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 |
|---|---|---|---|---|---|---|
Bus stop \(B\) is located on a straight highway between stops \(A\) and \(C\). After some time driving from \(A\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. After the same amount of time, the bus again finds i... | 180 | 0 | 8,192 | -1 | 8,192 | |
Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who cou... | 298 | Let $S$ be the percent of people who study Spanish, $F$ be the number of people who study French, and let $S \cup F$ be the number of students who study both. Then $\left\lceil 80\% \cdot 2001 \right\rceil = 1601 \le S \le \left\lfloor 85\% \cdot 2001 \right\rfloor = 1700$, and $\left\lceil 30\% \cdot 2001 \right\rceil... | 0.75 | 5,703.6875 | 5,062.416667 | 7,627.5 |
Given the function $f(x)=\cos x\cdot \sin \left(x+\frac{\pi }{3}\right)-\sqrt{3}\cos ^{2}x+\frac{\sqrt{3}}{4}$, where $x\in R$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[-\frac{\pi }{4},\frac{\pi }{4}\right]$. | -\frac{1}{2} | 0.5 | 7,109.4375 | 7,034.125 | 7,184.75 | |
Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$, or as a $b^{}_{}$ when it should be an $a^{}... | 532 | Consider $n$ letter strings instead. If the first letters all get transmitted correctly, then the $a$ string will be first. Otherwise, the only way is for both of the first letters to be the same, and then we consider the next $n-1$ letter string following the first letter. This easily leads to a recursion: $p_n=\frac2... | 0.125 | 7,635.5 | 6,763.5 | 7,760.071429 |
Consider a rectangular region of 2x1 unit squares at the center of a large grid of unit squares. Each subsequent ring forms around this rectangle by one unit thickness. Determine the number of unit squares in the $50^{th}$ ring around this central rectangle. | 402 | 0.375 | 5,943.5 | 4,683.166667 | 6,699.7 | |
The instantaneous rate of change of carbon-14 content is $-\frac{\ln2}{20}$ (becquerel/year) given that at $t=5730$. Using the formula $M(t) = M_0 \cdot 2^{-\frac{t}{5730}}$, determine $M(2865)$. | 573\sqrt{2}/2 | 0 | 5,819.375 | -1 | 5,819.375 | |
Given the sequence $1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2, \cdots$ where the number of 2s between consecutive 1s increases by 1 each time, calculate the sum of the first 1234 terms. | 2419 | 0 | 6,803.9375 | -1 | 6,803.9375 | |
The line $y=kx+b$ is tangent to the curve $y=x^2+\frac{a}{x}+1$ at the point $(2,3)$. Find the value of $b$. | -7 | 0.5625 | 2,477.4375 | 2,109.888889 | 2,950 | |
Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal t... | 112.5 | 0 | 8,192 | -1 | 8,192 | |
A triangle has vertices $P_{}^{}=(-8,5)$, $Q_{}^{}=(-15,-19)$, and $R_{}^{}=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0_{}^{}$. Find $a+c_{}^{}$.
[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-1... | 89 | [asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R); MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f);MP("P'",U,SE,f); D(P--Q--R--cycle);D(U);D(P--U); D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); [/asy]
U... | 0.9375 | 4,626.4375 | 4,388.733333 | 8,192 |
What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice? | 17 | 1 | 1,413.6875 | 1,413.6875 | -1 | |
The constant term in the expansion of $(x^2+ \frac{1}{x^2}-2)^3$ is ______. | -20 | 0.9375 | 5,642.125 | 5,472.133333 | 8,192 | |
Find the value of the algebraic cofactor of element $a$ in the determinant \\( \begin{vmatrix} 3 & a & 5 \\\\ 0 & -4 & 1 \\\\ -2 & 1 & 3\\end{vmatrix} \\). | -2 | 0.875 | 2,322.6875 | 2,199.5 | 3,185 | |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases}x= \frac{ \sqrt{2}}{2}t \\ y=3+ \frac{ \sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$... | \frac{2 \sqrt{5}}{3} | 0 | 5,683.1875 | -1 | 5,683.1875 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A}{\sqrt{3} \sin C}$.
(1) Find the value of $b$.
(2) If $\cos B + \sqrt{3} \sin B = 2$, find the maximum area of triangle $ABC$. | \frac{3\sqrt{3}}{4} | 0 | 5,785.125 | -1 | 5,785.125 | |
If $x (x+y) = 27$ and $y (x+y) = 54$, what is the value of $(x+y)^2$? | 81 | 1 | 1,713.625 | 1,713.625 | -1 | |
Let $N$ be the number of ordered pairs of integers $(x, y)$ such that
\[
4x^2 + 9y^2 \le 1000000000.
\]
Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$ . What is the value of $10a + b$ ? | 52 | 0.25 | 7,772 | 6,512 | 8,192 | |
The expressions \[A=1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39\]and \[B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39\]are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between i... | 722 | 0.5 | 5,934.1875 | 3,676.375 | 8,192 | |
Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? | 2118 | 0 | 8,192 | -1 | 8,192 | |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.
(1) Find the general term formula of the sequence ${a_{n}}$;
(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfying $c_{k}⋅c_{k+1} < 0$ is called the signum... | 23 | 0 | 8,192 | -1 | 8,192 | |
Let $a,$ $b,$ $c$ be distinct real numbers such that
\[\frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} = 0.\]Find all possible values of
\[\frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2}.\]Enter all the possible values, separated by commas. | 0 | 0.0625 | 8,002.4375 | 5,159 | 8,192 | |
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i] | g(n)=\lceil\frac{2n+1}{3}\rceil |
Let \( n \geq 5 \) be a given integer. We are tasked with determining the greatest integer \( k \) for which there exists a polygon with \( n \) vertices (which can be either convex or non-convex, with a non-self-intersecting boundary) having \( k \) internal right angles.
### Approach
To solve this problem, we must... | 0 | 8,192 | -1 | 8,192 |
Find the smallest possible value of \(x\) in the simplified form \(x=\frac{a+b\sqrt{c}}{d}\) if \(\frac{7x}{8}-1=\frac{4}{x}\), where \(a, b, c,\) and \(d\) are integers. What is \(\frac{acd}{b}\)? | -105 | 1 | 3,329.8125 | 3,329.8125 | -1 | |
Given the following six statements:
(1) All women are good drivers
(2) Some women are good drivers
(3) No men are good drivers
(4) All men are bad drivers
(5) At least one man is a bad driver
(6) All men are good drivers.
The statement that negates statement (6) is: | (5) | To find the statement that negates statement (6), we need to understand the logical opposite of the statement:
\[\text{(6) All men are good drivers.}\]
The negation of "All men are good drivers" is "There exists at least one man who is not a good driver." This can be expressed as:
\[\text{Not all men are good drivers.... | 0.0625 | 1,605.6875 | 1,282 | 1,627.266667 |
Cagney can frost a cupcake every 15 seconds and Lacey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes. | 53 | 0.75 | 4,994.875 | 4,751.416667 | 5,725.25 | |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2903^{\circ}+\sin 2904^{\circ}+\cdots+\sin 6503^{\circ}\right)^{\cos 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}}\right)
$$ | 67 | 0.3125 | 6,503.625 | 4,554.6 | 7,389.545455 | |
Given the function $$f(x)=a+\sin(x-\frac{1}{2})$$, if $$f(0)+f(\frac{1}{2019})+f(\frac{2}{2019})+…+f(\frac{2017}{2019})+f(\frac{2018}{2019})+f(1)=1010$$, find the value of the real number $a$. | \frac{1}{2} | 0.4375 | 6,417.1875 | 4,592.714286 | 7,836.222222 | |
Find $x$ if
\[1 + 5x + 9x^2 + 13x^3 + \dotsb = 85.\] | \frac{4}{5} | 0.75 | 4,242.8125 | 3,509.083333 | 6,444 | |
A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form. | 2\sqrt{2} | 1 | 1,430.375 | 1,430.375 | -1 | |
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____. | \frac{2}{9} | 0.25 | 7,672.0625 | 6,385 | 8,101.083333 | |
What is the distance, in units, between the points $(3, -2)$ and $(7, 5)$? | \sqrt{65} | 1 | 1,552.0625 | 1,552.0625 | -1 | |
The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \text{ cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \text{ cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? | 663 | 1. **Set up the equations based on the problem statement:**
Let $t$ be the length of each side of the equilateral triangle, and $s$ be the length of each side of the square. The perimeter of the triangle is $3t$ and the perimeter of the square is $4s$. According to the problem, the perimeter of the triangle exceeds ... | 1 | 2,574.125 | 2,574.125 | -1 |
What is the smallest three-digit multiple of 13? | 104 | 1 | 499.5625 | 499.5625 | -1 | |
Compute the definite integral:
$$
\int_{0}^{2 \operatorname{arctg} \frac{1}{2}} \frac{(1-\sin x) dx}{\cos x(1+\cos x)}
$$ | 2 \ln \frac{3}{2} - \frac{1}{2} | 0.0625 | 7,429 | 4,496 | 7,624.533333 | |
In a rectangle, the perimeter of quadrilateral $PQRS$ is given. If the horizontal distance between adjacent dots in the same row is 1 and the vertical distance between adjacent dots in the same column is 1, what is the perimeter of quadrilateral $PQRS$? | 14 | The perimeter of quadrilateral $PQRS$ equals $PQ+QR+RS+SP$. Since the dots are spaced 1 unit apart horizontally and vertically, then $PQ=4, QR=4$, and $PS=1$. Thus, the perimeter equals $4+4+RS+1$ which equals $RS+9$. We need to determine the length of $RS$. If we draw a horizontal line from $S$ to point $T$ on $QR$, w... | 0 | 4,755 | -1 | 4,755 |
Starting with the number 100, Shaffiq repeatedly divides his number by two and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1? | 6 | 1 | 4,215.5625 | 4,215.5625 | -1 | |
If the complex number $Z=(1+ai)i$ is an "equal parts complex number", determine the value of the real number $a$. | -1 | 0.9375 | 3,055.6875 | 2,792.466667 | 7,004 | |
In a 12-hour interval (from 0 hours to 12 hours), how many minutes are there when the value of the hour is greater than the value of the minutes? | 66 | 0.125 | 6,362.125 | 6,197 | 6,385.714286 | |
In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same ... | 36 | 0.0625 | 7,317.1875 | 7,434 | 7,309.4 | |
Given that the vertices of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ lie on the surface of a sphere $O$, the lateral area of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ is $6$, and the base area is $\sqrt{3}$, calculate the surface area of the sphere $O$. | \frac{19\pi}{3} | 0.25 | 5,275.625 | 4,015 | 5,695.833333 | |
A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer? | 99 | 0.9375 | 4,979.5 | 4,765.333333 | 8,192 | |
Point $C$ is the midpoint of $\overline{AB}$, point $D$ is the midpoint of $\overline{AC}$, point $E$ is the midpoint of $\overline{AD}$, and point $F$ is the midpoint of $\overline{AE}$. If $AF=3$, what is the number of units in the length of $\overline{AB}$? | 48 | 1 | 1,644 | 1,644 | -1 | |
A large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\%$ of the area of the large square region. Find the ratio $\frac{w}{t}$ in this scenario... | \frac{1}{9} | 0 | 8,099.75 | -1 | 8,099.75 | |
On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\frac{AM}{AC} = \frac{CN}{CE} = \lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\lambda$. | \frac{\sqrt{3}}{3} | 0 | 5,309.8125 | -1 | 5,309.8125 | |
Find $\sec 135^\circ.$ | -\sqrt{2} | 1 | 1,736.25 | 1,736.25 | -1 | |
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it? | \frac{\pi \sqrt{15}}{16} | 0 | 6,324 | -1 | 6,324 | |
Fill in the four boxes with the operations "+", "-", "*", and "$\div$" each exactly once in the expression 10 □ 10 □ 10 □ 10 □ 10 to maximize the value. What is the maximum value? | 109 | 0 | 8,192 | -1 | 8,192 | |
In digital communication, signals are sequences composed of the digits "$0$" and "$1$". Signals are transmitted continuously $n$ times, with each transmission of "$0$" and "$1$" being equally likely. Let $X$ denote the number of times the signal "$1$" is transmitted.<br/>① When $n=6$, $P\left(X\leqslant 2\right)=$____;... | 1250 | 0.8125 | 4,267.25 | 3,985.769231 | 5,487 | |
(The full score of this question is 14 points) It is known that A and B are two fixed points on a plane, and the moving point P satisfies $|PA| + |PB| = 2$.
(1) Find the equation of the trajectory of point P;
(2) Suppose the line $l: y = k (k > 0)$ intersects the trajectory of point P from (1) at points M and N, find t... | \frac{1}{2} | 0 | 8,030.5 | -1 | 8,030.5 | |
A number $x$ is equal to $6 \cdot 18 \cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | 441 | 1 | 2,524.8125 | 2,524.8125 | -1 | |
The parabola with the equation \( y = 25 - x^2 \) intersects the \( x \)-axis at points \( A \) and \( B \).
(a) Determine the length of \( AB \).
(b) Rectangle \( ABCD \) is formed such that \( C \) and \( D \) are below the \( x \)-axis and \( BD = 26 \). Determine the length of \( BC \).
(c) If \( CD \) is extend... | 14 | 0.8125 | 4,567.75 | 4,111.923077 | 6,543 | |
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 23 | 1. **Prime Factorization of $12!$:**
The prime factorization of $12!$ (12 factorial) is necessary to determine the number of divisors and the condition for these divisors to be perfect squares. We start by finding the prime factorization of $12!$:
\[
12! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times ... | 1 | 3,301.5 | 3,301.5 | -1 |
A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$. | 556 | Casework Solution: x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-1-1 (1 way to order) $\dbinom{8}{4} = 70$ ways total
x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-2 (3 ways to order) $\dbinom{7}{3} \times 3= 105$ ways total
x-distribution: 1-1-1-1 (1 way to order) y-distribution: 2-2 (1... | 0.125 | 7,589.4375 | 6,289 | 7,775.214286 |
Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive? | 12 | 0.3125 | 6,888.8125 | 5,825.2 | 7,372.272727 | |
Ten circles of diameter 1 are arranged in the first quadrant of a coordinate plane. Five circles are in the base row with centers at $(0.5, 0.5)$, $(1.5, 0.5)$, $(2.5, 0.5)$, $(3.5, 0.5)$, $(4.5, 0.5)$, and the remaining five directly above the first row with centers at $(0.5, 1.5)$, $(1.5, 1.5)$, $(2.5, 1.5)$, $(3.5, ... | 30 | 0 | 7,977.5625 | -1 | 7,977.5625 | |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$ . Find the maximum value of $n$ . | 64 | 0.0625 | 8,077 | 7,619 | 8,107.533333 | |
Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? | 4 | To solve this problem, we need to ensure that no element in the set $S$ is a multiple of any other element in $S$. We start by partitioning the set $\{1, 2, \dots, 12\}$ into subsets where each element in a subset is a multiple of the smallest element in that subset. This helps us to easily identify and avoid choosing ... | 0 | 8,192 | -1 | 8,192 |
I randomly choose an integer \( p \) between \( 1 \) and \( 20 \) inclusive. What is the probability that \( p \) is such that there exists an integer \( q \) so that \( p \) and \( q \) satisfy the equation \( pq - 6p - 3q = 3 \)? Express your answer as a common fraction. | \frac{3}{20} | 0 | 5,506.375 | -1 | 5,506.375 | |
How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0.1875 | 7,453.8125 | 5,819.666667 | 7,830.923077 | |
Let $\mathbf{a} = \begin{pmatrix} 7 \\ -4 \\ -4 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -1 \\ 2 \end{pmatrix}.$ Find the vector $\mathbf{b}$ such that $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are collinear, and $\mathbf{b}$ bisects the angle between $\mathbf{a}$ and $\mathbf{c}.$
[asy]
unitsize(0.5... | \begin{pmatrix} 1/4 \\ -7/4 \\ 1/2 \end{pmatrix} | 0 | 7,808 | -1 | 7,808 | |
Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$. | \[\frac{\ln^3(2)}{3}\] |
The problem requires evaluating the infinite series:
\[
\sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right).
\]
Firstly, observe the behavior of the logarithmic terms for large \( n \). Using the approximation \(\ln(1+x) \approx x\) for small \( x \), we ... | 0 | 8,192 | -1 | 8,192 |
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? | 29 | 0.8125 | 5,236.125 | 4,554 | 8,192 | |
There are 5 different types of books, with at least 3 copies of each. If we want to buy 3 books to gift to 3 students, with each student receiving one book, how many different ways are there to do this? | 125 | 0.4375 | 6,653 | 4,876.142857 | 8,035 | |
A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace. | 90 | 0 | 4,209 | -1 | 4,209 | |
The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is: | 9611 | 0 | 7,766.8125 | -1 | 7,766.8125 | |
If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-n representation of $b$ is | 80 | 1. **Understanding the Problem:**
Given an equation $x^2 - ax + b = 0$ with integer solutions $n$ and $m$, where $n > 8$. The coefficient $a$ in base-$n$ is represented as $18_n$. We need to find the base-$n$ representation of $b$.
2. **Using Vieta's Formulas:**
Vieta's formulas for a quadratic equation $x^2 - a... | 1 | 1,833 | 1,833 | -1 |
I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction. | \frac{2}{5} | 1 | 4,130.6875 | 4,130.6875 | -1 | |
Given a function $f(x)$ defined on $\mathbb{R}$, for any real numbers $x_1$, $x_2$, it satisfies $f(x_1+x_2)=f(x_1)+f(x_2)+2$. The sequence $\{a_n\}$ satisfies $a_1=0$, and for any $n\in\mathbb{N}^*$, $a_n=f(n)$. Find the value of $f(2010)$. | 4018 | 1 | 3,478.0625 | 3,478.0625 | -1 | |
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.
| 376 | 0.0625 | 7,603.125 | 6,678 | 7,664.8 | |
Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 0.$ Simplify
\[\frac{1}{b^2 + c^2 - a^2} + \frac{1}{a^2 + c^2 - b^2} + \frac{1}{a^2 + b^2 - c^2}.\] | 0 | 1 | 2,874.4375 | 2,874.4375 | -1 | |
Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino?
[asy]
size(4cm);
draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0));
[/asy] | 2^{n+1} - 2 |
We are tasked with determining the number of ways to tile a \(4 \times 4n\) grid using the \(L\)-shaped tetromino described in the problem. The shape of the \(L\)-shaped tetromino can cover precisely 4 unit squares.
### Step-by-step Analysis
1. **Understand the Requirements**:
- A \(4 \times 4n\) grid contains ... | 0 | 8,192 | -1 | 8,192 |
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 an "Eight Immortals Number." What is the smallest "Eight Immortals Number" greater than $2000$? | 2016 | 0.0625 | 7,900.5 | 6,755 | 7,976.866667 | |
A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew? | \frac{20}{11} | The probability of drawing $k$ marbles is the probability of drawing $k-1$ blue marbles and then the special marble, which is $p_{k}=\left(\frac{9}{20}\right)^{k-1} \times \frac{1}{20}$. The probability of drawing no ugly marbles is therefore $\sum_{k=1}^{\infty} p_{k}=\frac{1}{11}$. Then given that no ugly marbles wer... | 0 | 6,936.25 | -1 | 6,936.25 |
Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold. | 153846 | 0.875 | 4,506.5 | 3,980 | 8,192 | |
The traditional Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art" records: "There are 5 cows and 2 sheep, worth 19 taels of silver; 2 cows and 5 sheep, worth 16 taels of silver. How much is each cow and each sheep worth in silver?" According to the translation above, answer the following two ques... | 7.5 | 0 | 3,664.6875 | -1 | 3,664.6875 | |
Given the general term formula of the sequence $\{a_n\}$ as $a_n= \frac{3^{-n}+2^{-n}+(-1)^{n}(3^{-n}-2^{-n})}{2}$, where $n=1$, $2$, $...$, determine the value of $\lim_{n \rightarrow \infty }(a_1+a_2+...+a_n)$. | \frac{19}{24} | 0.25 | 7,415.625 | 5,563.5 | 8,033 | |
Simplify first, then evaluate: $\left(\frac{a+1}{2a-2}-\frac{5}{2{a}^{2}-2}-\frac{a+3}{2a+2}\right)÷\frac{{a}^{2}}{{a}^{2}-1}$, where the value of $a$ is chosen as an appropriate integer from the solution set of the inequality system $\left\{\begin{array}{l}{a-\sqrt{5}＜0}\\{\frac{a-1}{2}＜a}\end{array}\right.$. | -\frac{1}{8} | 0.75 | 6,157.5625 | 5,479.416667 | 8,192 | |
Points \( D \) and \( E \) are located on the diagonals \( A B_{1} \) and \( C A_{1} \) of the lateral faces of the prism \( A B C A_{1} B_{1} C_{1} \) such that the lines \( D E \) and \( B C_{1} \) are parallel. Find the ratio of the segments \( D E \) and \( B C_{1} \). | 1:2 | 0 | 4,597.375 | -1 | 4,597.375 | |
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand. | 2010 | 0 | 8,069.0625 | -1 | 8,069.0625 | |
Given the function $f(x)=\sin x+\lambda\cos x (\lambda\in\mathbb{R})$ is symmetric about $x=-\frac{\pi}{4}$, find the equation of one of the axes of symmetry of function $g(x)$ obtained by expanding the horizontal coordinate of each point of the graph of $f(x)$ by a factor of $2$ and then shifting it right by $\frac{\p... | \frac{11\pi}{6} | 0 | 7,286.625 | -1 | 7,286.625 | |
A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he feed al... | 144 | 0.6875 | 4,726.875 | 4,296.090909 | 5,674.6 | |
Evaluate the greatest integer less than or equal to \[\frac{5^{150} + 3^{150}}{5^{147} + 3^{147}}.\] | 124 | 0.4375 | 7,420.8125 | 6,429.285714 | 8,192 | |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{3} | 0.125 | 1,840.5 | 2,089.5 | 1,804.928571 | |
What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$? | 1010 | We are given the expression:
\[ 1+3+5+\cdots+2019 - (2+4+6+\cdots+2018) \]
1. **Identify the sequences**:
- The first sequence is the sum of all odd numbers from 1 to 2019.
- The second sequence is the sum of all even numbers from 2 to 2018.
2. **Sum of odd numbers**:
- The sequence of odd numbers from 1 to ... | 1 | 3,803.5625 | 3,803.5625 | -1 |
Five packages are delivered to five houses, one to each house. If the packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? | \frac{1}{6} | 0 | 4,503.1875 | -1 | 4,503.1875 | |
Find whole numbers $\heartsuit$ and $\clubsuit$ such that $\heartsuit \cdot \clubsuit = 48$ and $\heartsuit$ is even, then determine the largest possible value of $\heartsuit + \clubsuit$. | 26 | 0.5 | 2,934.625 | 2,192.875 | 3,676.375 | |
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The sequence satisfies the conditions $a\_1=23$, $a\_2=-9$, and $a_{n+2}=a\_n+6\times(-1)^{n+1}-2$ for all $n \in \mathbb{N}^*$.
(1) Find the general formula for the terms of the sequence $\{a\_n\}$;
(2) Find the value of $n$ when $S\_n$... | 11 | 0 | 8,192 | -1 | 8,192 | |
Among the 95 numbers $1^2, 2^2, 3^2, \ldots, 95^2$, how many of them have an odd digit in the tens place? | 19 | 0.125 | 7,852.8125 | 6,773 | 8,007.071429 | |
Determine the value of the sum $$\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2} \cdot 3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\cdots+\frac{29}{14^{2} \cdot 15^{2}}$$ | \frac{224}{225} | The sum telescopes as $$\left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)+\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)+\cdots+\left(\frac{1}{14^{2}}-\frac{1}{15^{2}}\right)=\frac{1}{1^{2}}-\frac{1}{15^{2}}=\frac{224}{225}$$ | 0.9375 | 3,373.3125 | 3,052.066667 | 8,192 |
In pentagon $MATHS$, $\angle M \cong \angle T \cong \angle H$ and $\angle A$ is supplementary to $\angle S$. How many degrees are in the measure of $\angle H$? | 120^\circ | 1 | 1,220.875 | 1,220.875 | -1 | |
On an algebra test, there were $5x$ problems. Lucky Lacy missed $x$ of them. What percent of the problems did she get correct? | 80 \% | 1 | 1,338.4375 | 1,338.4375 | -1 | |
$ABCDEFGH$ is a cube. Find $\cos \angle GAC$. | \frac{\sqrt{3}}{3} | 0 | 3,596.25 | -1 | 3,596.25 | |
How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct. | 24 | There are 4 ways to color the first vertex, then 3 ways to color the second vertex to be distinct from the first, and finally 2 ways to color the third vertex to be distinct from the earlier two vertices. Multiplying gives 24 ways. | 1 | 2,144.75 | 2,144.75 | -1 |
An infinite geometric series has a first term of \( 416 \) and a sum of \( 3120 \). What is its common ratio? | \frac{84}{97} | 0 | 1,713.9375 | -1 | 1,713.9375 | |
Given that $F\_1$ is the left focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{QP} = 4\overrightarrow{PF\_1}$, find the eccentric... | \frac{3}{2} | 0 | 8,192 | -1 | 8,192 | |
Given the parabola $C$: $y^2=2px (p > 0)$ with focus $F$ and directrix $l$. A line perpendicular to $l$ at point $A$ on the parabola $C$ at $A(4,y_0)$ intersects $l$ at $A_1$. If $\angle A_1AF=\frac{2\pi}{3}$, determine the value of $p$. | 24 | 0.4375 | 6,552.1875 | 5,322.428571 | 7,508.666667 | |
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ so that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$. Express your answer in simplest radical form.
[asy]
pair K,L,M,I,A,B,C,D;
D=(0,0);
C=(10,0);
B=(10,10);
A=(0,10);
I=(-8.7,5);
L=(18... | 32 + 16\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Find the greatest common divisor of the numbers $2002+2,2002^{2}+2,2002^{3}+2, \ldots$. | 6 | Notice that $2002+2$ divides $2002^{2}-2^{2}$, so any common divisor of $2002+2$ and $2002^{2}+2$ must divide $\left(2002^{2}+2\right)-\left(2002^{2}-2^{2}\right)=6$. On the other hand, every number in the sequence is even, and the $n$th number is always congruent to $1^{n}+2 \equiv 0$ modulo 3 . Thus, 6 divides every ... | 0.6875 | 5,663.3125 | 5,269.909091 | 6,528.8 |
Two people, A and B, are collaborating to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they reach half of the total amount, A's speed triples while B takes a 5-minute break and then continues at the original speed. By the time the document is completed, A an... | 18000 | 0 | 8,097.5625 | -1 | 8,097.5625 | |
In parallelogram \( A B C D \), the height drawn from vertex \( B \) of the obtuse angle to side \( DA \) divides it in a ratio of 5:3, starting from vertex \( D \). Find the ratio \( AC:BD \) if \( AD:AB=2 \). | 2:1 | 0.25 | 7,529.75 | 5,736.25 | 8,127.583333 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.