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 |
|---|---|---|---|---|---|---|
A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan can construct b... | 688 | Suppose that $x, y, z$ are the sides of the following solids. Then Volume $=xyz=128$. For the first solid, without loss of generality (with respect to assigning lengths to $x, y, z$), $xy=4$ and $yz=32$. Then $xy^{2}z=128$. Then $y=1$. Solving the remaining equations yields $x=4$ and $z=32$. Then the first solid has di... | 0 | 8,192 | -1 | 8,192 |
On the side $BC$ of the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 50^\circ$, $\angle CAD = 20^\circ$, and $AD = BD$. Find $\cos \angle C$. | \frac{\sqrt{3}}{2} | 0 | 7,569.3125 | -1 | 7,569.3125 | |
Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the cosine of the central angle subte... | 8\sqrt{39} | 0 | 8,192 | -1 | 8,192 | |
If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(10m,15n)$? | 60 | 0.625 | 6,503.625 | 5,490.6 | 8,192 | |
Sue owns 11 pairs of shoes: six identical black pairs, three identical brown pairs and two identical gray pairs. If she picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction. | \frac{7}{33} | 0.6875 | 4,847.375 | 4,237.909091 | 6,188.2 | |
Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$. | 11 | 0 | 7,573.1875 | -1 | 7,573.1875 | |
In an election, there are two candidates, A and B, who each have 5 supporters. Each supporter, independent of other supporters, has a \(\frac{1}{2}\) probability of voting for his or her candidate and a \(\frac{1}{2}\) probability of being lazy and not voting. What is the probability of a tie (which includes the case i... | 63/256 | 0.125 | 7,827.5625 | 6,016.5 | 8,086.285714 | |
Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers? | \frac{560}{2187} | 0.4375 | 6,081.5 | 3,368 | 8,192 | |
Given the lines $l_{1}$: $\left(3+a\right)x+4y=5-3a$ and $l_{2}$: $2x+\left(5+a\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $a$. | -7 | 0.25 | 6,301 | 5,557 | 6,549 | |
Distribute 5 volunteers from the Shanghai World Expo to work in the pavilions of China, the United States, and the United Kingdom. Each pavilion must have at least one volunteer, with the requirement that two specific volunteers, A and B, do not work in the same pavilion. How many different distribution schemes are pos... | 114 | 0.25 | 7,799.5 | 7,047.75 | 8,050.083333 | |
Let \( x = 19.\overline{87} \). If \( 19.\overline{87} = \frac{a}{99} \), find \( a \).
If \( \frac{\sqrt{3}}{b \sqrt{7} - \sqrt{3}} = \frac{2 \sqrt{21} + 3}{c} \), find \( c \).
If \( f(y) = 4 \sin y^{\circ} \) and \( f(a - 18) = b \), find \( b \). | 25 | 0 | 4,947.25 | -1 | 4,947.25 | |
There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this? | 761 | 0 | 7,754.25 | -1 | 7,754.25 | |
A marathon of 42 km started at 11:30 AM and the winner finished at 1:45 PM on the same day. What was the average speed of the winner, in km/h? | 18.6 | 0 | 2,910.5625 | -1 | 2,910.5625 | |
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term. | M > 1 |
Consider the sequence \( a_0, a_1, a_2, \ldots \) defined by:
\[
a_0 = M + \frac{1}{2}
\]
and
\[
a_{k+1} = a_k \lfloor a_k \rfloor \quad \text{for} \quad k = 0, 1, 2, \ldots
\]
We are tasked with finding all positive integers \( M \) such that at least one term in the sequence is an integer.
### Analysis of the S... | 0 | 7,942.25 | -1 | 7,942.25 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 0.8125 | 5,970 | 5,457.230769 | 8,192 | |
In a certain academic knowledge competition, where the total score is 100 points, if the scores (ξ) of the competitors follow a normal distribution (N(80,σ^2) where σ > 0), and the probability that ξ falls within the interval (70,90) is 0.8, then calculate the probability that it falls within the interval [90,100]. | 0.1 | 0.0625 | 6,864.4375 | 5,072 | 6,983.933333 | |
Given that $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 1 = 0,$ find $7 \alpha^5 + 8 \beta^4.$ | 1448 | 0 | 8,192 | -1 | 8,192 | |
The sequence of integers $ a_1 $ , $ a_2 $ , $ \dots $ is defined as follows: $ a_1 = 1 $ and $ n> 1 $ , $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different.
Define ... | 3006 | 0 | 8,192 | -1 | 8,192 | |
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how ... | 2001 | To solve this problem, we need to calculate the total cost of labeling all the lockers and match it with the given cost of $137.94. We will calculate the cost for each range of locker numbers based on the number of digits in the locker numbers.
1. **Calculate the cost for lockers with 1-digit numbers (1 to 9):**
- ... | 0.6875 | 3,515.625 | 3,306.454545 | 3,975.8 |
The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to the large car is 5:2. Find the speeds of bot... | 18 | 0.125 | 5,027.4375 | 5,895.5 | 4,903.428571 | |
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$. | 1800 | 0.125 | 7,998.5 | 6,644 | 8,192 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 0 | 6,618.625 | -1 | 6,618.625 | |
The triangle shown is an equilateral triangle with side length 12 cm. A side of the triangle is the diameter of the circle. If the sum of the areas of the two small shaded regions in square centimeters in simplest radical form is $a\pi - b\sqrt{c}$, what is $a+b+c$? [asy]
import graph;
size(2inch);
pair A = dir(60);
pa... | 33 | 0.375 | 7,102 | 5,285.333333 | 8,192 | |
What is the least common multiple of 135 and 468? | 7020 | 1 | 2,818.9375 | 2,818.9375 | -1 | |
Find the value of the function \( f(x) \) at the point \( x_{0} = 4500 \), if \( f(0) = 1 \) and for any \( x \) the equality \( f(x + 3) = f(x) + 2x + 3 \) holds. | 6750001 | 1 | 4,109.625 | 4,109.625 | -1 | |
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to determine what fraction of the villagers are... | 1/2 | 0.125 | 8,172.3125 | 8,034.5 | 8,192 | |
Al, Betty, and Clare split $\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\$100$. What was Al's original portion? | 400 | 1 | 2,254.8125 | 2,254.8125 | -1 | |
Among the integers from 1 to 100, how many integers can be divided by exactly two of the following four numbers: 2, 3, 5, 7? | 27 | 0.0625 | 8,074.625 | 6,314 | 8,192 | |
Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers. | 32 | 0 | 8,192 | -1 | 8,192 | |
Completely factor the following expression: \[(9x^5+25x^3-4)-(x^5-3x^3-4).\] | 4x^3(2x^2+7) | 1 | 1,652.125 | 1,652.125 | -1 | |
The diagonals of trapezoid \(ABCD\) intersect at point \(M\). The areas of triangles \(ABM\) and \(CDM\) are 18 and 50 units, respectively. What is the area of the trapezoid? | 128 | 0.6875 | 6,035.4375 | 5,055.181818 | 8,192 | |
What is the sum of the digits of the greatest prime number that is a divisor of 8,191? | 10 | 0 | 4,174.0625 | -1 | 4,174.0625 | |
Given that $\sin(\alpha + \frac{\pi}{5}) = \frac{1}{3}$ and $\alpha$ is an obtuse angle, find the value of $\cos(\alpha + \frac{9\pi}{20})$. | -\frac{\sqrt{2} + 4}{6} | 0 | 7,290.3125 | -1 | 7,290.3125 | |
What is the total number of digits used when the first 3003 positive even integers are written? | 11460 | 0.6875 | 5,888.375 | 4,841.272727 | 8,192 | |
The maximum value of $k$ such that the inequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution. | \sqrt{6} | 0.5625 | 3,776.4375 | 3,346.111111 | 4,329.714286 | |
A regular octahedron has a side length of 1. What is the distance between two opposite faces? | \sqrt{6} / 3 | Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the octahedron project to the vertices of a regular hexagon $... | 0 | 6,972.375 | -1 | 6,972.375 |
For a positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? $5!\cdot 9!=12\cdot N!$ | 10 | 1. **Understanding the given equation**: We start with the equation:
\[
5! \cdot 9! = 12 \cdot N!
\]
We need to find the value of $N$ that satisfies this equation.
2. **Expanding and simplifying the left-hand side**: We know that $5! = 120$ and $9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5!$. Substituting these ... | 0.9375 | 3,381.125 | 3,060.4 | 8,192 |
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\]is a real number? | \frac{6}{25} | 0 | 8,183.125 | -1 | 8,183.125 | |
There are 5 different books to be distributed among three people, with each person receiving at least 1 book and at most 2 books. Calculate the total number of different distribution methods. | 90 | 0.4375 | 6,698 | 5,383.142857 | 7,720.666667 | |
Let \( x_1, x_2, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In an \(80 \times 80\) table, numbers are arranged as follows: at the intersection of the \(i\)-th row and the \(k\)-th column, the number \(\log _{x_{k}} \frac{x_{i}}{16}\) is written. Find the minimum possible value of t... | -19200 | 0.125 | 8,140.0625 | 7,776.5 | 8,192 | |
On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ t... | 7.7 | 0 | 8,108.3125 | -1 | 8,108.3125 | |
In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, find the length of side $AB$. | \frac{50}{3} | 0.4375 | 7,104.3125 | 5,705.857143 | 8,192 | |
Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers? | 31 | 0 | 8,192 | -1 | 8,192 | |
Given complex numbers \( z, z_{1}, z_{2} \left( z_{1} \neq z_{2} \right) \) such that \( z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i} \), and \(\left|z-z_{1}\right|=\left|z-z_{2}\right|=4\), find \(|z|=\ \ \ \ \ .\) | 2\sqrt{3} | 0.8125 | 5,723.125 | 5,153.384615 | 8,192 | |
Given $α \in \left(0, \frac{\pi}{2}\right)$, $\cos \left(α+ \frac{\pi}{3}\right) = -\frac{2}{3}$, then $\cos α =$ \_\_\_\_\_\_. | \frac{\sqrt{15}-2}{6} | 0 | 4,901.4375 | -1 | 4,901.4375 | |
Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$. | 15 | 0.0625 | 7,915.4375 | 6,716 | 7,995.4 | |
Given a cube \( ABCD-A_1B_1C_1D_1 \) with edge length 1, a point \( M \) is taken on the diagonal \( A_1D \) and a point \( N \) is taken on \( CD_1 \) such that the line segment \( MN \) is parallel to the diagonal plane \( A_1ACC_1 \), find the minimum value of \( |MN| \). | \frac{\sqrt{3}}{3} | 0 | 4,863.9375 | -1 | 4,863.9375 | |
Among all the simple fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{3}{5}$. Provide the numerator of this fraction in your answer. | 59 | 0 | 8,192 | -1 | 8,192 | |
What is the value of the expression $2 \times 3 + 2 \times 3$? | 12 | Evaluating, $2 \times 3 + 2 \times 3 = 6 + 6 = 12$. | 1 | 710.5625 | 710.5625 | -1 |
A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi ve tiles wide. That red border is surrounded by a white border which is fi ve tiles wide. Finally, the white border is surrounded by a ... | 350 | 0.5 | 5,503.8125 | 4,366.5 | 6,641.125 | |
If $|x-2|=p$, where $x<2$, then what is $x-p$ in terms of $p$? | 2-2p | 1 | 1,378.125 | 1,378.125 | -1 | |
A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties? | 16 | 0.875 | 3,596.8125 | 2,940.357143 | 8,192 | |
What is \(\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) \)? | 1{,}010{,}000 | We are given the double summation:
\[
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j)
\]
We can simplify this by changing the order of summation and combining terms:
\[
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} \sum^{100}_{j=1} i + \sum^{100}_{i=1} \sum^{100}_{j=1} j
\]
Since $i$ is constant for the inner sum ... | 0 | 4,494.75 | -1 | 4,494.75 |
Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself. | 1094 | 0.0625 | 8,000.75 | 5,132 | 8,192 | |
Round $3.45$ to the nearest tenth. | 3.5 | 1 | 1,504.3125 | 1,504.3125 | -1 | |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which... | 20 | 0.3125 | 7,635.1875 | 6,410.2 | 8,192 | |
Aunt Wang needs 3 minutes to cut a paper-cut for window decoration. After cutting each one, she rests for 1 minute. She starts cutting at 9:40. After cutting 10 paper-cuts, it is \_\_\_\_ hour \_\_\_\_ minute. | 10:19 | 0 | 434.4375 | -1 | 434.4375 | |
In a similar game setup, there are 30 boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400... | 18 | 0.125 | 7,703.5625 | 6,082.5 | 7,935.142857 | |
Let $\mathcal{P}$ be a parabola, and let $V_{1}$ and $F_{1}$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal{P}$ so that $\angle AV_{1}B=90^{\circ}$. Let $\mathcal{Q}$ be the locus of the midpoint of $AB$. It turns out that $\mathcal{Q}$ is also a parabola, and let $V_{2}$ and $F_{2}$ deno... | \frac{7}{8} | Since all parabolas are similar, we may assume that $\mathcal{P}$ is the curve $y=x^{2}$. Then, if $A=\left(a, a^{2}\right)$ and $B=\left(b, b^{2}\right)$, the condition that $\angle AV_{1}B=90^{\circ}$ gives $ab+a^{2}b^{2}=0$, or $ab=-1$. Then, the midpoint of $AB$ is $$\frac{A+B}{2}=\left(\frac{a+b}{2}, \frac{a^{2}+b... | 0.625 | 5,996 | 5,229.5 | 7,273.5 |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286$
| 220 | 0 | 6,710.5625 | -1 | 6,710.5625 | |
(1) Given $\cos \alpha =\frac{\sqrt{5}}{3}, \alpha \in \left(-\frac{\pi }{2},0\right)$, find $\sin (\pi -\alpha)$;
(2) Given $\cos \left(\theta+ \frac{\pi}{4}\right)= \frac{4}{5}, \theta \in \left(0, \frac{\pi}{2}\right)$, find $\cos \left(\frac{\pi }{4}-\theta \right)$. | \frac{3}{5} | 0.4375 | 6,723.125 | 5,004.571429 | 8,059.777778 | |
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and the sum of its first \(n\) terms \(S_{n}\) has a maximum value. Then, when \(S_{n}\) attains its smallest positive value, \(n =\) ______ . | 19 | 0 | 8,102.25 | -1 | 8,102.25 | |
Given a fixed circle $\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\odot Q$ is externally tangent to circle $\odot P$. Circle $\odot Q$ intersects $l$ at points $M$ and $N$. For any diameter $MN$, there is always a fixed point $A... | 60 | 0 | 8,192 | -1 | 8,192 | |
For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied? | 5 | 1 | 3,097.3125 | 3,097.3125 | -1 | |
Let $S$ be a set. We say $S$ is $D^\ast$ *-finite* if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$ , there exists a $y \in Y$ such that $f(y) \notin Y$ . The function $f$ is called a *witness* of $S$ . How many witnesses does $\{0,1,\cdots,5\}$ have... | 120 | 0.0625 | 7,907.625 | 5,400 | 8,074.8 | |
Determine the value of $$1 \cdot 2-2 \cdot 3+3 \cdot 4-4 \cdot 5+\cdots+2001 \cdot 2002$$ | 2004002 | 2004002. Rewrite the expression as $$2+3 \cdot(4-2)+5 \cdot(6-4)+\cdots+2001 \cdot(2002-2000)$$ $$=2+6+10+\cdots+4002$$ This is an arithmetic progression with $(4002-2) / 4+1=1001$ terms and average 2002, so its sum is $1001 \cdot 2002=2004002$. | 0.4375 | 7,597 | 7,456.571429 | 7,706.222222 |
Determine the exact value of the series
\[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\] | \frac{1}{4} | 0.5 | 7,255.9375 | 6,319.875 | 8,192 | |
Multiply $555.55$ by $\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth. | -148.15 | 0.6875 | 5,334.875 | 4,746.545455 | 6,629.2 | |
Find the time, in seconds, after 12 o'clock, when the area of $\triangle OAB$ will reach its maximum for the first time. | \frac{15}{59} | 0 | 7,134.875 | -1 | 7,134.875 | |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | 0 | 1 | 1,370.0625 | 1,370.0625 | -1 | |
Given the sequence ${a_n}$ that satisfies the equation $a_{n+1}+(-1)^{n}a_{n}=3n-1,(n∈N^{*})$, determine the sum of the first 40 terms of the sequence ${a_n}$. | 1240 | 0.0625 | 7,596.8125 | 4,049 | 7,833.333333 | |
The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere. | 14/9 | 0.0625 | 8,106.8125 | 6,829 | 8,192 | |
In a circle centered at $O$, point $A$ is on the circle, and $\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\angle ABC = 45^\circ$. The circle intersects $\overline{BO}$ at $D$. Chord $\overline{BC}$ also extends to meet the circle at another point, $E$. What is the value o... | \frac{2 - \sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given that for any positive integer \( n \), \( 9^{2n} - 8^{2n} - 17 \) is always divisible by \( m \), find the largest positive integer \( m \). | 2448 | 0.125 | 7,712.5625 | 6,347.5 | 7,907.571429 | |
In a pentagon ABCDE, there is a vertical line of symmetry. Vertex E is moved to \(E(5,0)\), while \(A(0,0)\), \(B(0,5)\), and \(D(5,5)\). What is the \(y\)-coordinate of vertex C such that the area of pentagon ABCDE becomes 65 square units? | 21 | 0.625 | 6,620.0625 | 5,676.9 | 8,192 | |
At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first ... | 17 | 0 | 8,192 | -1 | 8,192 | |
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 | 1. **Identify the constraints and setup the problem:**
- The chef has to prepare desserts for 7 days starting from Sunday.
- The desserts options are cake, pie, ice cream, or pudding.
- The same dessert cannot be served on consecutive days.
- Cake must be served on Friday due to a birthday.
2. **Determine ... | 0 | 8,179.125 | -1 | 8,179.125 |
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last number... | 5.5 | 0 | 8,079 | -1 | 8,079 | |
Let \(a\) and \(b\) be constants. The parabola \(C: y = (t^2 + t + 1)x^2 - 2(a + t)^2 x + t^2 + 3at + b\) passes through a fixed point \(P(1,0)\) for any real number \(t\). Find the value of \(t\) such that the chord obtained by intersecting the parabola \(C\) with the x-axis is the longest. | -1 | 0.0625 | 7,024.375 | 7,499 | 6,992.733333 | |
Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away... | 13 | 0.0625 | 8,164.3125 | 7,749 | 8,192 | |
Given the function $f(x)=ax^{3}-4x+4$, where $a\in\mathbb{R}$, $f′(x)$ is the derivative of $f(x)$, and $f′(1)=-3$.
(1) Find the value of $a$;
(2) Find the extreme values of the function $f(x)$. | -\frac{4}{3} | 0.75 | 2,637.5625 | 2,656.833333 | 2,579.75 | |
In trapezoid $PQRS$, leg $\overline{QR}$ is perpendicular to bases $\overline{PQ}$ and $\overline{RS}$, and diagonals $\overline{PR}$ and $\overline{QS}$ are perpendicular. Given that $PQ=\sqrt{23}$ and $PS=\sqrt{2023}$, find $QR^2$. | 100\sqrt{46} | 0 | 8,192 | -1 | 8,192 | |
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Find $f(11)+f(13)+f(14)$. | 2 | 1 | 2,385.625 | 2,385.625 | -1 | |
Call a positive integer $n$ quixotic if the value of $\operatorname{lcm}(1,2,3, \ldots, n) \cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)$ is divisible by 45 . Compute the tenth smallest quixotic integer. | 573 | Let $L=\operatorname{lcm}(1,2,3, \ldots, n)$, and let $E=L\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)$ denote the expression. In order for $n$ to be quixotic, we need $E \equiv 0(\bmod 5)$ and $E \equiv 0(\bmod 9)$. We consider these two conditions separately. Claim: $E \equiv 0(\bmod 5)$ if and only if $... | 0 | 8,192 | -1 | 8,192 |
Find the smallest prime number that can be expressed as the sum of five different prime numbers. | 43 | 0.5 | 7,392.75 | 6,892.125 | 7,893.375 | |
Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with all triplets except his/her siblings and wi... | 394 | 0 | 7,989.75 | -1 | 7,989.75 | |
Given the function $f(x)=\sin x\cos x- \sqrt {3}\cos ^{2}x.$
(I) Find the smallest positive period of $f(x)$;
(II) When $x\in[0, \frac {π}{2}]$, find the maximum and minimum values of $f(x)$. | - \sqrt {3} | 0 | 4,965.5 | -1 | 4,965.5 | |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ . | -13 | 0 | 8,192 | -1 | 8,192 | |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | 702 | Suppose there are $n$ squares in every column of the grid, so there are $\frac{52}{24}n = \frac {13}6n$ squares in every row. Then $6|n$, and our goal is to maximize the value of $n$.
Each vertical fence has length $24$, and there are $\frac{13}{6}n - 1$ vertical fences; each horizontal fence has length $52$, and ther... | 0 | 7,833.75 | -1 | 7,833.75 |
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around ... | \frac{12}{13} | Let's analyze the problem step by step:
1. **Define the problem in terms of distance and speed:**
Let the total distance from the trailhead to the fire tower be $2d$. Chantal walks the first half of the trail, a distance of $d$, at a speed of $4$ miles per hour. The second half of the trail, also a distance of $d$,... | 0.875 | 3,506.5 | 3,261.357143 | 5,222.5 |
For any real number $x$, the symbol $[x]$ represents the largest integer not greater than $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. The function $y=[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Then, the value of $[\log _{3}1]+[\log _{3}2]+[\log _{3}3... | 12 | 1 | 5,007.1875 | 5,007.1875 | -1 | |
Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60 | 60 | 0 | 8,192 | -1 | 8,192 | |
Coordinate System and Parametric Equation
Given the ellipse $(C)$: $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\triangle PAB$. | 6(\sqrt{2} + 1) | 0.0625 | 7,480.875 | 6,955 | 7,515.933333 | |
Let \( M = \{1, 2, \ldots, 20\} \) and \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). When \( i \neq j \), the intersection of \( A_i \) and \( A_j \) has at most two elements. Find the maximum value of \( n \). | 1350 | 0 | 8,192 | -1 | 8,192 | |
A right circular cone is placed on a table, pointing upwards. The vertical cross-section triangle, perpendicular to the base, has a vertex angle of 90 degrees. The diameter of the cone's base is 16 inches. A sphere is placed inside the cone so that it touches the sides of the cone and rests on the table. Find the volum... | \frac{256}{3}\pi | 0 | 8,192 | -1 | 8,192 | |
When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: | 5 | 1. **Factor out the greatest common factor**:
The polynomial $x^9 - x$ can be rewritten by factoring out the common factor of $x$, giving:
\[
x^9 - x = x(x^8 - 1)
\]
2. **Factorize $x^8 - 1$ using difference of squares**:
The expression $x^8 - 1$ can be factored using the difference of squares formula,... | 0.625 | 6,765.75 | 5,910 | 8,192 |
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\overrightarrow{m}=(1-\cos(A+B), \cos \frac{A-B}{2})$ and $\overrightarrow{n}=(\frac{5}{8}, \cos \frac{A-B}{2})$, and it's known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{9}{8}$.
1.... | -\frac{3}{8} | 0 | 8,192 | -1 | 8,192 | |
The coefficient of $x^{2}$ in the expansion of $\left( \frac {3}{x}+x\right)\left(2- \sqrt {x}\right)^{6}$ is ______. | 243 | 0.5625 | 6,864.4375 | 6,165.111111 | 7,763.571429 | |
Given that point $O$ is the origin of coordinates, point $A$ in the first quadrant lies on the graph of the inverse proportional function $y=\frac{1}{x}$ for $x>0$, and point $B$ in the second quadrant lies on the graph of the inverse proportional function $y=-\frac{4}{x}$ for $x<0$, and $O A$ is perpendicular to $O B$... | $\frac{1}{2}$ | 0 | 5,487.25 | -1 | 5,487.25 | |
The area of the lunar crescent shape bounded by the portion of the circle of radius 5 and center (0,0), the portion of the circle with radius 2 and center (0,2), and the line segment from (0,0) to (5,0). | \frac{21\pi}{4} | 0 | 7,564.6875 | -1 | 7,564.6875 |
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