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 |
|---|---|---|---|---|---|---|
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | 12 | 0.9375 | 5,284.3125 | 5,090.466667 | 8,192 | |
There are \( k \) people and \( n \) chairs in a row, where \( 2 \leq k < n \). There is a couple among the \( k \) people. The number of ways in which all \( k \) people can be seated such that the couple is seated together is equal to the number of ways in which the \( k-2 \) people, without the couple present, can b... | 12 | 0 | 8,192 | -1 | 8,192 | |
Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence $\{a_n\}$ is $S_n$, and $\frac{a_{n+1}}{a_n} < 1$, if $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, calculate $S_6$. | 126 | 0.1875 | 7,000.3125 | 3,817.666667 | 7,734.769231 | |
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers? | 85 | 0.5 | 7,621.9375 | 7,051.875 | 8,192 | |
Allen and Ben are painting a fence. The ratio of the amount of work Allen does to the amount of work Ben does is $3:5$. If the fence requires a total of $240$ square feet to be painted, how many square feet does Ben paint? | 150 | 1 | 1,003.625 | 1,003.625 | -1 | |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls. | \frac{35}{128} | 0.6875 | 4,905.0625 | 3,931.090909 | 7,047.8 | |
The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$. | 447 | Note that $\frac{33}{37}=\frac{891}{999} = 0.\overline{891}$. Also note that the period of $0.abab\overline{ab}+0.abcabc\overline{abc}$ is at most $6$. Therefore, we only need to worry about the sum $0.ababab+ 0.abcabc$. Adding the two, we get \[\begin{array}{ccccccc}&a&b&a&b&a&b\\ +&a&b&c&a&b&c\\ \hline &8&9&1&8&9&1\e... | 0.4375 | 6,988.4375 | 5,441 | 8,192 |
In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was co... | 766 | Suppose $1000$ workers can complete one quarter of the job in one day. After the first day, there were $900$ workers remaining so the second quarter was completed in $\frac{10}{9}$ days. Now there are only $800$ workers remaining so the third quarter can be completed in $\frac{10}{8}$ days. It has been $1+\frac{10}{9}+... | 0.25 | 7,631.125 | 5,948.5 | 8,192 |
A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area? | \frac{17}{2} | Sketch the lines $x=1, x=5, y=8, y=13, y=10-x$, and $y=15-x$. The triangle has to be contained in the hexagonal region contained in all these lines. If all the projections are correct, every other vertex of the hexagon must be a vertex of the triangle, which gives us two possibilities for the triangle. One of these tri... | 0 | 8,165.5625 | -1 | 8,165.5625 |
Two students, A and B, are doing rope skipping training. It is known that A takes the same amount of time to skip 300 times as B does to skip 270 times. A skips 19 more times per minute than B. Find out how many times each of them skips per minute. | 171 | 0.75 | 2,108.1875 | 2,541.833333 | 807.25 | |
Let $f$ be a function taking the positive integers to the positive integers, such that
(i) $f$ is increasing (i.e. $f(n + 1) > f(n)$ for all positive integers $n$)
(ii) $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$
Find the sum of al... | 900 | 0 | 8,192 | -1 | 8,192 | |
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Company XYZ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A, B$, and $C$. There are 1,5 , and 4 workers at $A, B$, and $C$, respectively. Find the minimum possible total dista... | 69 | We want to minimize $1 \cdot P A+5 \cdot P B+4 \cdot P C$. By the triangle inequality, $(P A+P B)+4(P B+P C) \geq A B+4 B C=13+56=69$, with equality precisely when $P=[A B] \cap[B C]=B$. | 0 | 8,192 | -1 | 8,192 |
The set of positive even numbers $\{2, 4, 6, \cdots\}$ is grouped in increasing order such that the $n$-th group has $3n-2$ numbers:
\[
\{2\}, \{4, 6, 8, 10\}, \{12, 14, 16, 18, 20, 22, 24\}, \cdots
\]
Determine which group contains the number 2018. | 27 | 0.25 | 7,575.375 | 5,858.75 | 8,147.583333 | |
7.61 log₂ 3 + 2 log₄ x = x^(log₉ 16 / log₃ x). | 16/3 | 0 | 8,077.8125 | -1 | 8,077.8125 | |
In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G? | 1.5 | 0 | 5,884 | -1 | 5,884 | |
In the diagram, $CP$ and $CQ$ trisect $\angle ACB$. $CM$ bisects $\angle PCQ$. Find the ratio of the measure of $\angle MCQ$ to the measure of $\angle ACQ$. | \frac{1}{4} | 0.6875 | 3,260.5 | 2,895.909091 | 4,062.6 | |
Vitya has five math lessons per week, one on each day from Monday to Friday. Vitya knows that with a probability of \(1 / 2\), the teacher will not check his homework at all during the week, and with a probability of \(1 / 2\), the teacher will check it, but only once on any of the math lessons, with equal chances on a... | 1/6 | 0.125 | 7,828.9375 | 6,664.5 | 7,995.285714 | |
A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a triangle is $\frac{m}{n}$, where $m, n$ are p... | 115 | Instead of choosing three random chords, we instead first choose 6 random points on the circle and then choosing a random pairing of the points into 3 pairs with which to form chords. If the chords form a triangle, take a chord $C$. Any other chord $C^{\prime}$ must have its endpoints on different sides of $C$, since $... | 0 | 8,186.625 | -1 | 8,186.625 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n = \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. For how many positive integers $b_1 \le 1500$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$? | 1000 | 0 | 7,818.125 | -1 | 7,818.125 | |
The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.
Calculate $({-15})÷({\frac{1}{3}-3-\frac{3}{2}})×6$.
Solution: Original expression $=({-15})÷({-\frac{{25}}{6}})×6\ldots \ldots $ First step $=\left(-15\right)\div \left(-25\right)\ldots \ldots $ Second st... | \frac{108}{5} | 1 | 2,913.3125 | 2,913.3125 | -1 | |
Two standard 6-sided dice are rolled. What is the probability that the sum rolled is a perfect square? | \dfrac{7}{36} | 1 | 2,027.8125 | 2,027.8125 | -1 | |
Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z = 15 \). What is the maximum value of
\[ X \cdot Y \cdot Z + X \cdot Y + Y \cdot Z + Z \cdot X? \] | 200 | 0.625 | 6,534.5 | 5,540 | 8,192 | |
Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C. | 1340 | 0.625 | 3,818.0625 | 2,713 | 5,659.833333 | |
A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$.
[asy]import graph;
draw(Circle((0,0),25),black);
draw((0,0)--(7,18),Arrow);
draw((0,0)--(0,25));
draw((... | 40\pi | 1 | 784.3125 | 784.3125 | -1 | |
Compute the sum \( S = \sum_{i=0}^{101} \frac{x_{i}^{3}}{1 - 3x_{i} + 3x_{i}^{2}} \) for \( x_{i} = \frac{i}{101} \). | 51 | 0.1875 | 7,805.3125 | 6,129.666667 | 8,192 | |
Given positive integers \( N \) and \( k \), we counted how many different ways the number \( N \) can be written in the form \( a + b + c \), where \( 1 \leq a, b, c \leq k \), and the order of the summands matters. Could the result be 2007? | 2007 | 0.0625 | 8,192 | 8,192 | 8,192 | |
On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always te... | 1009 | 0 | 8,192 | -1 | 8,192 | |
Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers. | 3\sqrt{3} | 0.1875 | 8,044.1875 | 7,403.666667 | 8,192 | |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | 12 | 1. **Count the number of factors of 10 in $90!$:**
The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:
\[
\left\lfloor \frac{90}{5} \right\rfloor + \left\lf... | 0 | 8,192 | -1 | 8,192 |
Given the graphs of $y=\sin (\frac{1}{2}x-\frac{\pi }{6})$, determine the horizontal shift required to obtain the graph of $y=\sin \frac{1}{2}x$. | \frac{\pi}{3} | 0.8125 | 5,285.625 | 4,614.923077 | 8,192 | |
In triangle $A B C, \angle A B C$ is obtuse. Point $D$ lies on side $A C$ such that \angle A B D$ is right, and point $E$ lies on side $A C$ between $A$ and $D$ such that $B D$ bisects \angle E B C$. Find $C E$, given that $A C=35, B C=7$, and $B E=5$. | 10 | Reflect $A$ and $E$ over $B D$ to $A^{\prime}$ and $E^{\prime}$ respectively. Note that the angle conditions show that $A^{\prime}$ and $E^{\prime}$ lie on $A B$ and $B C$ respectively. $B$ is the midpoint of segment $A A^{\prime}$ and $C E^{\prime}=$ $B C-B E^{\prime}=2$. Menelaus' theorem now gives $$\frac{C D}{D A} ... | 0 | 8,192 | -1 | 8,192 |
If the function $f(x)=\sin \omega x+\sqrt{3}\cos \omega x$ $(x\in \mathbb{R})$, and $f(\alpha)=-2,f(\beta)=0$, with the minimum value of $|\alpha -\beta|$ being $\frac{3\pi}{4}$, determine the value of the positive number $\omega$. | \frac{2}{3} | 0.625 | 7,399.0625 | 6,923.3 | 8,192 | |
If $a = -2$, the largest number in the set $\{ -3a, 4a, \frac{24}{a}, a^2, 1\}$ is | -3a | 1. **Substitute $a = -2$ into each expression in the set**:
- $-3a = -3(-2) = 6$
- $4a = 4(-2) = -8$
- $\frac{24}{a} = \frac{24}{-2} = -12$
- $a^2 = (-2)^2 = 4$
- The last element in the set is $1$.
2. **List the evaluated set**:
\[
\{ -3a, 4a, \frac{24}{a}, a^2, 1 \} = \{ 6, -8, -12, 4, 1 \}
\... | 0 | 1,428.8125 | -1 | 1,428.8125 |
Teacher Tan awarded a stack of exercise books to the students who were named "Outstanding Students" in the math Olympiad class. If each student is awarded 3 books, there are 7 books left over; if each student is awarded 5 books, there are 9 books short. How many students received the award? How many exercise books are ... | 31 | 0.5 | 820.625 | 1,048 | 593.25 | |
Estimate the product $(.331)^3$. | 0.037 | 0 | 7,517.25 | -1 | 7,517.25 | |
Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$. | \frac{1}{16} | 0.1875 | 7,072.9375 | 5,665.333333 | 7,397.769231 | |
A student, Alex, is required to do a specified number of homework assignments to earn homework points using a different system: for the first four points, each point requires one homework assignment; for the next four points (points 5-8), each requires two homework assignments; then, for points 9-12, each requires thre... | 60 | 0.6875 | 6,352.1875 | 5,515.909091 | 8,192 | |
A store sells two suits at the same time, both priced at 168 yuan. One suit makes a 20% profit, while the other incurs a 20% loss. Calculate the net profit or loss of the store. | 14 | 0.3125 | 822.25 | 874.2 | 798.636364 | |
A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \(120^\circ\) and the other two angles being \(60^\circ\) each. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. Determine t... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
In the figure, $m\angle A = 28^{\circ}$, $m\angle B = 74^\circ$ and $m\angle C = 26^{\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x + y$? [asy]
size(150);
draw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));
draw((0,5)--(2,2)--(5,5)--(12,-2)--(15,5),linewidth(.7));
l... | 128 | 0 | 8,188.875 | -1 | 8,188.875 | |
Let $\left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in\{0,1\}, i \in \mathbf{N}_{+}, i \leqslant 6\right\}$. For $\forall x, y \in S$, $x=\left(x_{1}, x_{2}, \cdots, x_{6}\right)$ and $y=\left(y_{1}, y_{2}, \cdots, y_{6}\right)$, define:
1. $x=y$ if and only if $\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_... | 32 | 0.5 | 6,063.625 | 4,111.875 | 8,015.375 | |
Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : \[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\] | \[
\boxed{\frac{(k^2 - 4)^2}{4k(k^2 + 4)}}
\] | To start, we add the two fractions and simplify. \begin{align*} k &= \frac{(x^2+y^2)^2 + (x^2-y^2)^2}{x^4-y^4} \\ &= \frac{2x^4 + 2y^4}{x^4 - y^4}. \end{align*} Dividing both sides by two yields \[\frac{k}{2} = \frac{x^4 + y^4}{x^4 - y^4}.\] That means \begin{align*} \frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 +... | 0 | 7,700.5625 | -1 | 7,700.5625 |
Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\angle BAD=92^\circ$ and $\angle ADC=68^\circ$, find $\angle EBC$. | 68^\circ | 1. **Identify Properties of Cyclic Quadrilateral**: Since $ABCD$ is a cyclic quadrilateral, by the Inscribed Angle Theorem, opposite angles in a cyclic quadrilateral sum to $180^\circ$. Thus, we have:
\[
\angle ABC + \angle ADC = 180^\circ
\]
2. **Calculate $\angle ABC$**: Given $\angle ADC = 68^\circ$, we su... | 0.875 | 4,136.1875 | 3,556.785714 | 8,192 |
Given that $S_n$ and $T_n$ represent the sum of the first n terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ respectively, and $\frac{S_n}{T_n} = \frac{2n+1}{n+3}$, determine the value of $\frac{a_7}{b_7}$. | \frac{27}{16} | 0.6875 | 5,884.125 | 5,286.818182 | 7,198.2 | |
Given the function $f(x)=2\sin x\cos x+1-2\sin^2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$. | -\frac{\sqrt{3}+1}{2} | 0 | 4,382.125 | -1 | 4,382.125 | |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 72 | 0 | 8,049.5 | -1 | 8,049.5 | |
Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n$, $m_{i,i} = 10$; for $1 \le i \le n - 1$, $m_{i+1,i} = m_{i,i+1} = 3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\sum_{n=1}^{\infty} \frac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, whe... | 73 | \[D_{1}=\begin{vmatrix} 10 \end{vmatrix} = 10, \quad D_{2}=\begin{vmatrix} 10 & 3 \\ 3 & 10 \\ \end{vmatrix} =(10)(10) - (3)(3) = 91, \quad D_{3}=\begin{vmatrix} 10 & 3 & 0 \\ 3 & 10 & 3 \\ 0 & 3 & 10 \\ \end{vmatrix}.\] Using the expansionary/recursive definition of determinants (also stated in the problem):
$D_{3}=\l... | 0.8125 | 4,619.4375 | 3,795 | 8,192 |
The maximum and minimum values of the function $y=2x^{3}-3x^{2}-12x+5$ on the interval $[0,3]$ need to be determined. | -15 | 1 | 2,360.8125 | 2,360.8125 | -1 | |
The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary? | 8\% | 1 | 1,856.9375 | 1,856.9375 | -1 | |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | \tfrac{39}{11} | 1 | 3,203.5625 | 3,203.5625 | -1 | |
In the expression \((x+y+z)^{2030}+(x-y-z)^{2030}\), the parentheses were expanded, and like terms were collected. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a nonzero coefficient? | 1032256 | 0.0625 | 8,031.4375 | 5,623 | 8,192 | |
Three candles can burn for 30, 40, and 50 minutes, respectively (but are not ignited simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How long are exactly two candles burning simultaneously? | 35 | 0.0625 | 7,738.1875 | 6,893 | 7,794.533333 | |
Given the function \( f(x)=\frac{\sin (\pi x)-\cos (\pi x)+2}{\sqrt{x}} \) for \( \frac{1}{4} \leqslant x \leqslant \frac{5}{4} \), find the minimum value of \( f(x) \). | \frac{4\sqrt{5}}{5} - \frac{2\sqrt{10}}{5} | 0 | 8,098.25 | -1 | 8,098.25 | |
Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\circ}$. If the area of $\triangle SAB$ is $8$, then the volume of this cone is ______. | 8\pi | 0.5625 | 4,710.1875 | 4,285.333333 | 5,256.428571 | |
Months of the year are usually labeled numerically by '01' for January, '02' for February, and so on, through to '12' for December. Lydia notices that during January, the number of letters in the name of the month is greater than the month's numerical label (i.e., $7>1$). For how many days during 2024 will the date hav... | 121 | 0.3125 | 7,092.1875 | 6,989 | 7,139.090909 | |
What is the smallest integer whose square is 78 more than three times the integer? | -6 | 0 | 8,082.1875 | -1 | 8,082.1875 | |
The famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously "replace intuition with concepts." The function named after him, $D\left(x\right)=\left\{\begin{array}{l}{1, x \text{ is... | (1)(4) | 0 | 8,041.0625 | -1 | 8,041.0625 | |
Let $\mathbf{a} = \begin{pmatrix} 1 \\ -2 \\ -5 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} \sqrt{7} \\ 4 \\ -1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 13 \\ -4 \\ 17 \end{pmatrix}.$ Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}... | 90^\circ | 0.8125 | 4,937.1875 | 4,186.076923 | 8,192 | |
How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3? | 26 | 0 | 7,916.375 | -1 | 7,916.375 | |
Given a convex quadrilateral \( ABCD \) with \( X \) being the midpoint of the diagonal \( AC \). It is found that \( CD \parallel BX \). Find \( AD \) given that \( BX = 3 \), \( BC = 7 \), and \( CD = 6 \). | 14 | 0.3125 | 7,424.875 | 5,737.2 | 8,192 | |
In triangle $\triangle ABC$, the length of the side opposite angle $A$ is equal to 2, the vector $\overrightarrow {m} = (2, 2\cos^2 \frac {B+C}{2}-1)$, and the vector $\overrightarrow {n} = (\sin \frac {A}{2}, -1)$.
(1) Find the size of angle $A$ when the dot product $\overrightarrow {m} \cdot \overrightarrow {n}$ re... | \sqrt{3} | 0.9375 | 5,874.3125 | 5,719.8 | 8,192 | |
Let \( n = 1990 \). Find the value of the following expression:
$$
\frac{1}{2^{n}}\left(1-3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990} \right)
$$ | -\frac{1}{2} | 0.5625 | 5,792.4375 | 4,476.777778 | 7,484 | |
A certain store sells a type of student backpack. It is known that the cost price of this backpack is $30$ yuan each. Market research shows that the daily sales quantity $y$ (in units) of this backpack is related to the selling price $x$ (in yuan) as follows: $y=-x+60$ ($30\leqslant x\leqslant 60$). Let $w$ represent t... | 225 | 1 | 2,692.875 | 2,692.875 | -1 | |
Among all right triangles \(ABC\) with \( \angle C = 90^\circ\), find the maximum value of \( \sin A + \sin B + \sin^2 A \). | \sqrt{2} + \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
When $10^{95} - 95 - 2$ is expressed as a single whole number, calculate the sum of the digits. | 840 | 0.375 | 6,437.375 | 4,328.333333 | 7,702.8 | |
In $\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$?
$\textbf{(A)}\ 5/2\qquad\textbf{(... | \frac{7}{2} | 0 | 5,831.0625 | -1 | 5,831.0625 | |
Given an arithmetic sequence $\{a_{n}\}$ with a common difference of $\frac{{2π}}{3}$, let $S=\{\cos a_{n}|n\in N^{*}\}$. If $S=\{a,b\}$, find the value of $ab$. | -\frac{1}{2} | 0.75 | 5,712.75 | 4,886.333333 | 8,192 | |
In the Cartesian coordinate plane $xOy$, the parametric equations of the curve $C_1$ are given by $$\begin{cases} x=2\cos\phi \\ y=2\sin\phi \end{cases}$$ where $\phi$ is the parameter. By shrinking the abscissa of points on curve $C_1$ to $\frac{1}{2}$ of the original length and stretching the ordinate to twice the or... | \frac{60}{19} | 0.5625 | 6,096.4375 | 5,194.111111 | 7,256.571429 | |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. What is the length of side $AB$? | $\sqrt{17}$ | 0 | 5,891.125 | -1 | 5,891.125 | |
Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$. | -4 | 0.375 | 6,900.625 | 6,261.666667 | 7,284 | |
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end... | 351 | Let $E_n$ denotes the number of sequences with length $n$ that ends at $E$. Define similarly for the other vertices. We seek for a recursive formula for $E_n$. \begin{align*} E_n&=P_{3_{n-1}}+P_{4_{n-1}} \\ &=P_{2_{n-2}}+P_{5_{n-2}} \\ &=P_{1_{n-3}}+P_{3_{n-3}}+S_{n-3}+P_{4_{n-3}} \\ &=(P_{3_{n-3}}+P_{4_{n-3}})+S_{n-3}... | 0 | 8,175.1875 | -1 | 8,175.1875 |
What is the sum of the greatest common divisor of $45$ and $4410$ and the least common multiple of $45$ and $4410$? | 4455 | 1 | 2,592.8125 | 2,592.8125 | -1 | |
In triangle $ABC,$ $AB = 3,$ $AC = 6,$ $BC = 8,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$ | \frac{\sqrt{34}}{12} | 0 | 5,157.75 | -1 | 5,157.75 | |
In a rhombus \( ABCD \), the angle at vertex \( A \) is \( 60^\circ \). Point \( N \) divides side \( AB \) in the ratio \( AN:BN = 2:1 \). Find the tangent of angle \( DNC \). | \frac{\sqrt{243}}{17} | 0 | 5,852.1875 | -1 | 5,852.1875 | |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \theta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \... | 229 | 0 | 7,643.3125 | -1 | 7,643.3125 | |
Let \( n \) be a positive integer, \( [x] \) be the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \).
(1) Find all positive integers \( n \) satisfying \( \sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n \);
(2) Find all positive integers \( n \) that make \( \sum_{k=1}^{2013}\... | 1006 | 0 | 8,192 | -1 | 8,192 | |
Two connected rooms have different sizes and temperatures. One room has a length of $5 \,\mathrm{m}$, a width of $3 \,\mathrm{m}$, a height of $4 \,\mathrm{m}$, and a temperature of $22 \,^{\circ}\mathrm{C}$; the other room has a length of $6 \,\mathrm{m}$, a width of $5 \,\mathrm{m}$, a height of $4 \,\mathrm{m}$, and... | 16 | 0.75 | 3,813.3125 | 3,593.583333 | 4,472.5 | |
Given the function $f(x)=a\frac{x+1}{x}+\ln{x}$, the equation of the tangent line at the point (1, f(1)) is y=bx+5.
1. Find the real values of a and b.
2. Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$, where $e$ is the base of the natural logarithm. | 3+\ln{2} | 0.125 | 3,481.8125 | 3,800 | 3,436.357143 | |
Find the minimum value of
\[f(x) = x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}}\]for $x > 0.$ | \frac{5}{2} | 0.9375 | 4,624.375 | 4,386.533333 | 8,192 | |
A, B, and C are three people passing a ball to each other. The first pass is made by A, who has an equal chance of passing the ball to either of the other two people. After three passes, the probability that the ball is still with A is _______. | \frac{1}{4} | 0.5625 | 7,108.4375 | 6,265.666667 | 8,192 | |
If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$. | -1 | 0.75 | 5,225.5625 | 4,236.75 | 8,192 | |
Point \( M \) lies on side \( BC \) of parallelogram \( ABCD \) with a \(45^{\circ}\) angle at vertex \( A \), such that \(\angle AMD = 90^{\circ}\) and the ratio \( BM : MC = 2 : 3 \). Find the ratio of the adjacent sides of the parallelogram. | \frac{2\sqrt{2}}{5} | 0 | 7,050.4375 | -1 | 7,050.4375 | |
In the number \( 2016****02* \), each of the 5 asterisks needs to be replaced with any of the digits \( 0, 2, 4, 5, 7, 9 \) (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? | 864 | 0.125 | 7,881.875 | 7,384 | 7,953 | |
In a class of 38 students, we need to randomly select 5 people to participate in a survey. The number of possible selections where student A is chosen but student B is not is ______. (Express the answer numerically) | 58905 | 0.9375 | 3,352.5 | 3,029.866667 | 8,192 | |
When \(2x^2\) is added to a quadratic polynomial \(f(x)\), its maximum value increases by 10, and when \(5x^2\) is subtracted from it, its maximum value decreases by \(\frac{15}{2}\). By how much will the maximum value of \(f(x)\) change if \(3x^2\) is added to it? | \frac{45}{2} | 0.75 | 6,642.8125 | 6,300.666667 | 7,669.25 | |
The right vertex of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $A$. A line $l$ passing through the origin intersects the ellipse $C$ at points $P$ and $Q$. If $|PQ| = a$ and $AP \perpendicular PQ$, then the eccentricity of the ellipse $C$ is ______. | \frac{2\sqrt{5}}{5} | 0 | 6,468.0625 | -1 | 6,468.0625 | |
If for any $x \in D$, the inequality $f_1(x) \leq f(x) \leq f_2(x)$ holds, then the function $f(x)$ is called a "compromise function" of the functions $f_1(x)$ to $f_2(x)$ over the interval $D$. It is known that the function $f(x) = (k-1)x - 1$, $g(x) = 0$, $h(x) = (x+1)\ln x$, and $f(x)$ is a "compromise function" of ... | \{2\} | 0.125 | 6,430.5625 | 6,494.5 | 6,421.428571 | |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | \frac{4(3+2\sqrt{2})}{1} | 0 | 7,137.875 | -1 | 7,137.875 | |
A certain district's education department wants to send 5 staff members to 3 schools for earthquake safety education. Each school must receive at least 1 person and no more than 2 people. How many different arrangements are possible? (Answer with a number) | 90 | 0.25 | 7,837.8125 | 6,798.25 | 8,184.333333 | |
A cubic polynomial $p(x)$ satisfies
\[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$ | -\frac{5}{12} | 0.0625 | 8,088.875 | 6,542 | 8,192 | |
The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$. | 106 | 0.125 | 8,110.625 | 7,663 | 8,174.571429 | |
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$. | 12 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=\sin (\omega x+ \frac {\pi}{3})$ ($\omega > 0$), if $f( \frac {\pi}{6})=f( \frac {\pi}{3})$ and $f(x)$ has a minimum value but no maximum value in the interval $( \frac {\pi}{6}, \frac {\pi}{3})$, determine the value of $\omega$. | \frac {14}{3} | 0.125 | 8,132.25 | 7,714 | 8,192 | |
The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. | 1000 | 0.1875 | 536.0625 | 496.666667 | 545.153846 | |
Find the equation of the line that passes through the intersection of the lines $2x+3y+5=0$ and $2x+5y+7=0$, and is parallel to the line $x+3y=0$. Also, calculate the distance between these two parallel lines. | \frac{2\sqrt{10}}{5} | 0 | 5,629.8125 | -1 | 5,629.8125 | |
If $x=\frac{a}{b}$, $a\neq b$ and $b\neq 0$, then $\frac{a+b}{a-b}=$ | \frac{x+1}{x-1} | 1. Given that $x = \frac{a}{b}$, we can express $a$ in terms of $x$ and $b$:
\[
a = bx
\]
2. Substitute $a = bx$ into the expression $\frac{a+b}{a-b}$:
\[
\frac{a+b}{a-b} = \frac{bx + b}{bx - b}
\]
3. Factor out $b$ from both the numerator and the denominator:
\[
\frac{bx + b}{bx - b} = \frac{... | 1 | 2,519.0625 | 2,519.0625 | -1 |
In the diagram, if the area of $\triangle ABC$ is 36 where $A(3, 15)$, $B(15, 0)$, and $C(0, q)$ lie on a Cartesian plane. Determine the value of $q$.
[asy]
size(5cm);defaultpen(fontsize(9));
pair a = (3, 15); pair b = (15, 0); pair c = (0, 12);pair d= (3, 0);
draw(a--b--c--cycle);
label("$A(3, 15)$", a, N);
label("$B... | 12.75 | 0 | 7,948.9375 | -1 | 7,948.9375 | |
Alex has 12 friends and 63 coins. What is the minimum number of additional coins he needs so that he can give each friend at least one coin and no two friends receive the same number of coins? | 15 | 1 | 1,734.375 | 1,734.375 | -1 | |
Find the smallest exact square with last digit not $0$ , such that after deleting its last two digits we shall obtain another exact square. | 121 | 0.125 | 8,112.9375 | 7,559.5 | 8,192 | |
Find all prime numbers whose representation in a base-14 numeral system has the form 101010...101 (alternating ones and zeros). | 197 | 0.3125 | 8,028.375 | 7,668.4 | 8,192 |
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