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 |
|---|---|---|---|---|---|---|
Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use? | 10 | 0.0625 | 7,960.25 | 8,148 | 7,947.733333 | |
A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is first seen 2 seconds after midnight, and the red signal is first seen 8 seconds after midnight. At what time will both signals be seen together for the first time? | 92 | 0.6875 | 5,409.625 | 5,102.636364 | 6,085 | |
Moore's law says that the number of transistors that can be placed inexpensively on a silicon chip doubles every two years. In 1990, a typical CPU contained about $1,\!000,\!000$ transistors. According to Moore's law, how many transistors did a typical CPU contain in the year 2000? | 32,\!000,\!000 | 0 | 2,043.3125 | -1 | 2,043.3125 | |
If $α∈(0,\frac{π}{2})$, and $tan2α=\frac{cosα}{2-sinα}$, determine the value of $\sin \alpha$. | \frac{1}{4} | 0.875 | 2,963.375 | 2,216.428571 | 8,192 | |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2016 | 1. **Observation of $f(n)$**: We start by observing the function $f(n)$, which counts the number of ways to write $n$ as a sum of powers of 2, considering the order of terms. For example, $f(4) = 6$ because we can write $4$ as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1... | 0 | 7,879.875 | -1 | 7,879.875 |
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$. | 17 | 0 | 8,192 | -1 | 8,192 | |
Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$. | 140 | Observe that $AD+BC=AF+FB=20$, and that $ABCD$ is a trapezoid with height $BC=14$. Hence the answer is $\frac{1}{2}(AD+BC)(14)=140$. | 0 | 8,147.625 | -1 | 8,147.625 |
Compute $\cos \left( \arcsin \frac{5}{13} \right).$ | \frac{12}{13} | 1 | 1,500.0625 | 1,500.0625 | -1 | |
Agricultural researchers randomly selected $50$ plants of a certain crop in an experimental field for research. The frequency distribution of the mass (in grams) of a single plant falling into each group is shown in the table below:
| Mass of a Single Plant (in grams) | Frequency |
|-----------------------------------... | 0.8413 | 1 | 3,606.625 | 3,606.625 | -1 | |
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \). | \frac{\sqrt{3}}{3} | 0 | 8,192 | -1 | 8,192 | |
If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= -2, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= -1,
\end{align*}
what is $ab + cd$? | -\frac{190}{9} | 0.9375 | 4,836.8125 | 4,613.133333 | 8,192 | |
(1) Use the Euclidean algorithm to find the greatest common divisor of 117 and 182, and verify it using the subtraction method.
(2) Use the Horner's method to evaluate the polynomial $f(x) = 1 - 9x + 8x^2 - 4x^4 + 5x^5 + 3x^6$ at $x = -1$. | 12 | 0.75 | 4,716 | 3,665 | 7,869 | |
Given two arbitrary positive integers \( n \) and \( k \), let \( f(n, k) \) denote the number of unit squares that one of the diagonals of an \( n \times k \) grid rectangle passes through. How many such pairs \( (n, k) \) are there where \( n \geq k \) and \( f(n, k) = 2018 \)? | 874 | 0.25 | 7,621.5 | 5,955.75 | 8,176.75 | |
How many integers between 1 and 3015 are either multiples of 5 or 7 but not multiples of 35? | 948 | 0 | 3,231.5625 | -1 | 3,231.5625 | |
During a performance, a panda named A Bao is performing on stilts. A Bao's height accounts for $\frac{1}{4}$ of the total height when he is standing on the stilts. During the performance, A Bao accidentally breaks $20$ dm off each stilt. After the break, A Bao's height accounts for $\frac{1}{3}$ of the new total height... | 160 | 0.5 | 874.625 | 868.125 | 881.125 | |
Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square. | \frac{2}{5} | 0 | 7,626.625 | -1 | 7,626.625 | |
Evaluate the infinite geometric series: $$\frac{5}{3} - \frac{3}{5} + \frac{9}{25} - \frac{27}{125} + \dots$$ | \frac{125}{102} | 0.0625 | 8,161.9375 | 8,192 | 8,159.933333 | |
Given that an office at a school needs to arrange a duty roster from the 1st to the 6th day with six designated individuals participating, find the total number of different arrangements possible, given that person A and person B cannot be adjacent, and person C and person D also cannot be adjacent. | 336 | 0.8125 | 3,498.0625 | 2,826.692308 | 6,407.333333 | |
If $θ∈[0,π]$, then the probability of $\sin (θ+ \frac {π}{3}) > \frac {1}{2}$ being true is ______. | \frac {1}{2} | 0.75 | 6,020.375 | 5,587 | 7,320.5 | |
An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction. | \frac{5}{2} | 0.1875 | 7,358.5 | 5,392.333333 | 7,812.230769 | |
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \[\be... | 144 | We replace the numbers which are 0 mod(3) to 0, 1 mod(3) to 1, and 2 mod(3) to 2.
Then, the problem is equivalent to arranging 4 0's,4 1's, and 4 2's into the grid (and then multiplying by $4!^3$ to account for replacing the remainder numbers with actual numbers) such that no 2 of the same numbers are adjacent.
Then,... | 0 | 7,975.625 | -1 | 7,975.625 |
Given an ellipse with its foci on the $y$-axis, an eccentricity of $\frac{2\sqrt{2}}{3}$, and one focus at $(0, 2\sqrt{2})$.
(1) Find the standard equation of the ellipse;
(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origin with a radius of $2$ at points $A$ and $B$. $... | \frac{\sqrt{62}}{2} | 0 | 7,867.25 | -1 | 7,867.25 | |
The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(1,3)$ to $(5,7)$. What is the value of b? | 8 | 1 | 1,823.625 | 1,823.625 | -1 | |
For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$. | 21 | 0.25 | 7,520 | 6,463 | 7,872.333333 | |
The product $1! \cdot 2! \cdot 3! \cdots \cdots \cdot 99! \cdot 100!$ has ___ consecutive zeros at the end. | 1124 | 0.1875 | 8,035.0625 | 7,355 | 8,192 | |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity o... | \frac{\sqrt{5} - 1}{2} | 0 | 8,192 | -1 | 8,192 | |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is \(5:4\). Find the minimum possible value of their common perimeter. | 524 | 0 | 6,613.375 | -1 | 6,613.375 | |
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is | 320 | 1. **Identify the dimensions of the rectangle and the midpoints:**
- The rectangle $ACDE$ has length $AC = 32$ and width $AE = 20$.
- $B$ is the midpoint of $\overline{AC}$, so $AB = BC = \frac{32}{2} = 16$.
- $F$ is the midpoint of $\overline{AE}$, so $AF = FE = \frac{20}{2} = 10$.
2. **Calculate the area of... | 0.6875 | 6,294.6875 | 5,432.272727 | 8,192 |
Find the smallest integer $k$ for which the conditions
(1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
(3) $a_9=k$
are satisfied by more than one sequence. | 748 | From $(2)$, $a_9=$ $a_8+a_7=2a_7+a_6=3a_6+2a_5=5a_5+3a_4=8a_4+5a_3=13a_3+8a_2=21a_2+13a_1$ $=k$
Suppose that $a_1=x_0$ is the smallest possible value for $a_1$ that yields a good sequence, and $a_2=y_0$ in this sequence. So, $13x_0+21y_0=k$.
Since $\gcd(13,21)=1$, the next smallest possible value for $a_1$ that yield... | 0 | 8,192 | -1 | 8,192 |
Let $n = 2^{35}3^{17}$. How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$? | 594 | 0.125 | 7,002.6875 | 5,432.5 | 7,227 | |
Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$ to town $Z$, a distance of $25$ km all do... | 33 | 0 | 6,803.625 | -1 | 6,803.625 | |
If $\Phi$ and $\varphi$ are the two distinct solutions to the equation $x^2=x+1$, then what is the value of $(\Phi-\varphi)^2$? | 5 | 1 | 1,786.875 | 1,786.875 | -1 | |
Determine the value of the infinite product $(2^{1/3})(4^{1/9})(8^{1/27})(16^{1/81}) \dotsm.$ Enter your answer in the form "\sqrt[a]{b}", which stands for $\sqrt[a]{b}.$ | \sqrt[4]{8} | 0.8125 | 4,211.125 | 3,343.230769 | 7,972 | |
Let $S-ABC$ be a triangular prism with circumscribed sphere centered at $O$. The midpoints of $SB$ and $AC$ are $N$ and $M$, respectively. The midpoint of line segment $MN$ is $P$, and it is given that $SA^{2} + SB^{2} + SC^{2} = AB^{2} + BC^{2} + AC^{2}$. If $SP = 3\sqrt{7}$ and $OP = \sqrt{21}$, find the radius of sp... | 2 \sqrt{21} | 0.1875 | 7,902.6875 | 6,649 | 8,192 | |
The decimal $3.834834834\ldots$ can be written as a fraction. When reduced to lowest terms, find the sum of the numerator and denominator of this fraction. | 4830 | 0 | 3,029.625 | -1 | 3,029.625 | |
The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\). | 2 \sqrt{10} | 0.25 | 7,643.5625 | 5,998.25 | 8,192 | |
Let $r_1$ , $r_2$ , $\ldots$ , $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$ . If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$ . | 240 | 0.1875 | 7,778.75 | 6,617 | 8,046.846154 | |
Calculate the value of
$$
\sqrt{2018 \times 2021 \times 2022 \times 2023 + 2024^{2}} - 2024^{2}
$$ | -12138 | 0.375 | 7,850.5625 | 7,281.5 | 8,192 | |
For each positive integer \( x \), let \( f(x) \) denote the greatest power of 3 that divides \( x \). For example, \( f(9) = 9 \) and \( f(18) = 9 \). For each positive integer \( n \), let \( T_n = \sum_{k=1}^{3^n} f(3k) \). Find the greatest integer \( n \) less than 1000 such that \( T_n \) is a perfect square. | 960 | 0 | 8,192 | -1 | 8,192 | |
What is the ratio of the volume of cone $C$ to the volume of cone $D$? Cone $C$ has a radius of 15.6 and a height of 29.5. Cone $D$ has a radius of 29.5 and a height of 15.6. | \frac{156}{295} | 0.3125 | 7,615.4375 | 6,347 | 8,192 | |
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle? | 1/5 | 0 | 8,192 | -1 | 8,192 | |
In the expansion of $(1+x)^3 + (1+x)^4 + \ldots + (1+x)^{12}$, the coefficient of the term containing $x^2$ is _____. (Provide your answer as a number) | 285 | 0.9375 | 4,906.625 | 4,687.6 | 8,192 | |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5? | 100 | 1 | 2,018.9375 | 2,018.9375 | -1 | |
Given the equations $\left(625\right)^{0.24}$ and $\left(625\right)^{0.06}$, find the value of their product. | 5^{6/5} | 0 | 7,894.375 | -1 | 7,894.375 | |
Given the set $\{-25, -4, -1, 3, 5, 9\}$, find the largest quotient and the smallest product that can be formed using two numbers chosen from the set. | -225 | 0.875 | 3,814.375 | 3,274.928571 | 7,590.5 | |
Define a "spacy" set of integers such that it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 181 | 0.1875 | 7,643.125 | 5,264.666667 | 8,192 | |
Set \( A = \{1, 2, \cdots, n\} \). If there exist nonempty sets \( B \) and \( C \) such that \( B \cap C = \emptyset \), \( B \cup C = A \), and the sum of the squares of the elements in \( B \) is \( M \), and the sum of the squares of the elements in \( C \) is \( N \), and \( M - N = 2016 \), find the smallest valu... | 19 | 0.25 | 7,766.4375 | 6,586.75 | 8,159.666667 | |
Given that $\sum_{k=1}^{36}\sin 4k=\tan \frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\frac{p}{q}<90,$ find $p+q.$ | 73 | 0 | 8,117.6875 | -1 | 8,117.6875 | |
The terms $x, x + 2, x + 4, \dots, x + 2n$ form an arithmetic sequence, with $x$ an integer. If each term of the sequence is cubed, the sum of the cubes is $-1197$. What is the value of $n$ if $n > 3$? | 6 | 0.0625 | 8,135.9375 | 7,295 | 8,192 | |
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad? | 64 | 1. **Define Variables:**
Let $b$ be the number of blueberries, $r$ be the number of raspberries, $g$ be the number of grapes, and $c$ be the number of cherries.
2. **Set Up Equations Based on Conditions:**
- Total number of fruits:
\[ b + r + g + c = 280 \]
- Raspberries are twice as many as blueberries... | 1 | 1,565.75 | 1,565.75 | -1 |
Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another. | 36 | 0.0625 | 8,015.375 | 8,192 | 8,003.6 | |
Evaluate \( \frac{18}{4.9 \times 106} \). | \frac{18}{519.4} | 0 | 283.8125 | -1 | 283.8125 | |
For the set $\{1, 2, 3, \ldots, 8\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Calculate the sum of all such alternating sums for $n=8$. | 1024 | 0 | 8,026.0625 | -1 | 8,026.0625 | |
A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle? | \pi | 1. **Visualize the Problem**: We start by visualizing a $1 \times 2$ rectangle inscribed in a semicircle, with the longer side of the rectangle lying along the diameter of the semicircle.
2. **Double the Figure**: To simplify the problem, we consider doubling the semicircle along its diameter to form a complete circle... | 0.9375 | 4,704 | 4,471.466667 | 8,192 |
Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$? | 3 | 1. **Identify the requirement for $N$:**
$N$ must be divisible by every positive integer less than $7$. This includes the integers $1, 2, 3, 4, 5,$ and $6$.
2. **Determine the least common multiple (LCM):**
To find $N$, we need the least common multiple of these numbers. The LCM of a set of numbers is the sm... | 1 | 3,128.75 | 3,128.75 | -1 |
Define an odd function \( f(x) \) on \( \mathbb{R} \) that satisfies \( f(x+1) \) is an even function, and when \( x \in [0,1] \), \( f(x) = x(3-2x) \). Then, find the value of \( f\left(\frac{31}{2}\right) \). | -1 | 0.5 | 6,673.375 | 5,555.125 | 7,791.625 | |
Compute $\dbinom{20}{5}$. | 11628 | 0 | 5,152.75 | -1 | 5,152.75 | |
What is the value of $\frac{11! - 10!}{9!}$? | 100 | 1. Start by simplifying the expression given in the problem:
\[
\frac{11! - 10!}{9!}
\]
Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$.
2. Factor out $10!$ from the numerator:
\[
\frac{11! - 10!}{9!} = \frac{11 \cdot 10! - 10!}{9!} = \frac{10!(11 - 1)... | 1 | 2,296.375 | 2,296.375 | -1 |
A group of 11 people, including Ivanov and Petrov, are seated in a random order around a circular table. Find the probability that there will be exactly 3 people sitting between Ivanov and Petrov. | 1/10 | 0 | 5,621 | -1 | 5,621 | |
In the arithmetic sequence $17, a, b, c, 41$, what is the value of $b$? | 29 | 1 | 1,403.4375 | 1,403.4375 | -1 | |
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers? | 20 | 0 | 8,192 | -1 | 8,192 | |
The line $x+2ay-1=0$ is parallel to $\left(a-1\right)x+ay+1=0$, find the value of $a$. | \frac{3}{2} | 0.8125 | 5,046.875 | 4,381.538462 | 7,930 | |
A merchant acquires goods at a discount of $30\%$ of the list price and intends to sell them with a $25\%$ profit margin after a $25\%$ discount on the marked price. Determine the required percentage of the original list price that the goods should be marked. | 124\% | 0 | 3,852.25 | -1 | 3,852.25 | |
Compute
\[\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}.\] | \frac{5}{4} | 0.4375 | 7,126.5625 | 5,756.714286 | 8,192 | |
Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, determine the value of $g(x) = f(x) + 2$ when $x = -1$. | -1 | 1 | 1,521.25 | 1,521.25 | -1 | |
Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6. | \frac{18}{7} | 0.6875 | 4,100.25 | 3,772.363636 | 4,821.6 | |
Given a set \( A = \{0, 1, 2, \cdots, 9\} \), and a family of non-empty subsets \( B_1, B_2, \cdots, B_j \) of \( A \), where for \( i \neq j \), \(\left|B_i \cap B_j\right| \leqslant 2\), determine the maximum value of \( k \). | 175 | 0 | 8,192 | -1 | 8,192 | |
The decimal representation of a fraction ends in $0.\overline{3}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction? | 3 | 1 | 1,419.8125 | 1,419.8125 | -1 | |
How many of the four integers $222, 2222, 22222$, and $222222$ are multiples of 3? | 2 | We could use a calculator to divide each of the four given numbers by 3 to see which calculations give an integer answer. Alternatively, we could use the fact that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sums of the digits of $222, 2222, 22222$, and $222222$ are ... | 1 | 2,569.375 | 2,569.375 | -1 |
Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$. | \sqrt{166} | 0.4375 | 6,878.5625 | 6,104.428571 | 7,480.666667 | |
There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find the remainder when $N$ is divided by $1000$. | 440 | Note that $30=2\cdot 3\cdot 5$. Since $\gcd(2, 3, 5)=1$, by CRT, for each value $k=0\ldots 29$ modulo $30$ there exists a unique ordered triple of values $(a, b, c)$ such that $k\equiv a\pmod{2}$, $k\equiv b\pmod{3}$, and $k\equiv c\pmod{5}$. Therefore, we can independently assign the residues modulo $2, 3, 5$, so $N=2... | 0.0625 | 8,051.8125 | 7,589 | 8,082.666667 |
A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked out? ... | 5 | 0.3125 | 5,764.8125 | 3,454 | 6,815.181818 | |
Given the function $f(x) = \frac{x+3}{x^2+1}$, and $g(x) = x - \ln(x-p)$.
(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\left(\frac{1}{3}, f\left(\frac{1}{3}\right)\right)$;
(II) Determine the number of zeros of the function $g(x)$, and explain the reason;
(III) It is known that the seq... | 6044 | 0 | 7,961.3125 | -1 | 7,961.3125 | |
The solutions to the equations $z^2 = 9 + 9\sqrt{7}i$ and $z^2 = 5 + 5\sqrt{2}i$, where $i = \sqrt{-1}$, form the vertices of a parallelogram in the complex plane. Determine the area of this parallelogram, which can be written in the form $p\sqrt{q} - r\sqrt{s}$, where $p, q, r, s$ are positive integers and neither $q$... | \frac{3}{4}\sqrt{35} - \frac{3}{4}\sqrt{15} | 0 | 8,192 | -1 | 8,192 | |
For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\... | 483 | By considering the graph of this function, it is shown that the graph is composed of trapezoids ranging from $a^2$ to $(a+1)^2$ with the top made of diagonal line $y=ax$. The width of each trapezoid is $3, 5, 7$, etc. Whenever $a$ is odd, the value of $A(n)$ increases by an integer value, plus $\frac{1}{2}$. Whenever $... | 0 | 8,192 | -1 | 8,192 |
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.
After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$? | 34 | 0 | 8,089.9375 | -1 | 8,089.9375 | |
Given that a regular tetrahedron has all edge lengths of $\sqrt {2}$, and all four vertices are on the same spherical surface, find the surface area of this sphere. | 3\pi | 0.875 | 4,221.4375 | 3,654.214286 | 8,192 | |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Additionally, at least one head of each dragon always tells the truth. One day, 530 dragons sat at a round table, and each of them said:
- 1st head: "To my left is a green ... | 176 | 0 | 8,192 | -1 | 8,192 | |
On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \... | \frac{19}{3} | 0.75 | 4,649.9375 | 4,133.833333 | 6,198.25 | |
Read the following material and complete the corresponding tasks.
平衡多项式
定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子.
例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x^{2}+7x+10)=-4$,所以多项式$x+1$,$x+2$,$x+5$,$x+6$是一... | -5 | 0.625 | 1,270 | 724.5 | 2,179.166667 | |
Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$: | $3x + y = 10$ | To determine which equation does not represent $y$ as either directly or inversely proportional to $x$, we need to analyze each equation to see if it can be expressed in the form $\frac{x}{y} = k$ (direct proportionality) or $xy = k$ (inverse proportionality).
1. **Equation (A): $x + y = 0$**
- Rearranging, we get ... | 0 | 6,161.9375 | -1 | 6,161.9375 |
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | \dfrac{1}{2} | 0.4375 | 6,596.125 | 5,873.571429 | 7,158.111111 | |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+m)a_n$, ($m\in R$), and $a_nb_n=\frac{1}{2}$. If for any $n\in N^*$, $\lambda>T_n$ always holds true, then the minimum value of the real num... | \frac{1}{2} | 0.25 | 7,555.6875 | 5,646.75 | 8,192 | |
Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$? | 165 | We are given an isosceles triangle with two angles measuring $70^\circ$ and $x^\circ$. We need to find the sum of all possible values of $x$.
An isosceles triangle has at least two sides of equal length, and the angles opposite these sides are equal. The sum of the angles in any triangle is $180^\circ$.
#### Case Ana... | 0.875 | 5,036.9375 | 4,626.428571 | 7,910.5 |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | 37 | 1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to... | 0 | 8,192 | -1 | 8,192 |
In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\). | 15 | 0.5625 | 5,948.8125 | 4,816.555556 | 7,404.571429 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} | 0.125 | 8,047.6875 | 7,037.5 | 8,192 | |
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of th... | 134 | Let there be $N$ members of the committee. Suppose candidate $n$ gets $a_n$ votes. Then $a_n$ as a percentage out of $N$ is $100\frac{a_n}{N}$. Setting up the inequality $a_n \geq 1 + 100\frac{a_n}{N}$ and simplifying, $a_n \geq \lceil(\frac{N}{N - 100})\rceil$ (the ceiling function is there because $a_n$ is an integer... | 0 | 8,192 | -1 | 8,192 |
Factor $9y^2-30y+25$. | (3y - 5)^2 | 1 | 1,854.4375 | 1,854.4375 | -1 | |
Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other ... | 116 | Each match eliminates exactly one player, so exactly $32-1=31$ matches are played, each of which consists of a different pair of players. Among the $\binom{32}{2}=\frac{32 \cdot 31}{2}=496$ pairs of players, each pair is equally likely to play each other at some point during the tournament. Therefore, the probability t... | 0 | 8,101.875 | -1 | 8,101.875 |
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is | $7 \sqrt{3}$ | 0 | 7,484.6875 | -1 | 7,484.6875 | |
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid. What is $DE$? | 13 | 1. **Identify the areas of the shapes**:
- The area of rectangle $ABCD$ is given by the formula for the area of a rectangle, $AB \times AD$. Given $AB = 5$ and $AD = 6$, the area of rectangle $ABCD$ is:
\[
AB \cdot AD = 5 \cdot 6 = 30
\]
- The area of right triangle $DCE$ is given by the formula f... | 0 | 8,192 | -1 | 8,192 |
Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is . | 14 | 0 | 7,490.1875 | -1 | 7,490.1875 | |
How many ways are there to cover a $3 \times 8$ rectangle with 12 identical dominoes? | 153 | Trivially there is 1 way to tile a $3 \times 0$ rectangle, and it is not hard to see there are 3 ways to tile a $3 \times 2$. Let $T_{n}$ be the number of tilings of a $3 \times n$ rectangle, where $n$ is even. From the diagram below we see the recursion $T_{n}=3 T_{n-2}+2\left(T_{n-4}+T_{n-6}+\ldots+T_{2}+T_{0}\right)... | 0.375 | 7,075.375 | 5,443.5 | 8,054.5 |
What is the 100th letter in the pattern ABCABCABC...? | A | 0.9375 | 442.125 | 437.333333 | 514 | |
Given real numbers \( x \) and \( y \) satisfy
\[
\left\{
\begin{array}{l}
x - y \leq 0, \\
x + y - 5 \geq 0, \\
y - 3 \leq 0
\end{array}
\right.
\]
If the inequality \( a(x^2 + y^2) \leq (x + y)^2 \) always holds, then the maximum value of the real number \( a \) is $\qquad$. | 25/13 | 0.375 | 7,729.9375 | 6,959.833333 | 8,192 | |
A student at Harvard named Kevin was counting his stones by 11. He messed up $n$ times and instead counted 9s and wound up at 2007. How many values of $n$ could make this limerick true? | 21 | The mathematical content is that $9 n+11 k=2007$, for some nonnegative integers $n$ and $k$. As $2007=9 \cdot 223, k$ must be divisible by 9. Using modulo 11, we see that $n$ is 3 more than a multiple of 11. Thus, the possibilities are $n=223,212,201, \ldots, 3$, which are 21 in number. | 0.75 | 5,169.0625 | 4,166.833333 | 8,175.75 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 0.8125 | 5,265.875 | 4,791 | 7,323.666667 | |
$F$ is the right focus of the hyperbola $C: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \left(a > 0, b > 0\right)$. A perpendicular line is drawn from point $F$ to asymptote $C$, with the foot of the perpendicular denoted as $A$, intersecting another asymptote at point $B$. If $2\overrightarrow {AF} = \overrightarr... | \dfrac{2\sqrt{3}}{3} | 0 | 5,456.875 | -1 | 5,456.875 | |
Let \( n \) be a natural number. Decompose \( n \) into sums of powers of \( p \) (where \( p \) is a positive integer greater than 1), in such a way that each power \( p^k \) appears at most \( p^2 - 1 \) times. Denote by \( C(n, p) \) the total number of such decompositions. For example, for \( n = 8 \) and \( p = 2 ... | 118 | 0 | 8,192 | -1 | 8,192 |
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