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 |
|---|---|---|---|---|---|---|
In a regular 2017-gon, all diagonals are drawn. Peter randomly selects some $\mathrm{N}$ diagonals. What is the smallest $N$ such that there are guaranteed to be two diagonals of the same length among the selected ones? | 1008 | 0.375 | 6,866.0625 | 6,494.166667 | 7,089.2 | |
During the preparation period of the Shanghai Expo, 5 volunteers and 2 foreign friends line up for a photo. The 2 foreign friends stand next to each other but not at either end of the line. Calculate the total number of different arrangements for the 7 individuals. | 960 | 0.5625 | 5,996.25 | 4,288.444444 | 8,192 | |
Convert the point $\left( 8, \frac{7 \pi}{6} \right)$ in polar coordinates to rectangular coordinates. | (-4 \sqrt{3},-4) | 1 | 1,909.6875 | 1,909.6875 | -1 | |
The average of \( p, q, r \) is 12. The average of \( p, q, r, t, 2t \) is 15. Find \( t \).
\( k \) is a real number such that \( k^{4} + \frac{1}{k^{4}} = t + 1 \), and \( s = k^{2} + \frac{1}{k^{2}} \). Find \( s \).
\( M \) and \( N \) are the points \( (1, 2) \) and \( (11, 7) \) respectively. \( P(a, b) \) is a... | 12 | 0.6875 | 4,415.8125 | 3,647.272727 | 6,106.6 | |
Find the area in the plane contained by the graph of
\[|2x + 3y| + |2x - 3y| \le 12.\] | 12 | 0 | 7,263.25 | -1 | 7,263.25 | |
The equation of the line joining the complex numbers $-2 + 3i$ and $1 + i$ can be expressed in the form
\[az + b \overline{z} = 10\]for some complex numbers $a$ and $b$. Find the product $ab$. | 13 | 0.5 | 6,239.0625 | 4,286.125 | 8,192 | |
Determine the values of $a$, $b$, and $r$ of the circle given by the equation $x^2 + 14y + 65 = -y^2 - 8x$. Let $a$ and $b$ be the coordinates of the center of the circle, and $r$ be its radius. What is the sum $a + b + r$? | -11 | 1 | 4,627.25 | 4,627.25 | -1 | |
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$. Let $P$ be a point on the ellipse, and $\triangle F\_1 P F\_2$ have centroid $G$ and incenter $I$. If $\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentricity $e$ of the ellipse.
A) $\frac{1}{2}$
B) $\fr... | \frac{1}{2} | 0 | 7,408.625 | -1 | 7,408.625 | |
What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $17,$ $6,$ and $2?$ | 103 | 1 | 3,786.3125 | 3,786.3125 | -1 | |
What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form. | \frac{10\sqrt{3}}{3} | 0 | 3,075.5625 | -1 | 3,075.5625 | |
Compute $\tan 120^\circ$. | -\sqrt{3} | 1 | 2,228.9375 | 2,228.9375 | -1 | |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product by at least 2? Express your answer as a common fraction. | \frac{4}{9} | 0 | 6,992.5 | -1 | 6,992.5 | |
Consider the function $y=a\sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x}$ ($a\in\mathbb{R}$), and let $t= \sqrt{1+x} + \sqrt{1-x}$ ($\sqrt{2} \leq t \leq 2$).
(1) Express $y$ as a function of $t$, denoted as $m(t)$.
(2) Let the maximum value of the function $m(t)$ be $g(a)$. Find $g(a)$.
(3) For $a \geq -\sqrt{2}$, find all... | a = 1 | 0.0625 | 8,140 | 7,360 | 8,192 | |
In $\triangle XYZ$, $\angle X = 90^\circ$ and $\tan Z = \sqrt{3}$. If $YZ = 150$, what is $XY$? | 75\sqrt{3} | 0.9375 | 1,885.375 | 1,866.666667 | 2,166 | |
Given $A=3x^{2}-x+2y-4xy$, $B=x^{2}-2x-y+xy-5$.
$(1)$ Find $A-3B$.
$(2)$ If $(x+y-\frac{4}{5})^{2}+|xy+1|=0$, find the value of $A-3B$.
$(3)$ If the value of $A-3B$ is independent of $y$, find the value of $x$. | \frac{5}{7} | 0.9375 | 3,299 | 3,160.733333 | 5,373 | |
Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$. | 8\sqrt{2}-10 | 1 | 5,575.9375 | 5,575.9375 | -1 | |
The values of a function $f(x)$ are given in the table below.
\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $f(x)$ & 3 & 1 & 5 & 4 & 2 \\ \hline
\end{tabular}If $f^{-1}$ exists, then what is $f^{-1}(f^{-1}(f^{-1}(1)))$? | 3 | 1 | 1,679.5625 | 1,679.5625 | -1 | |
Two standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction. | \frac{5}{12} | 1 | 2,270.6875 | 2,270.6875 | -1 | |
Petya plans to spend all 90 days of his vacation in the village, swimming in the lake every second day (i.e., every other day), going shopping for groceries every third day, and solving math problems every fifth day. (On the first day, Petya did all three tasks and got very tired.) How many "pleasant" days will Petya h... | 24 | 0.8125 | 6,286.25 | 5,846.461538 | 8,192 | |
There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$.
What is $t$? | 76 | 1 | 2,255.8125 | 2,255.8125 | -1 | |
Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$? | 2 | 1 | 1,768 | 1,768 | -1 | |
A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\_\_\_\_\_\_\_.$ | \frac{1}{16} | 0.25 | 7,776.5625 | 6,896.5 | 8,069.916667 | |
Calculate the value of $333 + 33 + 3$. | 369 | 0.9375 | 914.5625 | 429.4 | 8,192 | |
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? | 45 | 1. **Identify the given information and the goal:** We are given a rectangle $ABCD$ with $AB = 6$ and $BC = 3$. We need to find the degree measure of $\angle AMD$ given that $\angle AMD = \angle CMD$.
2. **Assign variables and use the Pythagorean Theorem:** Let $AM = x$. Then, $MB = 6 - x$. Since $D$ and $C$ are on th... | 0 | 5,073.625 | -1 | 5,073.625 |
In quadrilateral $ABCD$, $\angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD}$, $AB=18$, $BC=21$, and $CD=14$. Find the perimeter of $ABCD$. | 84 | From the problem statement, we construct the following diagram:
[asy] pointpen = black; pathpen = black + linewidth(0.65); pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18)); D(MP("A",A,W)--MP("B",B,N)--MP("C",C,E)--MP("D",D,E)--A--C); D(rightanglemark(A,C,D,40)); D(rightanglemark(A,B,C,40)); [... | 0.6875 | 4,793.9375 | 3,249.363636 | 8,192 |
A 50-card deck consists of 4 cards labeled " $i$ " for $i=1,2, \ldots, 12$ and 2 cards labeled " 13 ". If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label? | \frac{73}{1225} | All pairs of distinct cards (where we distinguish cards even with the same label) are equally likely. There are $\binom{2}{2}+12\binom{4}{2}=73$ pairs of cards with the same label and $\binom{50}{2}=100 \cdot \frac{49}{4}=1225$ pairs of cards overall, so the desired probability is $\frac{73}{1225}$. | 1 | 4,706.5 | 4,706.5 | -1 |
A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of m... | 2807 | 0.1875 | 6,576.875 | 2,724 | 7,466 | |
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is: | 6 | To solve this problem, we need to determine how many nickels can be placed around a central nickel such that each is tangent to the central nickel and to its two neighboring nickels.
1. **Understanding the Problem:**
- Each nickel is a circle, and we are asked to place other nickels around one central nickel.
- ... | 1 | 2,605.75 | 2,605.75 | -1 |
What is the sum of all values of $y$ for which the expression $\frac{y+6}{y^2-5y+4}$ is undefined? | 5 | 1 | 1,084.8125 | 1,084.8125 | -1 | |
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis | \frac{16\pi}{15} | 0.0625 | 7,767.625 | 6,226 | 7,870.4 | |
The graph of the function $f(x)=\sin(\omega x+\varphi)$, where $(\omega>0, |\varphi|<\frac{\pi}{2})$, passes through the point $(0,-\frac{1}{2})$. Find the minimum value of $\omega$ if the graph of this function is shifted to the right by $\frac{\pi}{3}$ units and becomes symmetric about the origin. | \frac{5}{2} | 0.8125 | 5,580.5 | 4,977.846154 | 8,192 | |
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]
for all $x,y\in\mathbb{R}$. | $f(x)= 0,f(x)= 2-x, f(x)=-x$ |
To find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying
\[
f(x^2) + f(xy) = f(x)f(y) + yf(x) + xf(x+y)
\]
for all \( x, y \in \mathbb{R} \), we will proceed by considering special cases and functional forms.
### Step 1: Substitute \( y = 0 \)
First, set \( y = 0 \) in the functional equation:
... | 0 | 8,192 | -1 | 8,192 |
Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying $$A+A^{k}=A^{T}$$ for some integer $k \geq n$. | A = 0 | Solution 1. Taking the transpose of the matrix equation and substituting we have $$A^{T}+\left(A^{T}\right)^{k}=A \Longrightarrow A+A^{k}+\left(A+A^{k}\right)^{k}=A \Longrightarrow A^{k}\left(I+\left(I+A^{k-1}\right)^{k}\right)=0$$ Hence $p(x)=x^{k}\left(1+\left(1+x^{k-1}\right)^{k}\right)$ is an annihilating polynomia... | 0.6875 | 6,796.375 | 6,162 | 8,192 |
Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-1$, $b=2$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. | -\frac{1}{2} | 1 | 2,477.125 | 2,477.125 | -1 | |
Given $sinα-\sqrt{3}cosα=1$, then the value of $sin({\frac{{7π}}{6}-2α})$ is ______. | \frac{1}{2} | 1 | 5,161.25 | 5,161.25 | -1 | |
Name the greatest whole number less than $100$ that has an odd number of positive factors. | 81 | 1 | 1,419 | 1,419 | -1 | |
Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \) is \( 3\sqrt{10} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( 3\sqrt{14} \). Find the distance b... | 3\sqrt{2} | 0.5 | 6,942 | 5,885.875 | 7,998.125 | |
How many square units are in the area of the parallelogram with vertices at (0, 0), (6, 0), (2, 8) and (8, 8)? | 48 | 1 | 3,304.9375 | 3,304.9375 | -1 | |
The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle. | 2:5 | 0 | 4,320.25 | -1 | 4,320.25 | |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\frac{20}{3}-x$. | 16 | 0.75 | 5,050.9375 | 4,797.833333 | 5,810.25 | |
In the quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(K\). Points \(L\) and \(M\) are the midpoints of sides \(BC\) and \(AD\), respectively. Segment \(LM\) contains point \(K\). The quadrilateral \(ABCD\) is such that a circle can be inscribed in it. Find the radius of this circle, given that... | 3/2 | 0 | 8,192 | -1 | 8,192 | |
Find $n$ such that $2^5 \cdot 3^2 \cdot n = 8!$. | 140 | 1 | 2,734.125 | 2,734.125 | -1 | |
Given a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The square is folded along lines \(EC\) and \(ED\) so that \(AE\) coincides with \(BE\), and point \(A\) coincides with point \(B\), named point \(O\). Calculate the volume of the tetrahedron \(O-CDE\). | \frac{\sqrt{3}}{3} | 0 | 8,136.8125 | -1 | 8,136.8125 | |
Let the function $f(x)=(x-3)^3 +x-1$. The sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference. If $f(a_1)+f(a_2) + \ldots +f(a_7) =14$, then $a_1 +a_2 +\ldots +a_7 =$ ______. | 21 | 0.5625 | 6,346.375 | 4,910.888889 | 8,192 | |
Given the function $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;
(2) If $$0 < \theta < \frac{\pi}{6}$$ and $$f(\theta) = \frac{4}{3}$$, calculate the value of $\cos 2\theta$. | \frac{\sqrt{15} + 2}{6} | 0 | 6,034.0625 | -1 | 6,034.0625 | |
The equation \[\frac{x^2}{36} + \frac{(y+5)^2}{16} = 0\]describes a degenerate ellipse, because the right-hand side is $0$ instead of $1$ (as in the standard form for an ellipse). Of all the points on the graph of this equation, what is the largest possible $y$-coordinate? | -5 | 1 | 1,762.1875 | 1,762.1875 | -1 | |
How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not? | 64 | 0.125 | 8,175.5 | 8,060 | 8,192 | |
The average density of pathogenic microbes in one cubic meter of air is 100. A sample of 2 cubic decimeters of air is taken. Find the probability that at least one microbe will be found in the sample. | 0.181 | 0.125 | 3,793.875 | 3,372 | 3,854.142857 | |
Point \( F \) is the midpoint of side \( BC \) of square \( ABCD \). A perpendicular \( AE \) is drawn to segment \( DF \). Find the angle \( CEF \). | 45 | 1 | 4,029.125 | 4,029.125 | -1 | |
What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | 10 | 0 | 5,716.3125 | -1 | 5,716.3125 | |
Given that positive numbers $x$ and $y$ satisfy $x + 4y = 2$, find the minimum value of $\frac{x + 40y + 4}{3xy}$. | 18 | 0.75 | 6,048.1875 | 5,333.583333 | 8,192 | |
The circle is divided into 30 equal parts by 30 points on the circle. Randomly selecting 3 different points, what is the probability that these 3 points form an equilateral triangle? | 1/406 | 1 | 5,342 | 5,342 | -1 | |
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$ . What is the minimum possible value of $a + b$ ? | 25 | 0 | 8,192 | -1 | 8,192 | |
Let \(a\) and \(b\) be angles such that
\[\cos (a - b) = \cos a - \cos b.\]
Find the maximum value of \(\cos a\). | \sqrt{\frac{3+\sqrt{5}}{2}} | 0 | 7,837.4375 | -1 | 7,837.4375 | |
A quadrilateral pyramid \( S A B C D \) is given, with the base being the parallelogram \( A B C D \). A plane is drawn through the midpoint of edge \( A B \) that is parallel to the lines \( A C \) and \( S D \). In what ratio does this plane divide edge \( S B \)? | 1 : 3 | 0 | 5,612.875 | -1 | 5,612.875 | |
A certain store in Hefei plans to sell a newly launched stationery item, with a purchase price of 20 yuan per item. During the trial marketing phase, it was found that when the selling price is 25 yuan per item, the daily sales volume is 150 items; for every 1 yuan increase in the selling price, the daily sales volume ... | 960 | 0 | 4,520.8125 | -1 | 4,520.8125 | |
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it is always true that \( x y z \mid (x+y+z)^{n} \). | 13 | 0.0625 | 8,165.0625 | 7,761 | 8,192 | |
Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$. | 88 | 0 | 4,061 | -1 | 4,061 | |
Four friends have a total of 8 identical pencils, and each friend has at least one pencil. Additionally, the first friend always has at least two pencils. In how many ways can this happen? | 20 | 0.625 | 6,670.375 | 5,763.4 | 8,182 | |
Given the function $f(x) = 2\sin(\frac{1}{3}x - \frac{π}{6})$, where $x \in \mathbb{R}$.
(1) Find the value of $f(\frac{5π}{4})$;
(2) Let $\alpha, \beta \in [0, \frac{π}{2}], f(3\alpha + \frac{π}{2}) = \frac{10}{13}, f(3\beta + 2π) = \frac{6}{5}$, find the value of $\cos(\alpha + \beta)$. | \frac{16}{65} | 0.8125 | 4,016.75 | 3,053.230769 | 8,192 | |
If the variance of the sample $a_1, a_2, \ldots, a_n$ is 3, then the variance of the sample $3a_1+1, 3a_2+2, \ldots, 3a_n+1$ is ____. | 27 | 0.125 | 7,492.0625 | 2,592.5 | 8,192 | |
How many solutions does the equation $\tan x = \tan(\tan x + x)$ have on the interval $0 \leq x \leq \tan^{-1} 500$? | 160 | 0.375 | 7,094.375 | 5,265 | 8,192 | |
Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$ | \frac{\sqrt{3}}{2} | 0 | 8,025.625 | -1 | 8,025.625 | |
If two distinct members of the set $\{ 4, 10, 15, 24, 30, 40, 60 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 120? Express your answer as a common fraction. | \frac{3}{7} | 0 | 7,996.4375 | -1 | 7,996.4375 | |
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class. | 71.47 | 0 | 422 | -1 | 422 | |
Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number? | \frac{5}{12} | To find the probability that Diana's number is larger than Apollo's number, we consider all possible outcomes when both Diana and Apollo roll a standard six-sided die.
1. **List the possible outcomes for each roll of Apollo:**
- If Apollo rolls a 1, Diana can roll a 2, 3, 4, 5, or 6 (5 possibilities).
- If Apoll... | 1 | 2,903.375 | 2,903.375 | -1 |
A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the "Creating a Civilized City" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an additional 20 helmets are sold per month. It is ... | 65 | 0.1875 | 4,744.8125 | 3,935.666667 | 4,931.538462 | |
If \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that m... | \frac{9}{16} | 0.9375 | 3,552.6875 | 3,243.4 | 8,192 | |
In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so that there are always at least two identical ... | 19 | 0.5625 | 5,612.1875 | 5,708.888889 | 5,487.857143 | |
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \] | 126 | 0.6875 | 4,827.4375 | 3,700.545455 | 7,306.6 | |
Let $a \clubsuit b = \frac{2a}{b} \cdot \frac{b}{a}$. What is $(5 \clubsuit (3 \clubsuit 6)) \clubsuit 1$? | 2 | 1 | 2,069.6875 | 2,069.6875 | -1 | |
In triangle $ABC,$ $b = 5,$ $c = 4,$ and $\cos (B - C) = \frac{31}{32}.$ Find $a.$
Note: $a$ is the side length opposite $\angle A,$ etc. | 6 | 0.75 | 5,975.8125 | 5,398.083333 | 7,709 | |
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | 182 | Let $p(n)$ denote the product of the distinct proper divisors of $n$. A number $n$ is nice in one of two instances:
It has exactly two distinct prime divisors.
If we let $n = pq$, where $p,q$ are the prime factors, then its proper divisors are $p$ and $q$, and $p(n) = p \cdot q = n$.
It is the cube of a prime number.
... | 0.4375 | 6,665.1875 | 5,401.428571 | 7,648.111111 |
A geometric sequence of positive integers is formed for which the first term is 2 and the fifth term is 162. What is the sixth term of the sequence? | 486 | 1 | 1,311 | 1,311 | -1 | |
How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
| 6 | 0 | 5,146.5 | -1 | 5,146.5 | |
Many states are considering a new license-plate pattern that consists of a sequence of four letters followed by a sequence of four digits. Assuming that each four-letter four-digit arrangement is equally likely, what is the probability that such a license plate will contain at least one palindrome (a four-letter arrang... | 2735 | 0.5625 | 6,830.9375 | 5,772.333333 | 8,192 | |
Kolya and his sister Masha went to visit someone. After walking a quarter of the way, Kolya remembered that they had forgotten the gift at home and turned back, while Masha continued walking. Masha arrived at the visit 20 minutes after leaving home. How many minutes later did Kolya arrive, given that they walked at the... | 10 | 0.1875 | 7,328.125 | 6,389.666667 | 7,544.692308 | |
Consider a sequence of real numbers \(\{a_n\}\) defined by \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{1 + n a_n}\) for \(n \geq 1\). Find the value of \(\frac{1}{a_{2005}} - 2000000\). | 9011 | 1 | 4,143.9375 | 4,143.9375 | -1 | |
Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.
[asy]
unitsize(1 cm);
real func (real x) {
return (x^2 + 3/4);
}
path parab = graph(func,-1.5,1.5);
draw(parab);
d... | \frac{3}{4} | 0 | 8,088.3125 | -1 | 8,088.3125 | |
Calculate $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$. | 30p \sqrt{p} | 1 | 2,517.375 | 2,517.375 | -1 | |
Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $... | 254 | 0.125 | 8,137.3125 | 7,754.5 | 8,192 | |
Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a,$ $b,$ $c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as fractions in lowest terms. For example, bot... | 392 | $0.abcd=\frac{\overline{abcd}}{9999}$, $9999=9\times 11\times 101$.
Then we need to find the number of positive integers less than $10000$ that can meet the requirement. Suppose the number is $x$.
Case $1$: $(9999, x)=1$. Clearly $x$ satisfies. \[\varphi \left( 9999 \right) =9999\times \left( 1-\frac{1}{3} \right) \t... | 0 | 8,192 | -1 | 8,192 |
What is the probability that a positive integer less than or equal to 24 is a factor of 24? Express your answer as a common fraction. | \frac{1}{3} | 1 | 1,241.9375 | 1,241.9375 | -1 | |
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$ . Determine the minimum value of $p + q$ . | 296 | 0 | 8,192 | -1 | 8,192 | |
Given the parabola $C_1$: $y^{2}=4x$ with focus $F$ that coincides with the right focus of the ellipse $C_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and the line connecting the intersection points of the curves $C_1$ and $C_2$ passes through point $F$, determine the length of the major axis of the ell... | 2\sqrt{2}+2 | 0 | 7,640.4375 | -1 | 7,640.4375 | |
A fair coin is to be tossed $10_{}^{}$ times. Let $\frac{i}{j}^{}_{}$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$. | 73 | We can also split the problem into casework.
Case 1: 0 Heads
There is only one possibility.
Case 2: 1 Head
There are 10 possibilities.
Case 3: 2 Heads
There are 36 possibilities.
Case 4: 3 Heads
There are 56 possibilities.
Case 5: 4 Heads
There are 35 possibilities.
Case 6: 5 Heads
There are 6 possibilities... | 1 | 4,086.9375 | 4,086.9375 | -1 |
Let \( p, q, r, s, \) and \( t \) be the roots of the polynomial
\[ x^5 + 10x^4 + 20x^3 + 15x^2 + 6x + 3 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{pt} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{qt} + \frac{1}{rs} + \frac{1}{rt} + \frac{1}{st}. \] | \frac{20}{3} | 0 | 6,442.125 | -1 | 6,442.125 | |
In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required to replace all the shades in all the chand... | 25 | 0.5 | 5,694.0625 | 5,264 | 6,124.125 | |
Solve the equation: $x^{2}-2x-8=0$. | -2 | 0.1875 | 2,028.75 | 2,322.333333 | 1,961 | |
For a natural number \( N \), if at least eight out of the nine natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "Ba Xian number". What is the smallest "Ba Xian number" greater than 2000? | 2016 | 0.125 | 8,120.4375 | 7,619.5 | 8,192 | |
The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is: | 7350 | 1. **Factorize 1260**: We start by factorizing 1260 into its prime factors:
\[
1260 = 2^2 \cdot 3^2 \cdot 5 \cdot 7
\]
2. **Condition for $N^3$**: For $N^3$ to be a perfect cube, each prime factor in its factorization must have an exponent that is a multiple of 3.
3. **Adjust exponents to multiples of 3**: ... | 0.9375 | 4,200.125 | 3,934 | 8,192 |
Real numbers $a, b, c$ satisfy the equations $a+b+c=26,1 / a+1 / b+1 / c=28$. Find the value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$ | 725 | Multiplying the two given equations gives $$\frac{a}{a}+\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{b}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}+\frac{c}{c}=26 \cdot 28=728$$ and subtracting 3 from both sides gives the answer, 725. | 0.875 | 4,461.625 | 4,284 | 5,705 |
Given the function $y=\sin (2x+1)$, determine the direction and magnitude of the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{1}{2} | 1 | 2,962 | 2,962 | -1 | |
The expression $\cos x + \cos 3x + \cos 7x + \cos 9x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\]for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 13 | 0.9375 | 3,216.875 | 2,885.2 | 8,192 | |
Find $\tan A$ in the right triangle shown below.
[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,15);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$41$", (B+C)/2,NE);
label("$40$", B/2,S);
[/asy] | \frac{9}{40} | 0.0625 | 6,915.5 | 7,694 | 6,863.6 | |
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\le... | (2n)! \cdot 2^{n^2} |
To solve this problem, we need to determine the number of ways to choose the sets \( S_{i,j} \) such that they satisfy the given conditions. First, consider a fixed set \(\{1, 2, \ldots, 2n\}\). We construct nested sets \( S_{i,j} \) with \( i + j \) elements, ensuring that \( S_{i,j} \subseteq S_{k,l} \) whenever \( ... | 0 | 8,192 | -1 | 8,192 |
Let \(Q\) be a point chosen uniformly at random inside the unit square with vertices at \((0,0), (1,0), (1,1)\), and \((0,1)\). Calculate the probability that the slope of the line determined by \(Q\) and the point \(\left(\frac{1}{4}, \frac{3}{4}\right)\) is greater than or equal to 1. | \frac{1}{8} | 0.0625 | 7,569.9375 | 5,347 | 7,718.133333 | |
Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle? | \sqrt{7} | 0.0625 | 8,192 | 8,192 | 8,192 | |
A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\fra... | 84 | 0.5625 | 6,198.875 | 5,419.444444 | 7,201 | |
Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:
(1) $a_0+a_1+...+a_5$;
(2) $|a_0|+|a_1|+...+|a_5|$;
(3) $a_1+a_3+a_5$;
(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$. | -243 | 0.125 | 6,191.875 | 6,171 | 6,194.857143 |
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