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 |
|---|---|---|---|---|---|---|
Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$).
If the graphs of $y=f(x^2)$ and $y=f(x^4)$ are drawn, at how many points do they intersect? | 3 | 1 | 2,927 | 2,927 | -1 | |
A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other? | 41 | If Bill and Karl are on the committee, there are $\binom{7}{3}=35$ ways for the other group members to be chosen. However, if Alice and Jane are on the committee with Bill and Karl, there are $\binom{5}{1}=5$ ways for the last member to be chosen, yielding 5 unacceptable committees. If Bill and Karl are not on the comm... | 0.375 | 7,596.5 | 6,604 | 8,192 |
Given the function $f(x) = 2\sin^2\left(\frac{\pi}{4} + x\right) - \sqrt{3}\cos{2x} - 1$, where $x \in \mathbb{R}$:
1. If the graph of function $h(x) = f(x + t)$ is symmetric about the point $\left(-\frac{\pi}{6}, 0\right)$, and $t \in \left(0, \frac{\pi}{2}\right)$, find the value of $t$.
2. In an acute triangle $ABC$... | \frac{\pi}{3} | 0 | 8,117.8125 | -1 | 8,117.8125 | |
Let $A B C$ be an equilateral triangle of side length 15 . Let $A_{b}$ and $B_{a}$ be points on side $A B, A_{c}$ and $C_{a}$ be points on side $A C$, and $B_{c}$ and $C_{b}$ be points on side $B C$ such that $\triangle A A_{b} A_{c}, \triangle B B_{c} B_{a}$, and $\triangle C C_{a} C_{b}$ are equilateral triangles wit... | 3 \sqrt{3} | Let $\triangle X Y Z$ be the triangle formed by lines $A_{b} A_{c}, B_{a} B_{c}$, and $C_{a} C_{b}$. Then, the desired circle is the incircle of $\triangle X Y Z$, which is equilateral. We have $$\begin{aligned} Y Z & =Y A_{c}+A_{c} A_{b}+A_{b} Z \\ & =A_{c} C_{a}+A_{c} A_{b}+A_{b} B_{a} \\ & =(15-3-5)+3+(15-3-4) \\ & ... | 0 | 8,192 | -1 | 8,192 |
Rectangle $EFGH$ has an area of $4032$. An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of rectangle $EFGH$. | 8\sqrt{2016} | 0 | 7,269 | -1 | 7,269 | |
In acute $\triangle A B C$ with centroid $G, A B=22$ and $A C=19$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\prime}$ be the reflection of $G$ over $B C$. If $E, F, G$, and $G^{\prime}$ lie on a circle, compute $B C$. | 13 | Note that $B, C, E, F$ lie on a circle. Moreover, since $B C$ bisects $G G^{\prime}$, the center of the circle that goes through $E, F, G, G^{\prime}$ must lie on $B C$. Therefore, $B, C, E, F, G, G^{\prime}$ lie on a circle. Specifically, the center of this circle is $M$, the midpoint of $B C$, as $M E=M F$ because $M... | 0 | 8,192 | -1 | 8,192 |
Evaluate $101 \times 101$ using a similar mental math technique. | 10201 | 0.8125 | 1,031.5625 | 1,154.769231 | 497.666667 | |
In recent years, the food delivery industry in China has been developing rapidly, and delivery drivers shuttling through the streets of cities have become a beautiful scenery. A certain food delivery driver travels to and from $4$ different food delivery stores (numbered $1, 2, 3, 4$) every day. The rule is: he first p... | 0.25 | 0.125 | 8,156.5625 | 7,908.5 | 8,192 | |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(2\overrightarrow{a}+\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 0.125 | 1,671.25 | 2,054 | 1,616.571429 | |
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$ . Determine the length of the hypotenuse. | 994010 | 0.625 | 6,635.8125 | 5,702.1 | 8,192 | |
Given functions $y_1=\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\neq 0$).<br/>$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.<br/>$(2)$ If point $C(-1,n)$ is translated $6$ units upwards ... | -6 | 0.8125 | 4,805.3125 | 4,023.769231 | 8,192 | |
One angle of a parallelogram is 120 degrees, and two consecutive sides have lengths of 8 inches and 15 inches. What is the area of the parallelogram? Express your answer in simplest radical form. | 60\sqrt{3} | 1 | 2,013.875 | 2,013.875 | -1 | |
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome? | \frac{11}{30} | To solve this problem, we first need to understand the structure of a 6-digit palindrome and then determine how many of these palindromes are divisible by 11 and also form a palindrome when divided by 11.
1. **Structure of a 6-digit palindrome**:
A 6-digit palindrome can be represented as $n = \overline{abcba}$, w... | 0 | 8,192 | -1 | 8,192 |
For a certain square, two vertices lie on the line $y = 2x - 17,$ and the other two vertices lie on the parabola $y = x^2.$ Find the smallest possible area of the square. | 80 | 0 | 8,125.75 | -1 | 8,125.75 | |
In $\triangle ABC$, it is known that $BC=1$, $B= \frac{\pi}{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$. Determine the length of $AC$. | \sqrt{13} | 0.9375 | 3,936 | 3,652.266667 | 8,192 | |
Given that $cos({\frac{π}{4}-α})=\frac{3}{5}$, $sin({\frac{{5π}}{4}+β})=-\frac{{12}}{{13}}$, $α∈({\frac{π}{4},\frac{{3π}}{4}})$, $β∈({0,\frac{π}{4}})$, calculate the value of $\sin \left(\alpha +\beta \right)$. | \frac{56}{65} | 0.4375 | 6,606.4375 | 5,522.857143 | 7,449.222222 | |
For how many pairs of nonzero integers $(c, d)$ with $-2015 \leq c, d \leq 2015$ do the equations $c x=d$ and $d x=c$ both have an integer solution? | 8060 | We need both $c / d$ and $d / c$ to be integers, which is equivalent to $|c|=|d|$, or $d= \pm c$. So there are 4030 ways to pick $c$ and 2 ways to pick $d$, for a total of 8060 pairs. | 0.625 | 5,966.5 | 5,063.7 | 7,471.166667 |
Two rectangles, one measuring $2 \times 4$ and another measuring $3 \times 5$, along with a circle of diameter 3, are to be contained within a square. The sides of the square are parallel to the sides of the rectangles and the circle must not overlap any rectangle at any point internally. What is the smallest possible ... | 49 | 0 | 8,192 | -1 | 8,192 | |
Evaluate $\lfloor\sqrt{80}\rfloor$. | 8 | 0.75 | 4,845.25 | 3,729.666667 | 8,192 | |
The repeating decimal \( 0.\dot{x}y\dot{3} = \frac{a}{27} \), where \( x \) and \( y \) are distinct digits. Find the integer \( a \). | 19 | 0.625 | 6,637.5 | 5,704.8 | 8,192 | |
What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? | 6 | 1. **Define the expression and simplify each square root:**
Let $x = \sqrt{(3-2\sqrt{3})^2} + \sqrt{(3+2\sqrt{3})^2}$.
Since squaring a real number and then taking the square root gives the absolute value of the original number, we have:
\[
\sqrt{(3-2\sqrt{3})^2} = |3-2\sqrt{3}|
\]
\[
\sqrt{(3+2\s... | 0 | 2,209.9375 | -1 | 2,209.9375 |
[asy]
draw((-7,0)--(7,0),black+linewidth(.75));
draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));
draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75));
[/asy]
Five equilateral triangles, each with side $2\sq... | 12\sqrt{3} | 1. **Calculate the area of one large equilateral triangle**:
The formula for the area of an equilateral triangle with side length $s$ is given by:
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
Substituting $s = 2\sqrt{3}$, we get:
\[
\text{Area} = \frac{\sqrt{3}}{4} (2\sqrt{3})^2 = \frac{\sqrt{3}}{4... | 0.1875 | 7,868.1875 | 6,465 | 8,192 |
The number of recommendation plans the principal can make for a certain high school with 4 students and 3 universities can accept at most 2 students from that school is to be determined. | 54 | 0.125 | 7,073.5 | 4,027.5 | 7,508.642857 | |
Harriet lives in a large family with 4 sisters and 6 brothers, and she has a cousin Jerry who lives with them. Determine the product of the number of sisters and brothers Jerry has in the house. | 24 | 0.4375 | 4,277.0625 | 4,134.285714 | 4,388.111111 | |
If four departments A, B, C, and D select from six tourist destinations, calculate the total number of ways in which at least three departments have different destinations. | 1080 | 0 | 7,738.125 | -1 | 7,738.125 | |
For the set \( \{1, 2, 3, \ldots, 8\} \) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \( \{1, 3, 4, 7, 8\} \) would be \( 8-7+4-3+1=3 \) and ... | 1024 | 0 | 8,192 | -1 | 8,192 | |
Given an even function $f(x)$ defined on $\mathbb{R}$, for $x \geq 0$, $f(x) = x^2 - 4x$
(1) Find the value of $f(-2)$;
(2) For $x < 0$, find the expression for $f(x)$;
(3) Let the maximum value of the function $f(x)$ on the interval $[t-1, t+1]$ (where $t > 1$) be $g(t)$, find the minimum value of $g(t)$. | -3 | 0.25 | 6,102.875 | 4,851.25 | 6,520.083333 | |
If Alex does not sing on Saturday, then she has a $70 \%$ chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a $50 \%$ chance of singing on Sunday, find the probability that she sings on Saturday. | \frac{2}{7} | Let $p$ be the probability that Alex sings on Saturday. Then, the probability that she sings on Sunday is $.7(1-p)$; setting this equal to .5 gives $p=\frac{2}{7}$. | 0.75 | 4,986.4375 | 3,917.916667 | 8,192 |
At a conference of $40$ people, there are $25$ people who each know each other, and among them, $5$ people do not know $3$ other specific individuals in their group. The remaining $15$ people do not know anyone at the conference. People who know each other hug, and people who do not know each other shake hands. Determi... | 495 | 0 | 7,863.3125 | -1 | 7,863.3125 | |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\displaystyle {{m+n\pi}\over
p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$. | 505 | 0.5625 | 6,018 | 6,023.333333 | 6,011.142857 | |
Solve the equation \(2 x^{3} + 24 x = 3 - 12 x^{2}\). | \sqrt[3]{\frac{19}{2}} - 2 | 0 | 7,649.6875 | -1 | 7,649.6875 | |
Two standard decks of cards are combined, making a total of 104 cards (each deck contains 13 ranks and 4 suits, with all combinations unique within its own deck). The decks are randomly shuffled together. What is the probability that the top card is an Ace of $\heartsuit$? | \frac{1}{52} | 0.5 | 3,748.9375 | 3,054.375 | 4,443.5 | |
Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many p... | 16 | 0.4375 | 611 | 543.428571 | 663.555556 | |
Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number? | 4423 | 0.0625 | 7,038.4375 | 5,925 | 7,112.666667 | |
Find the 20th term in the sequence: $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \ldots, \frac{1}{m+1}, \frac{2}{m+1}, \ldots, \frac{m}{m+1}, \ldots$ | \frac{6}{7} | 0.0625 | 4,669.4375 | 4,641 | 4,671.333333 | |
Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball dra... | 7/15 | This is a case of conditional probability; the answer is the probability that the first ball is red and the second ball is black, divided by the probability that the second ball is black. First, we compute the numerator. If the first ball is drawn from Urn A, we have a probability of $2 / 6$ of getting a red ball, then... | 0 | 7,839.8125 | -1 | 7,839.8125 |
If $a$, $b$, and $c$ are positive numbers such that $ab=36$, $ac=72$, and $bc=108$, what is the value of $a+b+c$? | 13\sqrt{6} | 0 | 4,928.75 | -1 | 4,928.75 | |
In a revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:
\[
\begin{array}{|c|c|}
\hline
\$0.50 & \$50,000 \\
\hline
\$5 & \$100,000 \\
\hline
\$20 & \$150,000 \\
\hline
\$50 & \$200,000 \\
\hline
\$100 & \$250,000 \\
\hline
\$250 & \$300,... | 20 | 0.0625 | 8,022.375 | 5,478 | 8,192 | |
In triangle $PQR,$ $\angle Q = 30^\circ,$ $\angle R = 105^\circ,$ and $PR = 4 \sqrt{2}.$ Find $QR.$ | 8 | 0.9375 | 3,650.8125 | 3,348.066667 | 8,192 | |
How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1032 | 0.9375 | 4,731.1875 | 4,902.133333 | 2,167 | |
Lingling and Mingming were racing. Within 5 minutes, Lingling ran 380.5 meters, and Mingming ran 405.9 meters. How many more meters did Mingming run than Lingling? | 25.4 | 0.875 | 254 | 253.142857 | 260 | |
A polynomial $P$ has four roots, $\frac{1}{4}, \frac{1}{2}, 2,4$. The product of the roots is 1, and $P(1)=1$. Find $P(0)$. | \frac{8}{9} | A polynomial $Q$ with $n$ roots, $x_{1}, \ldots, x_{n}$, and $Q\left(x_{0}\right)=1$ is given by $Q(x)=\frac{\left(x-x_{1}\right)\left(x-x_{2}\right) \cdots\left(x-x_{n}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right) \cdots\left(x_{0}-x_{4}\right)}$, so $P(0)=\frac{1}{\frac{3}{4} \cdot \frac{1}{2} \cdot(-1) \... | 1 | 3,768 | 3,768 | -1 |
Solve the equation
$$
\sin ^{4} x + 5(x - 2 \pi)^{2} \cos x + 5 x^{2} + 20 \pi^{2} = 20 \pi x
$$
Find the sum of its roots that belong to the interval $[-\pi ; 6 \pi]$, and provide the answer, rounding to two decimal places if necessary. | 31.42 | 0 | 7,242.75 | -1 | 7,242.75 | |
In this problem only, assume that $s_{1}=4$ and that exactly one board square, say square number $n$, is marked with an arrow. Determine all choices of $n$ that maximize the average distance in squares the first player will travel in his first two turns. | n=4 | Because expectation is linear, the average distance the first player travels in his first two turns is the average sum of two rolls of his die (which does not depend on the board configuration) plus four times the probability that he lands on the arrow on one of his first two turns. Thus we just need to maximize the pr... | 0.125 | 7,955.5625 | 6,882 | 8,108.928571 |
In the tetrahedron \( A B C D \),
$$
\begin{array}{l}
AB=1, BC=2\sqrt{6}, CD=5, \\
DA=7, AC=5, BD=7.
\end{array}
$$
Find its volume. | \frac{\sqrt{66}}{2} | 0 | 8,088.9375 | -1 | 8,088.9375 | |
In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances ... | 160 | 0.625 | 7,291.9375 | 6,995.3 | 7,786.333333 | |
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarro... | \sqrt{3} + 1 | 0 | 6,647.4375 | -1 | 6,647.4375 | |
Given the following pseudocode:
```
S = 0
i = 1
Do
S = S + i
i = i + 2
Loop while S ≤ 200
n = i - 2
Output n
```
What is the value of the positive integer $n$? | 27 | 0.4375 | 5,894.4375 | 4,589.857143 | 6,909.111111 | |
Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it by appropriately selecting stamps of diffe... | 14 | 0 | 8,192 | -1 | 8,192 | |
In triangle $\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $a\sin B=-\sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $S=2\sqrt{3}$, find the perimeter of $\triangle ABC$. | 6 + 2\sqrt{7} | 0.9375 | 4,033.0625 | 3,755.8 | 8,192 | |
A university has 120 foreign language teachers. Among them, 50 teach English, 45 teach Japanese, and 40 teach French. There are 15 teachers who teach both English and Japanese, 10 who teach both English and French, and 8 who teach both Japanese and French. Additionally, 4 teachers teach all three languages: English, Ja... | 14 | 0.9375 | 3,338.625 | 3,015.066667 | 8,192 | |
Determine how many divisors of \(9!\) are multiples of 3. | 128 | 0.875 | 4,836.0625 | 4,356.642857 | 8,192 | |
Find the smallest positive integer \( n > 1 \) such that the arithmetic mean of the squares of the integers \( 1^2, 2^2, 3^2, \ldots, n^2 \) is a perfect square. | 337 | 0.1875 | 8,004.5 | 7,192 | 8,192 | |
Given a sequence $\{a_n\}$ where the first term is 1 and the common difference is 2,
(1) Find the general formula for $\{a_n\}$;
(2) Let $b_n=\frac{1}{a_n \cdot a_{n-1}}$, and the sum of the first n terms of the sequence $\{b_n\}$ is $T_n$. Find the minimum value of $T_n$. | \frac{1}{3} | 0.125 | 7,701.125 | 6,379 | 7,890 | |
For how many integers $a$ with $1 \leq a \leq 10$ is $a^{2014}+a^{2015}$ divisible by 5? | 4 | First, we factor $a^{2014}+a^{2015}$ as $a^{2014}(1+a)$. If $a=5$ or $a=10$, then the factor $a^{2014}$ is a multiple of 5, so the original expression is divisible by 5. If $a=4$ or $a=9$, then the factor $(1+a)$ is a multiple of 5, so the original expression is divisible by 5. If $a=1,2,3,6,7,8$, then neither $a^{2014... | 1 | 4,014.8125 | 4,014.8125 | -1 |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm ... | 12 | 0 | 7,137.375 | -1 | 7,137.375 | |
Circle $\Gamma$ has diameter $\overline{AB}$ with $AB = 6$ . Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \neq C$ . Let $D$ be on $\Gamma$ so that $\overleftrightarrow{CD}$ is tangent to $\Gamma$ . Compute the distance from line $\overleftrightarrow{AD}$ to the circumcenter of $... | 4\sqrt{3} | 0.625 | 6,019.25 | 4,928.5 | 7,837.166667 | |
Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value? | 0.9232 | 0.125 | 7,090.9375 | 4,326.5 | 7,485.857143 | |
A circle of radius \( 2 \) cm is inscribed in \( \triangle ABC \). Let \( D \) and \( E \) be the points of tangency of the circle with the sides \( AC \) and \( AB \), respectively. If \( \angle BAC = 45^{\circ} \), find the length of the minor arc \( DE \). | \pi | 0 | 7,091.75 | -1 | 7,091.75 | |
Find the coordinates of the foci and the eccentricity of the hyperbola $x^{2}-2y^{2}=2$. | \frac{\sqrt{6}}{2} | 0 | 1,511.75 | -1 | 1,511.75 | |
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. | F_{n+1} |
Consider the problem of counting the number of permutations of the sequence \(1, 2, \ldots, n\) that satisfy the inequality:
\[
a_1 \le 2a_2 \le 3a_3 \le \cdots \le na_n.
\]
To solve this, we relate the problem to a known sequence, specifically, the Fibonacci numbers. This can be approached using a combinatorial arg... | 0 | 8,192 | -1 | 8,192 |
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any... | 4:1 |
#### Initial Scenario
Let the heights of the narrow cone and the wide cone be \( h_1 \) and \( h_2 \), respectively. The volumes of the cones before the marble is dropped are given by:
\[
\text{Volume of Narrow Cone} = \frac{1}{3}\pi(3)^2h_1 = 3\pi h_1
\]
\[
\text{Volume of Wide Cone} = \frac{1}{3}\pi(6)^2h_2 = 12\pi ... | 0 | 6,238.8125 | -1 | 6,238.8125 |
A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\pi$. | \frac{81}{32}\pi | 0.8125 | 4,000 | 3,787.461538 | 4,921 | |
In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. The maximum possible area of ... | 150 | Proceed as in Solution 2 until you find $\angle CPB = 150$. The locus of points $P$ that give $\angle CPB = 150$ is a fixed arc from $B$ to $C$ ($P$ will move along this arc as $D$ moves along $BC$) and we want to maximise the area of [$\triangle BPC$]. This means we want $P$ to be farthest distance away from $BC$ as p... | 0 | 8,192 | -1 | 8,192 |
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | 310 | 0.1875 | 7,918.1875 | 7,379 | 8,042.615385 | |
In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $B= \frac{π}{3}$, and $c\sin A= \sqrt{3}a\cos C$, find the area of $\triangle ABC$. | \sqrt{3} | 0.9375 | 4,105.375 | 3,832.933333 | 8,192 | |
Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$. | 30 | 0.5 | 7,069.0625 | 6,126.625 | 8,011.5 | |
Is the following number rational or irrational?
$$
\sqrt[3]{2016^{2} + 2016 \cdot 2017 + 2017^{2} + 2016^{3}} ?
$$ | 2017 | 0.3125 | 4,124.6875 | 3,210.8 | 4,540.090909 | |
Liu and Li, each with one child, go to the park together to play. After buying tickets, they line up to enter the park. For safety reasons, the first and last positions must be occupied by fathers, and the two children must stand together. The number of ways for these 6 people to line up is \_\_\_\_\_\_. | 24 | 0.3125 | 7,375.1875 | 6,728.8 | 7,669 | |
Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with posi... | 30 | 0.5 | 6,336.625 | 5,643.375 | 7,029.875 | |
The ellipse whose equation is
\[\frac{x^2}{25} + \frac{y^2}{9} = 1\]is graphed below. The chord $\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = \frac{3}{2},$ then find $BF.$
[asy]
unitsize (0.6 cm);
pair A, B, F;
F = (4,0);
A = (35/8,3*sqrt(15)/8);
B = (55/16,-9*sqrt(15)/16);
draw(xscale(5)*ys... | \frac{9}{4} | 0.25 | 7,437.375 | 6,129.5 | 7,873.333333 | |
In order to obtain steel for a specific purpose, the golden section method was used to determine the optimal addition amount of a specific chemical element. After several experiments, a good point on the optimal range $[1000, m]$ is in the ratio of 1618, find $m$. | 2000 | 0 | 8,176.875 | -1 | 8,176.875 | |
Evaluate the monotonic intervals of $F(x)=\int_{0}^{x}(t^{2}+2t-8)dt$ for $x > 0$.
(1) Determine the monotonic intervals of $F(x)$.
(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,3]$. | -\frac{28}{3} | 0.9375 | 3,442.8125 | 3,454.6 | 3,266 | |
If the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$. | -1 | 0.0625 | 5,212.5 | 8,192 | 5,013.866667 | |
Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, which either shows which of the two sides ... | 28 | 0.0625 | 7,733.125 | 4,317 | 7,960.866667 | |
What is the least possible value of
\[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number? | 2018 | 1. **Expression Simplification**:
Start by grouping the terms in the expression \((x+1)(x+2)(x+3)(x+4)+2019\):
\[
(x+1)(x+4)(x+2)(x+3) + 2019 = (x^2 + 5x + 4)(x^2 + 5x + 6) + 2019.
\]
Here, we used the fact that \((x+1)(x+4) = x^2 + 5x + 4\) and \((x+2)(x+3) = x^2 + 5x + 6\).
2. **Variable Substitution*... | 0.6875 | 5,917.875 | 4,884.181818 | 8,192 |
Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$. | -2-\sqrt{3} | 1 | 3,883.8125 | 3,883.8125 | -1 | |
We are given that $$54+(98\div14)+(23\cdot 17)-200-(312\div 6)=200.$$Now, let's remove the parentheses: $$54+98\div14+23\cdot 17-200-312\div 6.$$What does this expression equal? | 200 | 1 | 2,645.6875 | 2,645.6875 | -1 | |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is 120°, and the magnitude of $\overrightarrow {a}$ is 2. If $(\overrightarrow {a} + \overrightarrow {b}) \cdot (\overrightarrow {a} - 2\overrightarrow {b}) = 0$, find the projection of $\overrightarrow {b}$ on $\overrightarrow {a}$. | -\frac{\sqrt{33} + 1}{8} | 0 | 4,798.125 | -1 | 4,798.125 | |
In the diagram, point \( D \) is on side \( BC \) of \( \triangle ABC \). If \( BD = CD = AD \) and \( \angle ACD = 40^\circ \), what is the measure of \( \angle BAC \)? | 90 | 0.4375 | 5,748 | 3,749 | 7,302.777778 | |
What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen? | 16 | 0.0625 | 8,017.3125 | 7,683 | 8,039.6 | |
In a regular 2017-gon, all diagonals are drawn. Petya randomly selects some number $\mathrm{N}$ of diagonals. What is the smallest $N$ such that among the selected diagonals there are guaranteed to be two diagonals of the same length? | 1008 | 0.5 | 6,411.8125 | 6,381.875 | 6,441.75 | |
Given that $\dfrac {\pi}{4} < \alpha < \dfrac {3\pi}{4}$ and $0 < \beta < \dfrac {\pi}{4}$, with $\cos \left( \dfrac {\pi}{4}+\alpha \right)=- \dfrac {3}{5}$ and $\sin \left( \dfrac {3\pi}{4}+\beta \right)= \dfrac {5}{13}$, find the value of $\sin(\alpha+\beta)$. | \dfrac {63}{65} | 0.5 | 6,893.25 | 5,740.375 | 8,046.125 | |
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i] | n=1,2,3,4 |
To solve the given problem, we need to determine all positive integers \( n \) such that there exists a sequence of positive integers \( a_1, a_2, \ldots, a_n \) satisfying the recurrence relation:
\[
a_{k+1} = \frac{a_k^2 + 1}{a_{k-1} + 1} - 1
\]
for every \( k \) where \( 2 \leq k \leq n-1 \).
### Step-by-step So... | 0 | 8,192 | -1 | 8,192 |
Given that $a$, $b$, $c$ are the opposite sides of the acute angles $A$, $B$, $C$ of triangle $\triangle ABC$, $\overrightarrow{m}=(3a,3)$, $\overrightarrow{n}=(-2\sin B,b)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=0$.
$(1)$ Find $A$;
$(2)$ If $a=2$ and the perimeter of $\triangle ABC$ is $6$, find the area of... | 6 - 3\sqrt{3} | 0.25 | 7,621.75 | 5,911 | 8,192 | |
In response to the call for rural revitalization, Xiao Jiao, a college graduate who has successfully started a business in another place, resolutely returned to her hometown to become a new farmer and established a fruit and vegetable ecological planting base. Recently, in order to fertilize the vegetables in the base,... | 500 | 0.0625 | 647.6875 | 897 | 631.066667 | |
Five women of different heights are standing in a line at a social gathering. Each woman decides to only shake hands with women taller than herself. How many handshakes take place? | 10 | 0.25 | 7,535.6875 | 5,677.25 | 8,155.166667 | |
A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in ... | 17 | 0 | 8,192 | -1 | 8,192 | |
When $0.\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator? | 15 | 1 | 1,361 | 1,361 | -1 | |
Evaluate the expression $\frac{16^{24}}{64^{8}}$.
A) $16^2$
B) $16^4$
C) $16^8$
D) $16^{16}$
E) $16^{24}$ | 16^8 | 0 | 7,680 | -1 | 7,680 | |
A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment? | 1.14 | 0 | 8,192 | -1 | 8,192 | |
Find the number of diagonals of a polygon with 150 sides and determine if 9900 represents half of this diagonal count. | 11025 | 0.8125 | 6,018.25 | 6,002.538462 | 6,086.333333 | |
What is the value of ${\left[\log_{10}\left(5\log_{10}100\right)\right]}^2$? | 1 | 1. **Identify the expression to simplify:**
We start with the expression $[\log_{10}(5\log_{10}100)]^2$.
2. **Simplify $\log_{10}100$:**
We know that $100 = 10^2$, so:
\[
\log_{10}100 = \log_{10}(10^2) = 2 \cdot \log_{10}10 = 2
\]
because $\log_{10}10 = 1$ by the definition of logarithm.
3. **Su... | 1 | 2,037.1875 | 2,037.1875 | -1 |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 0.25 | 4,272.125 | 2,918.25 | 4,723.416667 | |
For how many pairs of consecutive integers in the set $\{1100, 1101, 1102, \ldots, 2200\}$ is no carrying required when the two integers are added? | 1100 | 0 | 8,192 | -1 | 8,192 | |
Find the least positive integer $n$ for which $\frac{n-13}{5n+6}$ is a non-zero reducible fraction. | 84 | 1. **Identify the condition for reducibility**: The fraction $\frac{n-13}{5n+6}$ is reducible if and only if the greatest common divisor (GCD) of the numerator $n-13$ and the denominator $5n+6$ is greater than 1.
2. **Apply the Euclidean algorithm**: To find the GCD of $5n+6$ and $n-13$, we use the Euclidean algorithm... | 0.875 | 4,836.6875 | 4,357.357143 | 8,192 |
Compute
\[\begin{pmatrix} 1 & 1 & -2 \\ 0 & 4 & -3 \\ -1 & 4 & 3 \end{pmatrix} \begin{pmatrix} 2 & -2 & 0 \\ 1 & 0 & -3 \\ 4 & 0 & 0 \end{pmatrix}.\] | \begin{pmatrix} -5 & -2 & -3 \\ -8 & 0 & -12 \\ 14 & 2 & -12 \end{pmatrix} | 1 | 2,742.25 | 2,742.25 | -1 | |
Natural numbers of the form $F_n=2^{2^n} + 1 $ are called Fermat numbers. In 1640, Fermat conjectured that all numbers $F_n$, where $n\neq 0$, are prime. (The conjecture was later shown to be false.) What is the units digit of $F_{1000}$? | 7 | 0.9375 | 3,810.9375 | 3,518.866667 | 8,192 | |
Given numbers \( x_{1}, \cdots, x_{1991} \) satisfy the condition
$$
\left|x_{1}-x_{2}\right|+\cdots+\left|x_{1990}-x_{1991}\right|=1991 ,
$$
where \( y_{k}=\frac{1}{k}\left(x_{1}+\cdots+x_{k}\right) \) for \( k = 1, \cdots, 1991 \). Find the maximum possible value of the following expression:
$$
\left|y_{1}-y_{2}\rig... | 1990 | 0 | 8,192 | -1 | 8,192 | |
Vasiliy came up with a new chess piece called the "super-bishop." One "super-bishop" ($A$) attacks another ($B$) if they are on the same diagonal, there are no pieces between them, and the next cell along the diagonal after the "super-bishop" $B$ is empty. For example, in the image, piece $a$ attacks piece $b$, but do... | 32 | 0.0625 | 8,144 | 7,424 | 8,192 |
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