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 |
|---|---|---|---|---|---|---|
Simplify completely: $$\sqrt[3]{40^3 + 70^3 + 100^3}.$$ | 10 \sqrt[3]{1407} | 0.375 | 7,116.3125 | 5,937 | 7,823.9 | |
A community plans to organize three activities, "Book Club," "Fun Sports," and "Environmental Theme Painting," to enrich the lives of residents. A total of 120 people have signed up for the activities, with each resident participating in at most two activities. It is known that 80 people participate in the "Book Club,"... | 20 | 1 | 3,026.8125 | 3,026.8125 | -1 | |
How many integers $n$ in the set $\{4,9,14,19, \ldots, 2014\}$ have the property that the sum of the decimal digits of $n$ is even? | 201 | We know that 2014 does not qualify the property. So, we'll consider $\{4,9,14, \ldots, 2009\}$ instead. Now, we partition this set into 2 sets: $\{4,14,24, \ldots, 2004\}$ and $\{9,19,29, \ldots, 2009\}$. For each so the first and second set are basically $x 4$ and $x 9$, where $x=0,1,2, \ldots, 200$, respectively. And... | 0 | 7,861.5 | -1 | 7,861.5 |
In a course conducted by Professor Jones, each student is on average absent for one day out of a 40-day course. What is the probability that out of any two randomly selected students, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth. | 4.9\% | 1 | 3,161.3125 | 3,161.3125 | -1 | |
Find all solutions to $x^{4}+2 x^{3}+2 x^{2}+2 x+1=0$ (including non-real solutions). | -1, i, -i | We can factor the polynomial as $(x+1)^{2}(x^{2}+1)$. Therefore, the solutions are $-1, i, -i$. | 0 | 3,905.625 | -1 | 3,905.625 |
Four points $B,$ $A,$ $E,$ and $L$ are on a straight line, as shown. The point $G$ is off the line so that $\angle BAG = 120^\circ$ and $\angle GEL = 80^\circ.$ If the reflex angle at $G$ is $x^\circ,$ then what does $x$ equal?
[asy]
draw((0,0)--(30,0),black+linewidth(1));
draw((10,0)--(17,20)--(15,0),black+linewidth(... | 340 | 0 | 6,269.3125 | -1 | 6,269.3125 | |
Calculate $\cos \frac{\pi}{9} \cdot \cos \frac{2\pi}{9} \cdot \cos \frac{4\pi}{9} = $ ______. | \frac{1}{8} | 0.6875 | 6,383.6875 | 5,561.727273 | 8,192 | |
Define $||x||$ $(x\in R)$ as the integer closest to $x$ (when $x$ is the arithmetic mean of two adjacent integers, $||x||$ takes the larger integer). Let $G(x)=||x||$. If $G(\frac{4}{3})=1$, $G(\frac{5}{3})=2$, $G(2)=2$, and $G(2.5)=3$, then $\frac{1}{G(1)}+\frac{1}{G(2)}+\frac{1}{G(3)}+\frac{1}{G(4)}=$______; $\frac{1... | \frac{1334}{15} | 0.0625 | 8,070.6875 | 6,251 | 8,192 | |
If the value of the expression $(\square + 121 \times 3.125) \div 121$ is approximately 3.38, what natural number should be placed in $\square$? | 31 | 0.3125 | 677.375 | 720.6 | 657.727273 | |
The numbers $1,2, \ldots, 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k<10$, there exists an integer $k^{\prime}>k$ such that there is at most one number between $k$ and $k^{\prime}$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime posit... | 1390 | Let $n=10$ and call two numbers close if there is at most one number between them and an circular permutation focused if only $n$ is greater than all numbers close to it. Let $A_{n}$ be the number of focused circular permutations of $\{1,2, \ldots, n\}$. If $n \geq 5$, then there are 2 cases: $n-1$ is either one or two... | 0 | 8,126 | -1 | 8,126 |
A series of numbers were written: \(100^{100}, 101^{101}, 102^{102}, \ldots, 234^{234}\) (i.e., the numbers of the form \(n^{n}\) for natural \(n\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an integer.) | 71 | 0.4375 | 5,770.625 | 5,269.857143 | 6,160.111111 | |
What is the smallest prime factor of 1739? | 1739 | 0 | 2,720.8125 | -1 | 2,720.8125 | |
Let's call a word any finite sequence of letters of the Russian alphabet. How many different four-letter words can be made from the letters of the word КАША? And from the letters of the word ХЛЕБ? Indicate the sum of the found numbers in the answer. | 36 | 0.5 | 3,947.125 | 2,509.75 | 5,384.5 | |
If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\). | \sqrt{10} | 0 | 8,192 | -1 | 8,192 | |
Emilia wishes to create a basic solution with $7 \%$ hydroxide $(\mathrm{OH})$ ions. She has three solutions of different bases available: $10 \%$ rubidium hydroxide $(\mathrm{Rb}(\mathrm{OH})$ ), $8 \%$ cesium hydroxide $(\mathrm{Cs}(\mathrm{OH})$ ), and $5 \%$ francium hydroxide $(\operatorname{Fr}(\mathrm{OH})$ ). (... | 1 \% | Suppose that Emilia uses $R$ liters of $\mathrm{Rb}(\mathrm{OH}), C$ liters of $\mathrm{Cs}(\mathrm{OH})$, and $F$ liters of $\mathrm{Fr}(\mathrm{OH})$, then we have $$\frac{10 \% \cdot R+8 \% \cdot C+5 \% \cdot F}{R+C+F}=7 \% \text { and } \frac{5 \% \cdot F}{R+C+F} \leq 2 \%$$ The equations simplify to $3 R+C=2 F$ an... | 0.0625 | 7,865.4375 | 5,710 | 8,009.133333 |
Triangle $A B C$ has $A B=10, B C=17$, and $C A=21$. Point $P$ lies on the circle with diameter $A B$. What is the greatest possible area of $A P C$? | \frac{189}{2} | To maximize $[A P C]$, point $P$ should be the farthest point on the circle from $A C$. Let $M$ be the midpoint of $A B$ and $Q$ be the projection of $M$ onto $A C$. Then $P Q=P M+M Q=\frac{1}{2} A B+\frac{1}{2} h_{B}$, where $h_{B}$ is the length of the altitude from $B$ to $A C$. By Heron's formula, one finds that th... | 0.3125 | 7,778.4375 | 6,868.6 | 8,192 |
A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\%$ higher than the percentage of older people. Additionally, the... | 120 | 0.5 | 5,443.125 | 3,972.75 | 6,913.5 | |
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample? | \frac{7}{4} | 0 | 5,795.5625 | -1 | 5,795.5625 | |
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(... | 288 | Draw an altitude from a vertex of the square base to the top edge. By using $30,60, 90$ triangle ratios, we obtain that the altitude has a length of $3 \sqrt{6}$, and that little portion that hangs out has a length of $3\sqrt2$. This is a triangular pyramid with a base of $3\sqrt6, 3\sqrt6, 3\sqrt2$, and a height of $3... | 0 | 8,192 | -1 | 8,192 |
Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$. | \frac{2865 - 120\sqrt{165}}{49} | 0 | 8,192 | -1 | 8,192 | |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
To solve this problem, we need to determine the smallest initial number $N$ such that Bernardo wins the game. We will analyze the sequence of operations and the conditions under which Bernardo wins.
1. **Sequence of Operations**:
- Bernardo receives $N$, doubles it: $2N$
- Silvia receives $2N$, adds 50: $2N + 5... | 0 | 8,012.375 | -1 | 8,012.375 |
The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction. | \frac{1}{4} | 0.625 | 4,837.3125 | 4,029.2 | 6,184.166667 | |
A shopkeeper set up incorrect scales in his shop, where one side of the balance beam is longer than the other. During one weighing, 3 cans balanced with 8 packets, and during another, 1 packet balanced with 6 cans. Given that the true weight of one can is 1 kg, how much do 8 packets weigh? | 12 | 0.3125 | 6,360.3125 | 4,554.6 | 7,181.090909 | |
Let triangle $ABC$ have incircle $\omega$, which touches $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. Then, let $\omega_{1}$ and $\omega_{2}$ be circles tangent to $AD$ and internally tangent to $\omega$ at $E$ and $F$, respectively. Let $P$ be the intersection of line $EF$ and the line passing through the cent... | 3600 | Let the centers of $\omega_{1}$ and $\omega_{2}$ be $O_{1}$ and $O_{2}$. Let $DE$ intersect $\omega_{1}$ again at $Q$, and let $DF$ intersect $\omega_{2}$ again at $R$. Note that since $\omega_{1}$ and $\omega_{2}$ must be tangent to $AD$ at the same point (by equal tangents), so $AD$ must be the radical axis of $\omeg... | 0 | 8,192 | -1 | 8,192 |
In the Cartesian coordinate system, with the origin O as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. The polar coordinate of point P is $(1, \pi)$. Given the curve $C: \rho=2\sqrt{2}a\sin(\theta+ \frac{\pi}{4}) (a>0)$, and a line $l$ passes through point P, whose parame... | 2\sqrt{3}-2 | 0 | 8,192 | -1 | 8,192 | |
Given that
\[
\frac{1}{x} + \frac{1}{y} = 4, \quad x + y = 5,
\]
compute \(x^2 + y^2\). | \frac{35}{2} | 0 | 2,215.875 | -1 | 2,215.875 | |
Find the area of the region enclosed by the graph of \( |x-75| + |y| = \left|\frac{x}{3}\right| \). | 703.125 | 0 | 8,027.5625 | -1 | 8,027.5625 | |
A basketball player scored a mix of free throws, 2-pointers, and 3-pointers during a game, totaling 7 successful shots. Find the different numbers that could represent the total points scored by the player, assuming free throws are worth 1 point each. | 15 | 0 | 7,191.25 | -1 | 7,191.25 | |
Given the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{20},$$ find the sum $$\beta_1 + \beta_2 + \beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\beta_k=r_k[\cos(2\pi\beta_k)+i\sin(2\pi\beta_k)]$, with $0<\beta_1\le\beta_2\le\ldots\le\beta_{41}<1$ and $r_k>0$. | \frac{3}{7} | 0 | 8,117 | -1 | 8,117 | |
Consider a $7 \times 7$ grid of squares. Let $f:\{1,2,3,4,5,6,7\} \rightarrow\{1,2,3,4,5,6,7\}$ be a function; in other words, $f(1), f(2), \ldots, f(7)$ are each (not necessarily distinct) integers from 1 to 7 . In the top row of the grid, the numbers from 1 to 7 are written in order; in every other square, $f(x)$ is ... | 1470 | Consider the directed graph with $1,2,3,4,5,6,7$ as vertices, and there is an edge from $i$ to $j$ if and only if $f(i)=j$. Since the bottom row is equivalent to the top one, we have $f^{6}(x)=x$. Therefore, the graph must decompose into cycles of length $6,3,2$, or 1 . Furthermore, since no other row is equivalent to ... | 0 | 5,904.4375 | -1 | 5,904.4375 |
The number
\[\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ\]is expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees. | 111^\circ | 0.375 | 7,167.125 | 5,459 | 8,192 | |
Given that the function $y = (m^2 + 2m - 2)x^{\frac{1}{m-1}}$ is a power function, find the value of $m$. | -3 | 0.0625 | 6,506.3125 | 4,398 | 6,646.866667 | |
A school selects 4 teachers from 8 to teach in 4 remote areas, with one teacher per area. Among them, A and B cannot go together, and A and C must either both go or both not go. Derive the total number of different dispatch plans. | 600 | 0.0625 | 7,845.75 | 6,567 | 7,931 | |
The perimeter of triangle \(ABC\) is 1. Circle \(\omega\) is tangent to side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the... | 0.5 | 0 | 8,192 | -1 | 8,192 | |
The angle can be represented by the two uppercase letters on its sides and the vertex letter. The angle in the diagram $\angle A O B$ symbol ("$\angle$" represents angle) can also be represented by $\angle O$ (when there is only one angle). In the triangle $\mathrm{ABC}$ below, given $\angle B A O = \angle C A O$, $\an... | 20 | 1 | 3,301.125 | 3,301.125 | -1 | |
If $\mathbf{A}^{-1} = \begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix},$ then find the inverse of $\mathbf{A}^2.$ | \begin{pmatrix} -1 & -5 \\ 1 & 4 \end{pmatrix} | 0.8125 | 4,213.5 | 3,295.384615 | 8,192 | |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$? | 4625 | 0.125 | 7,951.9375 | 6,271.5 | 8,192 | |
Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers? | 23 | 0.25 | 6,935.6875 | 5,126.75 | 7,538.666667 | |
The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
| 888 | 0 | 8,002.875 | -1 | 8,002.875 | |
The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property? | 46 | We are tasked with finding how many integers between 1000 and 2013 have the property that their units digit is the sum of the other digits. We will consider two cases based on the thousands digit, which can be either 1 or 2.
#### Case 1: Thousands digit is 1
The number is of the form $\overline{1bcd}$, where $b, c, d$... | 0.1875 | 7,923.8125 | 7,563 | 8,007.076923 |
Given that $α∈(0,π)$, and $\sin α + \cos α = \frac{\sqrt{2}}{2}$, find the value of $\sin α - \cos α$. | \frac{\sqrt{6}}{2} | 0 | 3,940.5625 | -1 | 3,940.5625 | |
Given an ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.
(Ⅰ) Find the... | \frac {1}{2} | 0.0625 | 8,000.0625 | 5,121 | 8,192 | |
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy $(ab+1)(bc+1)(ca+1)=84$. | 12 | The solutions are $(0,1,83)$ and $(1,2,3)$ up to permutation. First, we do the case where at least one of $a, b, c$ is 0. WLOG, say $a=0$. Then we have $1+bc=84 \Longrightarrow bc=83$. As 83 is prime, the only solution is $(0,1,83)$ up to permutation. Otherwise, we claim that at least one of $a, b, c$ is equal to 1. Ot... | 0 | 8,192 | -1 | 8,192 |
Determine all polynomials $P(x)$ with real coefficients such that
$P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$. | P(x) = 0 |
To solve the problem, we need to determine all polynomials \( P(x) \) with real coefficients satisfying the equation:
\[
P(x)^2 + P\left(\frac{1}{x}\right)^2 = P(x^2)P\left(\frac{1}{x^2}\right)
\]
for all \( x \).
### Step 1: Analyze the Equation
Let's start by inspecting the given functional equation. Set \( x = ... | 0 | 8,192 | -1 | 8,192 |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$. | \frac{9 \pi}{16} | 0 | 8,156.6875 | -1 | 8,156.6875 | |
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? | \pi + 6\sqrt{3} | To solve this problem, we need to understand the geometric construction and calculate the area enclosed by the curve.
1. **Understanding the Construction**:
- The curve is made up of 9 congruent circular arcs, each with a length of $\frac{2\pi}{3}$.
- The centers of these arcs are located at the vertices of a re... | 0 | 8,132.5 | -1 | 8,132.5 |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). \(D\) is a point chosen on \(BC\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\). \(DE + DF\) equals: | 24 | 0.875 | 6,201.8125 | 5,917.5 | 8,192 | |
Let $A B C D$ be a rectangle with $A B=8$ and $A D=20$. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to $A B$ and $A D$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles? | 112-25 \pi | Let $O_{1}$ and $O_{2}$ be the centers of the circles, and let $M$ be the midpoint of $\overline{C D}$. We can see that $\triangle O_{2} M C$ and $\triangle O_{2} M D$ are both 3-4-5 right triangles. Now let $C^{\prime}$ be the intersection of circle $O_{2}$ and $\overline{B C}$ (that isn't $C$ ), and let $D^{\prime}$ ... | 0 | 7,604.4375 | -1 | 7,604.4375 |
Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \times 14$ rectangular grid? | 16 | 0 | 7,953.6875 | -1 | 7,953.6875 | |
Let $x$ and $y$ be real numbers greater than 1 such that
\[(\log_2 x)^4 + (\log_3 y)^4 + 8 = 8 (\log_2 x)(\log_3 y).\]Compute $x^{\sqrt{2}} + y^{\sqrt{2}}.$ | 13 | 1 | 3,195.4375 | 3,195.4375 | -1 | |
(Caos) A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not... | 1416528 | ```
Answer: 1416528
# 1 = on ground, 0 = raised, 2 = back on ground
cache = {}
def pangzi(legs):
if legs == (2,2,2,2,2,2): return 1
elif legs.count(0) > 3: return 0
elif legs[0] + legs[1] + legs[2] == 0: return 0
elif legs[3] + legs[4] + legs[5] == 0: return 0
elif cache.has_key(legs): return cache[... | 0 | 8,192 | -1 | 8,192 |
Express $\frac{31}{2\cdot5^6}$ as a terminating decimal. | 0.000992 | 0.5625 | 5,026.5 | 3,880.444444 | 6,500 | |
Given real numbers $x$, $y$, $z$ satisfying $\begin{cases} xy+2z=1 \\ x^{2}+y^{2}+z^{2}=5 \end{cases}$, the minimum value of $xyz$ is \_\_\_\_\_\_. | 9 \sqrt {11}-32 | 0 | 8,192 | -1 | 8,192 | |
35 times 61,000 unit cubes are combined to form a large cube with an edge length of 10 units. After being painted, the large cube is then separated back into the original unit cubes. How many of these unit cubes have at least one face painted? | 488 | 0.0625 | 8,173.625 | 7,898 | 8,192 | |
In $\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$ so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find ... | 289 | By the Law of Cosines, we have: \[\cos(A) = \frac{10^2+10^2-12^2}{2*10*10} = \frac{7}{25}\] Let $M$ be midpoint of $AE$, then \[\frac{7}{25} = \frac{10-x}{2x} \iff x =\frac{250}{39}\] So, our answer is $250+39=\boxed{289}$. | 0.8125 | 5,950.125 | 5,432.769231 | 8,192 |
Add together all natural numbers less than 1980 for which the sum of their digits is even! | 979605 | 0 | 8,192 | -1 | 8,192 | |
Circles $A$ and $B$ each have a radius of 1 and are tangent to each other. Circle $C$ has a radius of 2 and is tangent to the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$
A) $1.16$
B) $3 \pi - 2.456$
C) $4 \pi - 4.912$
D) $2 \pi$
E) $\pi + 4.912$ | 4 \pi - 4.912 | 0 | 8,030.3125 | -1 | 8,030.3125 | |
Given that a class selects 4 athletes from 5 male and 4 female track and field athletes to participate in the competition, where the selection must include both male and female athletes, and at least one of the male athlete A or female athlete B must be selected, calculate the number of ways to select the athletes. | 86 | 0.0625 | 7,731.875 | 7,914 | 7,719.733333 | |
If $\cos \theta + \sin \theta = \frac{5}{4},$ then find $\sin 2 \theta.$ | \frac{9}{16} | 1 | 2,478.3125 | 2,478.3125 | -1 | |
Let $n$ be the answer to this problem. Hexagon $ABCDEF$ is inscribed in a circle of radius 90. The area of $ABCDEF$ is $8n$, $AB=BC=DE=EF$, and $CD=FA$. Find the area of triangle $ABC$. | 2592 | Let $O$ be the center of the circle, and let $OB$ intersect $AC$ at point $M$; note $OB$ is the perpendicular bisector of $AC$. Since triangles $ABC$ and $DEF$ are congruent, $ACDF$ has area $6n$, meaning that $AOC$ has area $3n/2$. It follows that $\frac{BM}{OM}=\frac{2}{3}$. Therefore $OM=54$ and $MB=36$, so by the P... | 0.25 | 7,316.6875 | 5,923.75 | 7,781 |
Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \(\pi\). | 16\pi (\sqrt{10} + 1) | 0.3125 | 887.625 | 820 | 918.363636 | |
The real numbers $a$ and $b$ satisfy
\[\begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix} \times \begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = \mathbf{0}.\]Enter the ordered pair $(a,b).$ | \left( \frac{8}{5}, -\frac{35}{2} \right) | 1 | 1,980.875 | 1,980.875 | -1 | |
Given that $a > b > 0$, and $a + b = 2$, find the minimum value of $$\frac {3a-b}{a^{2}+2ab-3b^{2}}$$. | \frac {3+ \sqrt {5}}{4} | 0 | 6,591.6875 | -1 | 6,591.6875 | |
Let \( x \) and \( y \) be positive integers such that
\[ x^2 + y^2 - 2017xy > 0 \]
and it is not a perfect square. Find the minimum value of \( x^2 + y^2 - 2017xy \). | 2019 | 0 | 8,192 | -1 | 8,192 | |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+m^2+n^2,$ for all positive integers $m$ and $n.$ Find $b_{12}.$ | 160 | 0 | 8,042.5 | -1 | 8,042.5 | |
For each positive integer $ n$, let $ c(n)$ be the largest real number such that
\[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\]
for all triples $ (f, a, b)$ such that
--$ f$ is a polynomial of degree $ n$ taking integers to integers, and
--$ a, b$ are integers with $ f(a) \neq f(b)$.
Find... | \frac{1}{L_n} |
For each positive integer \( n \), let \( c(n) \) be the largest real number such that
\[
c(n) \le \left| \frac{f(a) - f(b)}{a - b} \right|
\]
for all triples \( (f, a, b) \) such that:
- \( f \) is a polynomial of degree \( n \) taking integers to integers, and
- \( a, b \) are integers with \( f(a) \neq f(b) \).
To... | 0 | 8,187.0625 | -1 | 8,187.0625 |
In order to test students' mastery of high school mathematics knowledge, two opaque boxes, Box A and Box B, are prepared. Box A contains 2 conceptual description questions and 2 calculation questions; Box B contains 2 conceptual description questions and 3 calculation questions (all questions are different). Two studen... | \frac{3}{7} | 0.3125 | 6,145.5 | 4,397.2 | 6,940.181818 | |
A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median? | 6.5 | 0.1875 | 7,750 | 7,095.666667 | 7,901 | |
Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend from any other side. | 24 | 0 | 4,451.3125 | -1 | 4,451.3125 | |
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace$. Then the area of $S$ has the form $a\pi + \sqrt{b}$, wh... | 29 | We can describe the line parallel to the imaginary axis $x=\frac{1}{2}$ using polar coordinates as $r(\theta)=\dfrac{1}{2\cos{\theta}},$
which rearranges to $z=\left(\dfrac{1}{2\cos{\theta}}\right)(cis{\theta})\implies \frac{1}{z}=2\cos{\theta}cis(-\theta).$
Denote the area of $S$ as $[S].$ Now, dividing the hexagon ... | 0 | 8,192 | -1 | 8,192 |
If $x=3$, what is the value of $-(5x - 6x)$? | 3 | When $x=3$, we have $-(5x - 6x) = -(-x) = x = 3$. Alternatively, when $x=3$, we have $-(5x - 6x) = -(15 - 18) = -(-3) = 3$. | 0.9375 | 429.25 | 444.466667 | 201 |
Suppose a sequence starts with 1254, 2547, 5478, and ends with 4781. Let $T$ be the sum of all terms in this sequence. Find the largest prime factor that always divides $T$. | 101 | 0 | 6,146.25 | -1 | 6,146.25 | |
A shooter fires at a target until the first hit, with a hit rate of 0.6 for each shot. If there are 4 bullets in total, what is the expected number of remaining bullets $\xi$? | 2.376 | 0 | 8,077.5625 | -1 | 8,077.5625 | |
The coefficients of the polynomial
\[a_{10} x^{10} + a_9 x^9 + a_8 x^8 + \dots + a_2 x^2 + a_1 x + a_0 = 0\]are all integers, and its roots $r_1,$ $r_2,$ $\dots,$ $r_{10}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{10} + a_1 x^9 + a_2 x^8 + \dots + a_8 x^2 + a_9 x + a_{10} = 0\]are also $r_1,$... | 11 | 0.3125 | 7,635.5 | 6,411.2 | 8,192 | |
What is the smallest natural number that is divisible by 2022 and starts with 2023? | 20230110 | 0.3125 | 7,738.1875 | 6,739.8 | 8,192 | |
In an arithmetic sequence $\{a_n\}$, $S_n$ represents the sum of the first $n$ terms. Given that $a_4 + a_8 = 4$, find the value of $S_{11} + a_6$. | 24 | 1 | 2,689.0625 | 2,689.0625 | -1 | |
There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on? | 499 | 0.25 | 7,691.5625 | 6,621.75 | 8,048.166667 | |
Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$. | 250\pi | 0.6875 | 4,649.375 | 3,039.090909 | 8,192 | |
The fenced area of a yard is an 18.5-foot by 14-foot rectangular region with a 3.5-foot by 3.5-foot square cutout. Calculate the area of the region within the fence, in square feet. | 246.75 | 1 | 2,013.0625 | 2,013.0625 | -1 | |
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$? | 7 | 1 | 3,706.125 | 3,706.125 | -1 | |
Jill has 8 red marbles and 4 blue marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are red? Express your answer as a decimal rounded to the nearest... | 0.171 | 0.25 | 7,805.5 | 6,646 | 8,192 | |
The convex pentagon $ABCDE$ has $\angle A = \angle B = 120^\circ$, $EA = AB = BC = 2$ and $CD = DE = 4$. What is the area of $ABCDE$?
[asy]
unitsize(1 cm);
pair A, B, C, D, E;
A = (0,0);
B = (1,0);
C = B + dir(60);
D = C + 2*dir(120);
E = dir(120);
draw(A--B--C--D--E--cycle);
label("$A$", A, SW);
label("$B$... | 7 \sqrt{3} | 0.375 | 7,388.25 | 6,048.666667 | 8,192 | |
The equations $x^3 + Cx + 20 = 0$ and $x^3 + Dx^2 + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | 15 | 0.5 | 6,387.75 | 4,815.5 | 7,960 | |
In the company, there are elves, fairies, and dwarves. Each elf is friends with all fairies except for three of them, and each fairy is friends with twice as many elves. Each elf is friends with exactly three dwarves, and each fairy is friends with all the dwarves. Each dwarf is friends with exactly half of the total n... | 12 | 0.0625 | 7,326.875 | 5,141 | 7,472.6 | |
A school organizes a table tennis competition. Class A has 5 male students and 3 female students registered; Class B has 6 male students and 2 female students registered. If 2 students are selected from each of Class A and Class B, there are a total of $\boxed{345}$ different ways to select 4 students such that exactly... | 345 | 0.9375 | 4,773.6875 | 4,545.8 | 8,192 | |
Let \( p, q, r, \) and \( s \) be the roots of the polynomial
\[ x^4 + 10x^3 + 20x^2 + 15x + 6 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}. \] | \frac{10}{3} | 0.5 | 6,777.3125 | 5,367.625 | 8,187 | |
Twelve tiles numbered $1$ through $12$ are turned up at random, and an eight-sided die is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a perfect square. | \frac{13}{96} | 0 | 7,847.5 | -1 | 7,847.5 | |
Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$. | 501 |
Denote \( S \) as a subset of \( \{ 1, 2, 3, \ldots, 1000 \} \) with the property that no sum of two different elements in \( S \) is itself an element of \( S \). We wish to find the maximum number of elements in \( S \).
To address this problem, consider the possibility of selecting elements from \( \{ 1, 2, 3, \ld... | 0 | 7,628.9375 | -1 | 7,628.9375 |
Guangcai Kindergarten has a total of 180 books, of which 40% are given to the senior class. The remaining books are divided between the junior and middle classes in a ratio of 4:5. How many books does each of the junior and middle classes get? | 60 | 0.125 | 560.75 | 583 | 557.571429 | |
Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$.
| 472 | 0 | 8,099.375 | -1 | 8,099.375 | |
Calculate using factorization:<br/>$(1)\frac{2021×2023}{2022^2-1}$;<br/>$(2)2\times 101^{2}+2\times 101\times 98+2\times 49^{2}$. | 45000 | 0.75 | 4,638.8125 | 3,454.416667 | 8,192 | |
Given an equilateral triangle $PQR$ with a side length of 8 units, a process similar to the previous one is applied, but here each time, the triangle is divided into three smaller equilateral triangles by joining the midpoints of its sides, and the middle triangle is shaded each time. If this procedure is repeated 100 ... | 8\sqrt{3} | 0 | 7,879.125 | -1 | 7,879.125 | |
A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\%$, what is the volume of the new pyramid, in cubic inches? | 360 | 1 | 2,566.25 | 2,566.25 | -1 | |
Given the function $f(x)=\cos^2x+\cos^2\left(x-\frac{\pi}{3}\right)-1$, where $x\in \mathbb{R}$,
$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;
$(2)$ The function $f(x)$ is translated to the right by $\frac{\pi}{3}$ units to obtain the function $g(x)$. Find the expression ... | - \frac{\sqrt{3}}{4} | 0 | 5,638.3125 | -1 | 5,638.3125 | |
A can is in the shape of a right circular cylinder. The circumference of the base of the can is 12 inches, and the height of the can is 5 inches. A spiral strip is painted on the can in such a way that it winds around the can exactly once as it reaches from the bottom of the can to the top. It reaches the top of the ca... | 13 | 1 | 1,518.6875 | 1,518.6875 | -1 | |
What is the hundreds digit of $(20! - 15!)?$ | 0 | To find the hundreds digit of $(20! - 15!)$, we need to analyze the factorials and their properties modulo $1000$.
1. **Factorial Properties**:
- $n!$ (where $n \geq 5$) contains at least one factor of $5$ and at least one factor of $2$, making it divisible by $10$.
- $n!$ (where $n \geq 10$) contains at least t... | 0.625 | 5,808.875 | 4,379 | 8,192 |
Consider the function:
\[ f(x) = \left\{
\begin{aligned}
3x + 1 & \quad \text{ if } x \leq 2 \\
x^2 & \quad \text{ if } x > 2
\end{aligned}
\right.\]
The function has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(0) + \dots + f^{-1}(8) + f^{-1}(9)$. | 22 + 2\sqrt{2} | 0 | 7,738.5 | -1 | 7,738.5 | |
Given the function $f(x)=2\sin x( \sqrt {3}\cos x+\sin x)-2$.
1. If point $P( \sqrt {3},-1)$ is on the terminal side of angle $α$, find the value of $f(α)$.
2. If $x∈[0, \frac {π}{2}]$, find the minimum value of $f(x)$. | -2 | 1 | 4,817.125 | 4,817.125 | -1 | |
A certain scenic area has two attractions that require tickets for visiting. The three ticket purchase options presented at the ticket office are as follows:
Option 1: Visit attraction A only, $30$ yuan per person;
Option 2: Visit attraction B only, $50$ yuan per person;
Option 3: Combined ticket for attractions ... | 188.1 | 0.1875 | 6,974.375 | 5,262.666667 | 7,369.384615 | |
How many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? | 1 | 0.875 | 3,683.5 | 3,039.428571 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.