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 |
|---|---|---|---|---|---|---|
Each of the numbers \( m \) and \( n \) is the square of an integer. The difference \( m - n \) is a prime number. Which of the following could be \( n \)? | 900 | 0 | 8,112.6875 | -1 | 8,112.6875 | |
The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers. | 5\pi | 0.875 | 1,053.5 | 1,137.714286 | 464 | |
Given the equation of a circle $x^2 + y^2 - 6x - 8y = 0$, if the longest chord AC and the shortest chord BD both pass through the point (-1, 4) on the circle, find the area of the quadrilateral ABCD. | 30 | 0.5625 | 6,406.3125 | 5,017.444444 | 8,192 | |
Pete's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars. What is the maximum amount Pete can withdraw from the account if he has no other money? | 498 | 0 | 3,184.8125 | -1 | 3,184.8125 | |
A certain bookstore currently has $7700$ yuan in funds, planning to use all of it to purchase a total of $20$ sets of three types of books, A, B, and C. Among them, type A books cost $500$ yuan per set, type B books cost $400$ yuan per set, and type C books cost $250$ yuan per set. The bookstore sets the selling prices... | 10 | 0.25 | 7,220.5625 | 5,285.25 | 7,865.666667 | |
Determine the value of $a$ such that the equation
\[\frac{x - 3}{ax - 2} = x\]
has exactly one solution. | \frac{3}{4} | 0.3125 | 7,935.5 | 7,371.2 | 8,192 | |
Call an integer $n>1$ radical if $2^{n}-1$ is prime. What is the 20th smallest radical number? If $A$ is your answer, and $S$ is the correct answer, you will get $\max \left(25\left(1-\frac{|A-S|}{S}\right), 0\right)$ points, rounded to the nearest integer. | 4423 | The answer is 4423. | 0.0625 | 5,450.6875 | 2,202 | 5,667.266667 |
The number of the distinct solutions to the equation $|x-|2x+1||=3$ is | 2 | To solve the equation $|x - |2x + 1|| = 3$, we need to consider the different cases that arise from the absolute values.
#### Step 1: Analyze the inner absolute value
The expression $|2x + 1|$ can be simplified by considering two cases:
- **Case 1:** $2x + 1 \geq 0 \Rightarrow x \geq -\frac{1}{2}$
- Here, $|2x + 1| ... | 0.9375 | 4,409.25 | 4,157.066667 | 8,192 |
A table consisting of 1861 rows and 1861 columns is filled with natural numbers from 1 to 1861 such that each row contains all numbers from 1 to 1861. Find the sum of the numbers on the diagonal that connects the top left and bottom right corners of the table if the filling of the table is symmetric with respect to thi... | 1732591 | 0.5625 | 5,873.875 | 4,070.888889 | 8,192 | |
Let $a,$ $b,$ and $c$ be nonzero real numbers, and let
\[x = \frac{b}{c} + \frac{c}{b}, \quad y = \frac{a}{c} + \frac{c}{a}, \quad z = \frac{a}{b} + \frac{b}{a}.\]Simplify $x^2 + y^2 + z^2 - xyz.$ | 4 | 0.4375 | 7,405.75 | 6,394.857143 | 8,192 | |
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then w... | 1 + 1 + \frac{1}{2} + \ldots + \frac{1}{7} |
Let \( k \) be a positive real number. \( A \) and \( B \) play the following game: at the start, there are 80 zeroes arranged around a circle. Each turn, \( A \) increases some of these 80 numbers such that the total sum added is 1. Next, \( B \) selects ten consecutive numbers with the largest sum and reduces them a... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x$,
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) When $x\in[0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$. | 2+\sqrt{2} | 0 | 5,535.125 | -1 | 5,535.125 | |
A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. Initially, there are 20 people in total in... | 4100 | 0.375 | 4,622.875 | 4,526.833333 | 4,680.5 | |
What is the least integer whose square is 48 more than its double? | -6 | 1 | 2,115 | 2,115 | -1 | |
In the trapezoid shown, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $7:3$. If $AB + CD = 210$ cm, how long is segment $\overline{AB}$? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
pair A = (0,0), B = (5,0), C = (3,2), D = (1,2);
draw(A--B--C--D--cycle--C);
label("$A$",A,SW)... | 147\text{ cm} | 1 | 3,202.25 | 3,202.25 | -1 | |
A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form. | 2\sqrt{6} | 0 | 8,192 | -1 | 8,192 | |
Calculate \(7 \cdot 9\frac{2}{5}\). | 65\frac{4}{5} | 0.6875 | 508.4375 | 563.545455 | 387.2 | |
(1) In $\triangle ABC$, if $2\lg \tan B=\lg \tan A+\lg \tan C$, then the range of values for $B$ is ______.
(2) Find the maximum value of the function $y=7-4\sin x\cos x+4\cos ^{2}x-4\cos ^{4}x$ ______. | 10 | 0.375 | 7,768.0625 | 7,061.5 | 8,192 | |
Snow White entered a room where 30 chairs were arranged around a circular table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit in such a way that there was no one next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs must h... | 10 | 0.3125 | 6,749.375 | 4,372.2 | 7,829.909091 | |
Find the values of $a$ and $b$ such that $a + b^2$ can be calculated, where $x = a \pm b i$ are the solutions to the equation $5x^2 + 7 = 2x - 10$. Express your answer as a fraction. | \frac{89}{25} | 0.875 | 2,404.9375 | 2,466.428571 | 1,974.5 | |
The line with equation $y = x$ is translated 3 units to the right and 2 units down. What is the $y$-intercept of the resulting line? | -5 | The line with equation $y = x$ has slope 1 and passes through $(0,0)$. When this line is translated, its slope does not change. When this line is translated 3 units to the right and 2 units down, every point on the line is translated 3 units to the right and 2 units down. Thus, the point $(0,0)$ moves to $(3,-2)$. Ther... | 0.9375 | 4,258.625 | 3,996.4 | 8,192 |
$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} | 0 | 7,385.875 | -1 | 7,385.875 | |
Given circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.
$(1)$ Find the equation of $C$.
$(2)$ Let $l$ be a line tangent to both circle $P$ and circle... | \dfrac {18}{7} | 0.125 | 7,929.875 | 8,192 | 7,892.428571 | |
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i] | k=m-1 |
We need to determine the greatest \( k \) such that the sequence defined by:
\[
x_i =
\begin{cases}
2^i & \text{if } 0 \leq i \leq m - 1, \\
\sum_{j=1}^m x_{i-j} & \text{if } i \geq m,
\end{cases}
\]
contains \( k \) consecutive terms divisible by \( m \).
Firstly, we observe the initial terms of the sequence \(... | 0 | 8,192 | -1 | 8,192 |
The perimeter of the polygon shown is | 28 | To solve this problem, we need to determine the perimeter of the polygon. The solution suggests visualizing the polygon as part of a rectangle and then using the properties of the rectangle to find the perimeter of the polygon.
1. **Visualize the Polygon as Part of a Rectangle**:
- Assume the polygon is part of a r... | 0 | 8,139 | -1 | 8,139 |
When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops? | 70 | 1 | 3,448.5 | 3,448.5 | -1 | |
How many different 8-digit positive integers exist if the digits from the second to the eighth can only be 0, 1, 2, 3, or 4? | 703125 | 0.9375 | 1,770.875 | 1,772.066667 | 1,753 | |
What is $x-y$ if a town has 2017 houses, 1820 have a dog, 1651 have a cat, 1182 have a turtle, $x$ is the largest possible number of houses that have a dog, a cat, and a turtle, and $y$ is the smallest possible number of houses that have a dog, a cat, and a turtle? | 563 | Since there are 1182 houses that have a turtle, then there cannot be more than 1182 houses that have a dog, a cat, and a turtle.
Since there are more houses with dogs and more houses with cats than there are with turtles, it is possible that all 1182 houses that have a turtle also have a dog and a cat.
Therefore, t... | 0.4375 | 6,932.9375 | 5,314.142857 | 8,192 |
A right pyramid has a square base that measures 10 cm on each side. Its peak is 12 cm above the center of its base. What is the sum of the lengths of the pyramid's eight edges? Express your answer to the nearest whole number.
[asy]
size(150);
draw((0,0)--(3,3)--(13,3)--(10,0)--cycle,linewidth(1));
draw((0,0)--(6.5,15)... | 96 | 1 | 4,618.6875 | 4,618.6875 | -1 | |
Given the function $y=\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$. | \frac{\pi}{12} | 0.8125 | 2,850.8125 | 2,765.538462 | 3,220.333333 | |
Calculate $(2.1)(50.5 + 0.15)$ after increasing $50.5$ by $5\%$. What is the product closest to? | 112 | 0.0625 | 7,017.25 | 8,192 | 6,938.933333 | |
Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. | 677 | This method uses complex numbers with $A$ as the origin. Let $A=0$, $B=\sqrt{111}$, $C = \sqrt{111}\theta$, where $\theta = e^{i \pi/3} = \frac{1}{2} + \frac{\sqrt{3}}{2}i$.
Also, let $x$ be $D_1$ or $D_2$. Then
$|x|=\sqrt{111}, |x-\sqrt{111}|=\sqrt{11}$
Therefore, $11 = |x-\sqrt{111}|^2 = |x|^2 + 111 -2\sqrt{111}Re(... | 0 | 8,192 | -1 | 8,192 |
Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$. | 800 | 0.875 | 3,633.6875 | 3,414 | 5,171.5 | |
Let $f(x) = |x+1| - |x-4|$.
(1) Find the range of real number $m$ such that $f(x) \leq -m^2 + 6m$ always holds;
(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$. | \frac{1}{2} | 0.9375 | 4,214.3125 | 3,949.133333 | 8,192 | |
Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$. | 20 | 0.1875 | 7,862.6875 | 6,624.666667 | 8,148.384615 | |
For each pair of real numbers $a \ne b$, define the operation $\star$ as \[
(a \star b) = \frac{a + b}{a - b}.
\]What is the value of $((1 \star 2) \star 3)$? | 0 | 0.9375 | 3,531.5 | 3,220.8 | 8,192 | |
A valuable right-angled triangular metal plate $A B O$ is placed in a plane rectangular coordinate system (as shown in the diagram), with $A B = B O = 1$ (meter) and $A B \perp O B$. Due to damage in the shaded part of the triangular plate, a line $M N$ passing through the point $P\left(\frac{1}{2}, \frac{1}{4}\right)$... | -\frac{1}{2} | 0 | 8,060.1875 | -1 | 8,060.1875 | |
What is the least common multiple of the numbers 1584 and 1188? | 4752 | 0.875 | 2,543.625 | 2,818.285714 | 621 | |
Evaluate
\[\begin{vmatrix} 1 & x & y \\ 1 & x + y & y \\ 1 & x & x + y \end{vmatrix}.\] | xy | 0.5 | 6,466.6875 | 4,741.375 | 8,192 | |
On a line, two red points and several blue points are marked. It turns out that one of the red points is contained in exactly 56 segments with blue endpoints, and the other red point is contained in 50 segments with blue endpoints. How many blue points are marked? | 15 | 0 | 7,110.125 | -1 | 7,110.125 | |
For each positive integer \(1 \leqq k \leqq 100\), let \(a_{k}\) denote the sum \(\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}\). Calculate the value of
\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \ldots + a_{100}^{2}. \] | 200 | 0 | 8,192 | -1 | 8,192 | |
Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ. | 32 | 0 | 7,970.375 | -1 | 7,970.375 | |
Calculate the probability that athlete A cannot run the first leg and athlete B cannot run the last leg in a 4x100 meter relay race selection from 6 short-distance runners, including athletes A and B, to form a team of 4 runners. | \frac{7}{10} | 0.5 | 7,114.125 | 6,438.625 | 7,789.625 | |
The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = x, \quad
g(x,y) = g(y,x), \quad
(x + y) g(x,y) = yg(x, x + y).
\]
Calculate $g(18,63)$. | 126 | 0.25 | 5,635.625 | 3,650 | 6,297.5 | |
Given that \( x \) and \( y \) are real numbers greater than 10, the leading digit of \( \lg x \) is \( a \) and the trailing digit is \( b \); the leading digit of \( \lg y \) is \( c \) and the trailing digit is \( d \). Additionally, it is known that \( |1 - a| + \sqrt{c - 4} = 1 \) and \( b + d = 1 \). Find the val... | 10^7 | 0 | 8,192 | -1 | 8,192 | |
If $x$, $y$, and $z$ are positive integers such that $\gcd(x,y) = 360$ and $\gcd(x,z) = 1176$, find the smallest possible value of $\gcd(y,z)$. | 24 | 0.625 | 6,650.4375 | 6,033.9 | 7,678 | |
Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$ | 2\sqrt{5} | 0.625 | 6,880.0625 | 6,092.9 | 8,192 | |
Find the value of $y$ if $y$ is positive and $y \cdot \lfloor y \rfloor = 132$. Express your answer as a decimal. | 12 | 0 | 8,192 | -1 | 8,192 | |
Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$. | \sqrt{2} | 0.125 | 7,677.25 | 4,790.5 | 8,089.642857 | |
The figure shown consists of a right triangle and two squares. If the figure's total area equals 850 square inches, what is the value of $x$ in inches? [asy]
unitsize(5mm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((0,5)--(0,-2)--(-2,-2)--(-2,0)--(5,0)--(5,5)--cycle--(-2,0));
draw(scale(0.2)*((-1,0)--(-1,1)--(1... | 5 | 1 | 2,448.3125 | 2,448.3125 | -1 | |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[2\mathbf{a} + 3\mathbf{b} + 4\mathbf{c} = \mathbf{0}.\]Compute $2(\mathbf{a} \cdot \mathbf{b}) + 3(\mathbf{a} \cdot \mathbf{c}) + 4(\mathbf{b} \cdot \mathbf{c}).$ | -673/4 | 0 | 7,927.25 | -1 | 7,927.25 | |
How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) | 2 | 1. **Identify the form of the polynomial and the transformation of roots:**
Let $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers. We are given that if $r$ is a root, then $\frac{-1+i\sqrt{3}}{2} \cdot r$ is also a root. We recognize $\frac{-1+i\sqrt{3}}{2}$ as a complex ... | 0 | 8,172.875 | -1 | 8,172.875 |
Let $\pi$ be a permutation of the numbers from 1 through 2012. What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$? | 1006 | Since any proper divisor of $n$ must be less than or equal to $n / 2$, none of the numbers greater than 1006 can divide any other number less than or equal to 2012. Since there are at most 1006 values of $n$ for which $\pi(n) \leq 1006$, this means that there can be at most 1006 values of $n$ for which $\pi(n)$ divides... | 0 | 8,192 | -1 | 8,192 |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | 5 | 1 | 2,088.9375 | 2,088.9375 | -1 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a=2$, $b= \sqrt {3}$, $B= \frac {\pi}{3}$, then $A=$ \_\_\_\_\_\_. | \frac{\pi}{2} | 1 | 2,096.6875 | 2,096.6875 | -1 | |
The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this? | 665 | To find the total duration of the tennis match in minutes, we need to convert the hours into minutes and then add the remaining minutes.
1. **Convert hours to minutes**:
- There are 60 minutes in one hour.
- Therefore, for 11 hours, the total minutes are:
\[
11 \text{ hours} \times 60 \text{ minutes/h... | 1 | 1,129 | 1,129 | -1 |
Let $r=3^s-s$ and $s=2^n+1$. What is the value of $r$ when $n=2$? | 238 | 1 | 1,528.25 | 1,528.25 | -1 | |
Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \). | -2 | 0 | 6,649.0625 | -1 | 6,649.0625 | |
In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn? | 19 | 0 | 7,652.875 | -1 | 7,652.875 | |
Calculate the area of the shape bounded by the lines given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=t-\sin t \\
y=1-\cos t
\end{array}\right. \\
& y=1 \quad (0<x<2\pi, \, y \geq 1)
\end{aligned}
$$ | \frac{\pi}{2} + 2 | 0 | 7,846.4375 | -1 | 7,846.4375 | |
Given the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$, a line $L$ passing through the right focus $F$ of the ellipse intersects the ellipse at points $A$ and $B$, and intersects the $y$-axis at point $P$. Suppose $\overrightarrow{PA} = λ_{1} \overrightarrow{AF}$ and $\overrightarrow{PB} = λ_{2} \overrightarrow{BF}... | -\frac{50}{9} | 0.125 | 8,051.0625 | 7,064.5 | 8,192 | |
Given a rectangular yard containing two congruent isosceles right triangles in the form of flower beds and a trapezoidal remainder, with the parallel sides of the trapezoid having lengths $15$ and $25$ meters. | \frac{1}{5} | 0 | 6,818.4375 | -1 | 6,818.4375 | |
Find the number of six-digit palindromes. | 9000 | 0 | 2,115 | -1 | 2,115 | |
What is the area of the gray region, in square units, if the radius of the larger circle is four times the radius of the smaller circle and the diameter of the smaller circle is 2 units? Express your answer in terms of $\pi$. [asy]
size(150);
pair A, B;
A=(0,0);
B=(-4,1);
fill(circle(A, 8), gray(.7));
fill(circle(B, 2)... | 15\pi | 0.6875 | 4,338.5 | 3,191.818182 | 6,861.2 | |
A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\%$ longer than the minor axis, the length of the major axis is: | 10.5 | 0.125 | 3,611.5 | 2,778 | 3,730.571429 | |
Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: "The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together... | 18 | 0.8125 | 4,518.75 | 3,671.076923 | 8,192 | |
Eight students participate in a pie-eating contest. The graph shows the number of pies eaten by each participating student. Sarah ate the most pies and Tom ate the fewest. Calculate how many more pies than Tom did Sarah eat and find the average number of pies eaten by all the students.
[asy]
defaultpen(linewidth(1pt)+... | 4.5 | 0.8125 | 2,317.9375 | 2,252.538462 | 2,601.333333 | |
Find the root of the following equation to three significant digits:
$$
(\sqrt{5}-\sqrt{2})(1+x)=(\sqrt{6}-\sqrt{3})(1-x)
$$ | -0.068 | 0.3125 | 7,842.8125 | 7,074.6 | 8,192 | |
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \). | 4/15 | 0 | 8,192 | -1 | 8,192 | |
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$ . Given that $m$ and $n$ are both integers, compute $100m+n$ .
*Proposed by Evan Chen* | 39515 | 0.9375 | 3,745.0625 | 3,448.6 | 8,192 | |
If \( 3x + 4 = x + 2 \), what is the value of \( x \)? | -1 | If \( 3x + 4 = x + 2 \), then \( 3x - x = 2 - 4 \) and so \( 2x = -2 \), which gives \( x = -1 \). | 1 | 1,522.5 | 1,522.5 | -1 |
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction. | \frac{1}{6561} | 0.375 | 5,805 | 4,224 | 6,753.6 | |
What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \times 5?$ | \frac{1}{5} | 1. **Evaluate the expression inside the parentheses**:
\[
2^0 - 1 + 5^2 - 0
\]
- \(2^0 = 1\) because any non-zero number raised to the power of 0 is 1.
- \(5^2 = 25\) because squaring 5 gives 25.
- Therefore, the expression simplifies to:
\[
1 - 1 + 25 - 0 = 25
\]
2. **Apply the expon... | 1 | 1,204.1875 | 1,204.1875 | -1 |
A, B, C, and D obtained the top four positions (without ties) in the school, and they made the following statements:
A: "I am neither first nor second."
B: "My position is adjacent to C's position."
C: "I am neither second nor third."
D: "My position is adjacent to B's position."
It is known that A, B, C, and D respec... | 4213 | 0.4375 | 6,215.4375 | 3,749.714286 | 8,133.222222 | |
There are 8 keys numbered 1 to 8 and 8 boxes numbered 1 to 8. Each key can only open the box with the same number. All keys are placed in these boxes and locked up so that each box contains one key. How many different ways are there to place the keys in the boxes such that at least two boxes have to be opened to unlock... | 35280 | 0.125 | 7,912.3125 | 5,954.5 | 8,192 | |
Regarding the value of \\(\pi\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\(\pi\\) through designing the following experiment: ask \\(200\\) students, ea... | \dfrac {78}{25} | 0.0625 | 7,385.125 | 3,913 | 7,616.6 | |
In right triangle $ABC$ with $\angle B = 90^\circ$, sides $AB=1$ and $BC=3$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. Calculate the length of segment $BD$.
A) $\frac{1}{2}$
B) $\frac{3}{4}$
C) $1$
D) $\frac{5}{4}$
E) $2$ | \frac{3}{4} | 0 | 8,192 | -1 | 8,192 | |
How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | 600 | 0.125 | 7,953.6875 | 6,874.5 | 8,107.857143 | |
Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills. The total value of these bills is $\$700$. How many $\$5$ bills does Arturo have? | 20 | Since Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\$5$ bill, one $\$10$ bill, and one $\$20$ bill. The value of the bills in each group is $\$5 + \$10 + \$20 = \$35$. Since the total value of Arturo's bills i... | 1 | 1,040 | 1,040 | -1 |
Suppose that \( f(x) \) is a function defined for every real number \( x \) with \( 0 \leq x \leq 1 \) with the properties that
- \( f(1-x)=1-f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \),
- \( f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \), an... | \frac{3}{4} | 0.0625 | 7,957.625 | 5,816 | 8,100.4 | |
The angle bisectors of triangle \( A B C \) intersect at point \( I \), and the external angle bisectors of angles \( B \) and \( C \) intersect at point \( J \). The circle \( \omega_{b} \) with center at point \( O_{b} \) passes through point \( B \) and is tangent to line \( C I \) at point \( I \). The circle \( \o... | 1/3 | 0 | 8,113.9375 | -1 | 8,113.9375 | |
For all $n \geq 1$, let \[ a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}. \] Determine \[ \lim_{n \to \infty} \frac{a_n}{n^3}. \] | \frac{8}{\pi^3} | The answer is $\frac{8}{\pi^3}$.
By the double angle and sum-product identities for cosine, we have \begin{align*} 2\cos^2\left(\frac{(k-1)\pi}{2n}\right) - 2\cos^2 \left(\frac{k\pi}{2n}\right) &= \cos\left(\frac{(k-1)\pi}{n}\right) - \cos\left(\frac{k\pi}{n}\right) \\ &= 2\sin\left(\frac{(2k-1)\pi}{2n}\right) \sin\le... | 0 | 8,192 | -1 | 8,192 |
Consider the polynomial $49x^3 - 105x^2 + 63x - 10 = 0$ whose roots are in arithmetic progression. Determine the difference between the largest and smallest roots.
A) $\frac{2}{7}$
B) $\frac{1}{7}$
C) $\frac{3\sqrt{11}}{7}$
D) $\frac{2\sqrt{11}}{7}$
E) $\frac{4\sqrt{11}}{7}$ | \frac{2\sqrt{11}}{7} | 0 | 8,148.4375 | -1 | 8,148.4375 | |
Given the parametric equations of curve C as $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta \end{cases}(\theta\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\begin{cases} x'= \frac {1}{2}x \\ y'= \frac {1}{ \sqrt {3... | \frac {3 \sqrt {3}}{5} | 0 | 6,874.6875 | -1 | 6,874.6875 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, then find the measure of angle $B$. | \frac{3\pi}{10} | 0.3125 | 6,019.8125 | 4,460.2 | 6,728.727273 | |
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? | 20 | 1. **Calculate the Mean**:
The mean of the list $10, 2, 5, 2, 4, 2, x$ is calculated as follows:
\[
\text{Mean} = \frac{10 + 2 + 5 + 2 + 4 + 2 + x}{7} = \frac{25 + x}{7}.
\]
2. **Determine the Mode**:
The mode is the number that appears most frequently in the list. Here, the number $2$ appears three t... | 0.4375 | 7,318.5625 | 6,195.571429 | 8,192 |
I have 5 marbles numbered 1 through 5 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles? | 6 | 1 | 2,936.8125 | 2,936.8125 | -1 | |
A 30 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 11 feet from the base of the building. If the top of the ladder slips 6 feet, then the foot of the ladder will slide how many feet? | 9.49 | 0.0625 | 7,767.4375 | 8,192 | 7,739.133333 | |
Let set $\mathcal{A}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$. | 3501 | 0.4375 | 6,189.1875 | 5,061 | 7,066.666667 | |
Let $(b_1, b_2, ... b_{12})$ be a list of the 12 integers from 4 to 15 inclusive such that for each $2 \le i \le 12$, either $b_i + 1$ or $b_i - 1$ or both appear somewhere before $b_i$ in the list. How many such lists are there? | 2048 | 0.1875 | 8,154.3125 | 7,991 | 8,192 | |
A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | 10 | 0.9375 | 4,837.875 | 4,614.266667 | 8,192 | |
How many more digits does the base-3 representation of $987_{10}$ have than the base-8 representation of $987_{10}$? | 3 | 1 | 2,969.625 | 2,969.625 | -1 | |
How many triangles are in the figure below? [asy]
draw((0,0)--(30,0)--(30,20)--(0,20)--cycle);
draw((15,0)--(15,20));
draw((0,0)--(15,20));
draw((15,0)--(0,20));
draw((15,0)--(30,20));
draw((30,0)--(15,20));
draw((0,10)--(30,10));
draw((7.5,0)--(7.5,20));
draw((22.5,0)--(22.5,20));
[/asy] | 36 | 0 | 8,192 | -1 | 8,192 | |
The specific heat capacity of a body with mass \( m = 3 \) kg depends on the temperature in the following way: \( c = c_{0}(1 + \alpha t) \), where \( c_{0} = 200 \) J/kg·°C is the specific heat capacity at \( 0^{\circ} \mathrm{C} \), \( \alpha = 0.05 \,^{\circ} \mathrm{C}^{-1} \) is the temperature coefficient, and \(... | 112.5 | 0 | 4,712.3125 | -1 | 4,712.3125 | |
Consider a rectangle \(ABCD\) which is cut into two parts along a dashed line, resulting in two shapes that resemble the Chinese characters "凹" and "凸". Given that \(AD = 10\) cm, \(AB = 6\) cm, and \(EF = GH = 2\) cm, find the total perimeter of the two shapes formed. | 40 | 0.4375 | 6,504.5 | 5,378.142857 | 7,380.555556 | |
Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^5 - 3x^2 + 4x^4 - 2x^3 + x$ when $x=2$. | 102 | 0.8125 | 4,234.1875 | 3,320.846154 | 8,192 | |
There are five positive integers that are divisors of each number in the list $$30, 90, -15, 135, 45.$$ Find the sum of these five positive integers. | 24 | 0.125 | 8,143.3125 | 7,802.5 | 8,192 | |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2a\sin A = (2\sin B - \sqrt{3}\sin C)b + (2\sin C - \sqrt{3}\sin B)c$.
(1) Find the measure of angle $A$.
(2) If $a=2$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$. | \sqrt{3} | 0.0625 | 7,760.3125 | 8,192 | 7,731.533333 | |
Calculate $(42 \div (12 - 10 + 3))^{2} \cdot 7$. | 493.92 | 0.8125 | 304.0625 | 296.230769 | 338 | |
Given that $a+b+c=0$, calculate the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}+\frac{|ab|}{ab}+\frac{|ac|}{ac}+\frac{|bc|}{bc}+\frac{|abc|}{abc}$. | -1 | 0.5625 | 6,978.125 | 6,034 | 8,192 |
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