + \sqrt[3]{13x + 1} = 4 \sqrt[3]{x}.\]Enter all the solutions, separated by commas. | 0, \frac{1}{14}, -\frac{1}{12} | 0 | 6,861.9375 | -1 | 6,861.9375 | |
Ten 6-sided dice are rolled. What is the probability that exactly three of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | .155 | 0.5625 | 7,102.75 | 6,255.555556 | 8,192 | |
Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. | 3\pi | 1 | 3,967.4375 | 3,967.4375 | -1 | |
Given that the solution set of the inequality $x^{2}-2x+1-m^{2} \lt 0$ is $A$;
$(1)$ Find $A$;
$(2)$ If $0 \lt m \lt 1$, and $A=\{x\left|\right.a \lt x \lt b\}$, find the minimum value of $\frac{1}{{8a+2b}}-\frac{1}{{3a-3b}}$. | \frac{2}{5} | 1 | 4,087.5 | 4,087.5 | -1 | |
Determine the value of the expression
\[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))),\]assuming it is positive. | 5 | 1 | 1,735.0625 | 1,735.0625 | -1 | |
Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$? | -112 | 1 | 2,173.25 | 2,173.25 | -1 | |
Express as a common fraction: $(0.\overline{09})(0.\overline{7})$. | \frac{7}{99} | 1 | 1,813.1875 | 1,813.1875 | -1 | |
In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelogram. Let the total numb... | \frac{1}{3} | 0.3125 | 7,119.0625 | 6,300 | 7,491.363636 | |
Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then | 120 < n \leq 130 | 1. **Identify the conditions**: We need to find the smallest nonprime integer $n$ greater than $1$ that has no prime factors less than $10$. This means all prime factors of $n$ must be $10$ or greater.
2. **Prime factors greater than $10$**: The smallest prime number greater than $10$ is $11$.
3. **Forming the small... | 0 | 3,958.25 | -1 | 3,958.25 |
Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)? | 2998.5 | 0 | 6,842.625 | -1 | 6,842.625 | |
Let $p$ and $q$ be the two distinct solutions to the equation $$(x-3)(x+3) = 21x - 63.$$If $p > q$, what is the value of $p - q$? | 15 | 1 | 1,809.9375 | 1,809.9375 | -1 | |
Let $S_{7}$ denote all the permutations of $1,2, \ldots, 7$. For any \pi \in S_{7}$, let $f(\pi)$ be the smallest positive integer $i$ such that \pi(1), \pi(2), \ldots, \pi(i)$ is a permutation of $1,2, \ldots, i$. Compute \sum_{\pi \in S_{7}} f(\pi)$. | 29093 | Extend the definition of $f$ to apply for any permutation of $1,2, \ldots, n$, for any positive integer $n$. For positive integer $n$, let $g(n)$ denote the number of permutations \pi$ of $1,2, \ldots, n$ such that $f(\pi)=n$. We have $g(1)=1$. For fixed $n, k$ (with $k \leq n$ ), the number of permutations \pi$ of $1,... | 0 | 7,861.5 | -1 | 7,861.5 |
Let $E(n)$ denote the sum of the even digits of $n$. Modify $E(n)$ such that if $n$ is prime, $E(n)$ is counted as zero, and if $n$ is not prime, $E(n)$ is counted twice. Calculate $E'(1)+E'(2)+E'(3)+\cdots+E'(200)$.
A) 1200
B) 1320
C) 1360
D) 1400
E) 1500 | 1360 | 0 | 8,192 | -1 | 8,192 | |
If the system of equations
\begin{align*}
8x - 6y &= c, \\
10y - 15x &= d.
\end{align*}
has a solution $(x,y)$ where $x$ and $y$ are both nonzero, find $\frac{c}{d},$ assuming $d$ is nonzero. | -\frac{4}{5} | 0 | 7,179.6875 | -1 | 7,179.6875 | |
A certain number is written in the base-12 numeral system. For which divisor \( m \) is the following divisibility rule valid: if the sum of the digits of the number is divisible by \( m \), then the number itself is divisible by \( m \)? | 11 | 0.625 | 6,957.25 | 6,216.4 | 8,192 | |
Given that at the beginning of the year, 40% of students answered "Yes", 40% answered "No", and 20% were undecided, while at the end of the year, 60% answered "Yes", 30% answered "No", and 10% remained undecided, find the difference between the maximum and minimum possible values of y%, the percentage of students who c... | 60\% | 0.0625 | 7,591.375 | 4,005 | 7,830.466667 | |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$. | \frac{1}{2} | 0 | 6,894.375 | -1 | 6,894.375 | |
In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis, using the same unit of length in both coordinate systems. Given that circle $C$ has a center at point ($2$, $\frac{7π}{6}$) in the polar coordinate ... | \sqrt{2} | 0.8125 | 5,821.5 | 5,274.461538 | 8,192 | |
There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? | 18 | 0 | 8,192 | -1 | 8,192 | |
Given that sinα + cosα = $\frac{7}{5}$, find the value of tanα. | \frac{3}{4} | 0.375 | 6,433.0625 | 6,104.5 | 6,630.2 | |
Let \( S = \{1, 2, 3, \ldots, 30\} \). Determine the number of vectors \((x, y, z, w)\) with \(x, y, z, w \in S\) such that \(x < w\) and \(y < z < w\). | 90335 | 0.125 | 7,603.75 | 7,528.5 | 7,614.5 | |
A line parallel to leg \(AC\) of right triangle \(ABC\) intersects leg \(BC\) at point \(K\) and the hypotenuse \(AB\) at point \(N\). On leg \(AC\), a point \(M\) is chosen such that \(MK = MN\). Find the ratio \(\frac{AM}{MC}\) if \(\frac{BK}{BC} = 14\). | 27 | 0.125 | 7,110.0625 | 4,419.5 | 7,494.428571 | |
A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z? | 6 | Label the lengths of the vertical and horizontal segments as $a, b, c, d$. Rectangle W is $b$ by $c$, so its perimeter is $2 b+2 c$, which equals 2. Rectangle X is $b$ by $d$, so its perimeter is $2 b+2 d$, which equals 3. Rectangle Y is $a$ by $c$, so its perimeter is $2 a+2 c$, which equals 5. Rectangle Z is $a$ by $... | 0.625 | 4,552.4375 | 2,751.4 | 7,554.166667 |
$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF... | 14 | 0.3125 | 7,760.5625 | 6,811.4 | 8,192 | |
Given a triangle \( \triangle ABC \) with sides \( a, b, c \) and corresponding medians \( m_a, m_b, m_c \), and angle bisectors \( w_a, w_b, w_c \). Let \( w_a \cap m_b = P \), \( w_b \cap m_c = Q \), and \( w_c \cap m_a = R \). Denote the area of \( \triangle PQR \) by \( \delta \) and the area of \( \triangle ABC \)... | \frac{1}{6} | 0 | 8,192 | -1 | 8,192 | |
A science student is asked to find the coefficient of the $x^2$ term in the expansion of $(x^2-3x+2)^4$. The coefficient is \_\_\_\_\_\_. (Answer with a number) | 248 | 0.6875 | 6,933.3125 | 6,361.181818 | 8,192 | |
Given that $\tan \alpha = -\frac{1}{3}$ and $\cos \beta = \frac{\sqrt{5}}{5}$, with $\alpha, \beta \in (0, \pi)$, find:
1. The value of $\tan(\alpha + \beta)$;
2. The maximum value of the function $f(x) = \sqrt{2} \sin(x - \alpha) + \cos(x + \beta)$. | \sqrt{5} | 0.6875 | 5,063.875 | 4,622.272727 | 6,035.4 | |
Let a line passing through the origin \\(O\\) intersect a circle \\((x-4)^{2}+y^{2}=16\\) at point \\(P\\), and let \\(M\\) be the midpoint of segment \\(OP\\). Establish a polar coordinate system with the origin \\(O\\) as the pole and the positive half-axis of \\(x\\) as the polar axis.
\\((\\)Ⅰ\\()\\) Find the pol... | 3+ \dfrac {3}{2} \sqrt {3} | 0 | 8,165.5625 | -1 | 8,165.5625 | |
When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers? | 4 | 0.75 | 5,513 | 4,620 | 8,192 | |
We are approaching a 120-meter high skyscraper on a horizontal road. After traveling 300 meters, we see the building at an angle of elevation that is $45^\circ$ greater than at the start of our journey. How close have we approached the skyscraper? | 60 | 0.625 | 4,316.375 | 3,943 | 4,938.666667 | |
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 35, what is the probability that this number will be divisible by 11?
A) $\frac{1}{4}$
B) $\frac{1}{8}$
C) $\frac{1}{5}$
D) $\frac{1}{10}$
E) $\frac{1}{15}$ | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
What is the positive difference between the two largest prime factors of $159137$? | 14 | 0.875 | 3,883.5625 | 3,268.071429 | 8,192 | |
Calculate:
$$\frac{\left(1+\frac{1}{2}\right)^{2} \times\left(1+\frac{1}{3}\right)^{2} \times\left(1+\frac{1}{4}\right)^{2} \times\left(1+\frac{1}{5}\right)^{2} \times \cdots \times\left(1+\frac{1}{10}\right)^{2}}{\left(1-\frac{1}{2^{2}}\right) \times\left(1-\frac{1}{3^{2}}\right) \times\left(1-\frac{1}{4^{2}}\right)... | 55 | 0.8125 | 4,359.6875 | 4,346.307692 | 4,417.666667 | |
Given \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=1}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). | 51 | 0 | 8,192 | -1 | 8,192 | |
Determine the area enclosed by the curves \( y = \sin x \) and \( y = \left(\frac{4}{\pi}\right)^{2} \sin \left(\frac{\pi}{4}\right) x^{2} \) (the latter is a quadratic function). | 1 - \frac{\sqrt{2}}{2}\left(1 + \frac{\pi}{12}\right) | 0 | 7,726.0625 | -1 | 7,726.0625 | |
In a class, each student has either 5 or 6 friends (friendship is mutual), and any two friends have a different number of friends. What is the minimum number of students, greater than 0, that can be in the class? | 11 | 0.625 | 6,175.625 | 5,244.7 | 7,727.166667 | |
A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \). | 763 | 0.625 | 6,068.5 | 4,882.4 | 8,045.333333 | |
Max bought a new dirt bike and paid $10\%$ of the cost upfront, which was $\$150$. What was the price of the bike? | \$ 1500 | 1 | 1,204.875 | 1,204.875 | -1 | |
Suppose that $a$ and $b$ are integers with $4<a<b<22$. If the average (mean) of the numbers $4, a, b, 22$ is 13, how many possible pairs $(a, b)$ are there? | 8 | Since the average of the four numbers $4, a, b, 22$ is 13, then $\frac{4+a+b+22}{4}=13$ and so $4+a+b+22=52$ or $a+b=26$. Since $a>4$ and $a$ is an integer, then $a \geq 5$. Since $a+b=26$ and $a<b$, then $a$ is less than half of 26, or $a<13$. Since $a$ is an integer, then $a \leq 12$. Therefore, we have $5 \leq a \le... | 1 | 3,555.25 | 3,555.25 | -1 |
A valid license plate in Xanadu consists of two letters followed by three digits. How many valid license plates are possible? | 676,\!000 | 0.125 | 1,605.5625 | 579 | 1,752.214286 | |
The distribution of populations in a group of counties is shown in this pie chart. What percent of the counties have fewer than 100,000 residents?
[asy]
draw(Circle((0,0),25),linewidth(1));
draw((-24,-7)--(0,0)--(24,-7),linewidth(1));
draw((0,0)--(7,-24),linewidth(1));
label("59\%",(0,12));
label("25\%",(-10,-10));
la... | 84\% | 0.6875 | 544.125 | 584.363636 | 455.6 | |
Let $g(x) = 3x^4 + 2x^3 - x^2 - 4x + s$. For what value of $s$ is $g(-1) = 0$? | -4 | 1 | 1,674.25 | 1,674.25 | -1 | |
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$ ,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$ , and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$ .
Find $f(1990, 31).$ | \frac{30! \cdot 1989!}{2020!} | 0 | 7,954.625 | -1 | 7,954.625 | |
The number of cans in the layers of a display in a supermarket form an arithmetic sequence. The bottom layer has 28 cans; the next layer has 25 cans and so on until there is one can at the top of the display. How many cans are in the entire display? | 145 | 1 | 1,725.9375 | 1,725.9375 | -1 | |
A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop? | 45 | 0.1875 | 7,278 | 6,615.333333 | 7,430.923077 | |
How many positive perfect squares less than \(10^8\) are multiples of 36? | 1666 | 0.8125 | 5,545.375 | 4,934.615385 | 8,192 | |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 18 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $G$ is the midpoint of $CD$. Determine the area of triangle $EFG$. | 52.5 | 0 | 6,042.5625 | -1 | 6,042.5625 | |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | 63 | 0 | 8,192 | -1 | 8,192 | |
Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid below so that a path is formed from the upper left-hand corner $A$ to the lower righthand corner $B$. In a path, consecutive dominoes must touch at their sides and not just their corners. No domino may be placed diagonally;... | 35 | 0 | 8,028.75 | -1 | 8,028.75 | |
A palindrome is a string that does not change when its characters are written in reverse order. Let S be a 40-digit string consisting only of 0's and 1's, chosen uniformly at random out of all such strings. Let $E$ be the expected number of nonempty contiguous substrings of $S$ which are palindromes. Compute the value ... | 113 | Note that $S$ has $41-n$ contiguous substrings of length $n$, so we see that the expected number of palindromic substrings of length $n$ is just $(41-n) \cdot 2^{-\lfloor n / 2\rfloor}$. By linearity of expectation, $E$ is just the sum of this over all $n$ from 1 to 40. However, it is much easier to just compute $$\sum... | 0.0625 | 7,656.9375 | 7,177 | 7,688.933333 |
Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[3.14] = 3$. Then, find the value of $\left[\frac{2017 \times 3}{11}\right] + \left[\frac{2017 \times 4}{11}\right] + \left[\frac{2017 \times 5}{11}\right] + \left[\frac{2017 \times 6}{11}\right] + \left[\frac{2017 \times 7}{11}\right] + \left[\frac... | 6048 | 0.875 | 4,885.8125 | 4,413.5 | 8,192 | |
In right $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 12$, $BC = 35$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\o... | 11 | This solution is not a real solution and is solving the problem with a ruler and compass.
Draw $AC = 4.8, BC = 14, AB = 14.8$. Then, drawing the tangents and intersecting them, we get that $IA$ is around $6.55$ and $IB$ is around $18.1$. We then find the ratio to be around $\frac{39.45}{14.8}$. Using long division, we... | 0 | 8,192 | -1 | 8,192 |
The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$.
Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval | [\frac{5}{12},\frac{2}{3}] | 1. **Identify the given probabilities**:
- Probability that event $A$ occurs, $P(A) = \frac{3}{4}$.
- Probability that event $B$ occurs, $P(B) = \frac{2}{3}$.
2. **Determine the upper bound for $p$**:
- Since the probability that both events $A$ and $B$ occur cannot exceed the probability of either event occ... | 0.75 | 1,390.3125 | 1,628.666667 | 675.25 |
Given that $α\in\mathbb{R}$ and $\sin α + 2\cos α = \frac{\sqrt{10}}{2}$, find the value of $\tan α$. | -\frac{1}{3} | 0.875 | 6,884.5625 | 6,697.785714 | 8,192 | |
Let $a \oslash b = (\sqrt{2a+b})^3$. If $4 \oslash x = 27$, find the value of $x$. | 1 | 1 | 1,775.125 | 1,775.125 | -1 | |
In a right triangle $PQR$, where $\angle P = 90^\circ$, suppose $\cos Q = \frac{4}{5}$. The length of side $PQ$ (adjacent to $\angle Q$) is $12$. What is the length of $PR$ (hypotenuse)? | 15 | 0.375 | 2,381.6875 | 1,600 | 2,850.7 | |
When simplified, what is the value of $$(10^{0.5})(10^{0.3})(10^{0.2})(10^{0.1})(10^{0.9})?$$ | 100 | 0.9375 | 3,546.9375 | 3,237.266667 | 8,192 | |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | 5 | 1 | 1,529.0625 | 1,529.0625 | -1 | |
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$. | 134 | 0 | 8,192 | -1 | 8,192 | |
Petya wrote a natural number \( A \) on the board. If you multiply it by 8, you get the square of a natural number. How many such three-digit numbers \( B \) exist for which \( A \cdot B \) is also a square of a natural number? | 15 | 0.3125 | 7,378.5625 | 5,975.6 | 8,016.272727 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.