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 |
|---|---|---|---|---|---|---|
Find the sum: $(-39) + (-37) + \cdots + (-1)$. | -400 | 1 | 2,596.125 | 2,596.125 | -1 | |
From the set of numbers \( 1, 2, 3, 4, \cdots, 1982 \), remove some numbers so that in the remaining numbers, no number is equal to the product of any two other numbers. What is the minimum number of numbers that need to be removed to achieve this? How can this be done? | 43 | 0 | 7,968.9375 | -1 | 7,968.9375 | |
Three concentric circles have radii $5$ meters, $15$ meters, and $25$ meters respectively. Calculate the total distance a beetle travels, which starts at a point $P$ on the outer circle, moves inward along a radius to the middle circle, traces a one-third arc of the middle circle, then travels radially to the inner cir... | 15\pi + 70 | 0 | 4,331.8125 | -1 | 4,331.8125 | |
A fair die is rolled six times. The probability of rolling at least a five at least five times is | \frac{13}{729} | 1. **Determine the probability of rolling at least a five on a single roll**:
A fair die has six faces, and rolling at least a five corresponds to rolling either a five or a six. There are 2 favorable outcomes (5 and 6) out of 6 possible outcomes. Thus, the probability of rolling at least a five on one roll is:
... | 0.4375 | 6,388.5625 | 4,069.857143 | 8,192 |
Suppose you have two bank cards for making purchases: a debit card and a credit card. Today you decided to buy airline tickets worth 20,000 rubles. If you pay for the purchase with the credit card (the credit limit allows it), you will have to repay the bank within $\mathrm{N}$ days to stay within the grace period in w... | 31 | 0 | 8,029.625 | -1 | 8,029.625 | |
There are two rows of seats, with 6 seats in the front row and 7 seats in the back row. Arrange seating for 2 people in such a way that these 2 people cannot sit next to each other. Determine the number of different seating arrangements. | 134 | 0 | 8,192 | -1 | 8,192 | |
In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge between the losing teams on Saturday to de... | 16 | 0.25 | 7,130.875 | 7,035 | 7,162.833333 | |
On a $9 \times 9$ chessboard, determine the total number of rectangles and the number of those rectangles that are squares. Then, find the simplified fraction of the ratio of the number of squares to the total number of rectangles and compute $m + n$ where the fraction is expressed as $\frac{m}{n}$ with $m$ and $n$ as ... | 154 | 0.9375 | 2,554.25 | 2,464.133333 | 3,906 | |
How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10? | 1845 | 0.25 | 7,846.875 | 7,153.25 | 8,078.083333 | |
For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. | 125 | Since the problem works for all positive integers $a$, let's plug in $a=2$ and see what we get. Since $\sigma({2^n}) = 2^{n+1}-1,$ we have $2^{n+1} \equiv 2 \pmod{2021}.$ Simplifying using CRT and Fermat's Little Theorem, we get that $2^n \equiv 0 \pmod{42}$ and $2^n \equiv 0 \pmod{46}.$ Then, we can look at $a=2022$ j... | 0.0625 | 6,949.4375 | 6,627 | 6,970.933333 |
Mike and Harry play a game on an 8x8 board. Mike chooses \( k \) squares and writes an \( M \) in each of them. Harry then chooses \( k+1 \) squares and writes an \( H \) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming " \( H M M \) " or " \( M M H \) ", when read either horizo... | 16 | 0.0625 | 8,122.3125 | 8,192 | 8,117.666667 | |
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls? | $\dfrac{10}{21}$ | 0 | 7,955 | -1 | 7,955 | |
Say that a complex number $z$ is three-presentable if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$? | \frac{80}{9}\pi | 0.8125 | 4,454.25 | 3,918.846154 | 6,774.333333 | |
If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, the value of $(p + q)r$ is: | 15 | 1. **Define the polynomials and their roots:**
Let $f(x) = x^3 + 3x^2 + 9x + 3$ and $g(x) = x^4 + 4x^3 + 6px^2 + 4qx + r$. Assume $f(x)$ has roots $r_1, r_2, r_3$. Since $f(x)$ divides $g(x)$, $g(x)$ must have roots $r_1, r_2, r_3$, and an additional root $r_4$.
2. **Apply Vieta's formulas to $f(x)$:**
- The sum... | 0.9375 | 3,361.4375 | 3,039.4 | 8,192 |
The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{3}}{2} | 0 | 3,973.25 | -1 | 3,973.25 | |
Robert reads 90 pages per hour. How many 270-page books can he read in six hours? | 2 | 1 | 337.875 | 337.875 | -1 | |
The expression $a^3-a^{-3}$ equals: | \left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) | 1. **Identify the form of the expression**: We are given the expression $a^3 - a^{-3}$ and need to simplify it. We recognize that this expression can be rewritten using the difference of cubes formula.
2. **Recall the difference of cubes formula**: The formula for the difference of cubes is:
\[
x^3 - y^3 = (x - ... | 0.125 | 6,987.125 | 6,234.5 | 7,094.642857 |
Evaluate the sum $$\frac{1}{2\lfloor\sqrt{1}\rfloor+1}+\frac{1}{2\lfloor\sqrt{2}\rfloor+1}+\frac{1}{2\lfloor\sqrt{3}\rfloor+1}+\cdots+\frac{1}{2\lfloor\sqrt{100}\rfloor+1}$$ | 190/21 | The first three terms all equal $1 / 3$, then the next five all equal $1 / 5$; more generally, for each $a=1,2, \ldots, 9$, the terms $1 /(2\lfloor\sqrt{a^{2}}\rfloor+1)$ to $1 /(2\lfloor\sqrt{a^{2}+2 a}\rfloor+1)$ all equal $1 /(2 a+1)$, and there are $2 a+1$ such terms. Thus our terms can be arranged into 9 groups, e... | 0.6875 | 6,009.1875 | 5,624.909091 | 6,854.6 |
Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$. | 5 \sqrt{2} | 0.6875 | 6,084.8125 | 5,787.090909 | 6,739.8 | |
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$ , with $A$ closer to $B$ than $C$ , such that $2016 \cdot AB = BC$ . Line $XY$ intersects line $AC$ ... | 2017 | 0 | 8,065.5625 | -1 | 8,065.5625 | |
A certain tour group checked the weather conditions on the day of the outing. A weather forecasting software predicted that the probability of rain during the time periods $12:00$ to $13:00$ and $13:00$ to $14:00$ on the day of the outing are $0.5$ and $0.4$ respectively. Then, the probability of rain during the time p... | 0.7 | 0.75 | 4,203.375 | 3,959.416667 | 4,935.25 | |
Given $x+x^{-1}=3$, calculate the value of $x^{ \frac {3}{2}}+x^{- \frac {3}{2}}$. | \sqrt{5} | 0 | 5,295.3125 | -1 | 5,295.3125 | |
Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let $p$ be the probability that the triangle formed by the chosen vertices is acute. Compute $10 p$. | 5 | The only way for the three vertices to form an acute triangle is if they consist of two adjacent vertices and the vertex opposite their side. Since there are 5 ways to choose this and $\binom{5}{3}=10$ ways to choose the three vertices, we have $p=\frac{5}{10}=\frac{1}{2}$. | 0.0625 | 7,691.25 | 4,908 | 7,876.8 |
For a bijective function $g : R \to R$ , we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$ , where $g^{-1}$ is the inverse of $g$ . Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$ . | 10 | 0 | 5,327.3125 | -1 | 5,327.3125 | |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with a focal distance of $2\sqrt{3}$, the line $l_1: y = kx$ ($k \neq 0$) intersects the ellipse at points A and B. A line $l_2$ passing through point B with a slope of $\frac{1}{4}k$ intersects the ellipse at another point D, and $AD \perp AB$.
1. ... | \frac{9}{8} | 0.0625 | 8,152.3125 | 7,557 | 8,192 | |
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labe... | c < \sqrt{2} |
To solve this problem, we need to determine all real numbers \( c > 0 \) such that there exists a labeling of the lattice points \( (x, y) \in \mathbf{Z}^2 \) with positive integers while satisfying the given conditions:
- Only finitely many distinct labels occur.
- For each label \( i \), the distance between any two... | 0 | 8,192 | -1 | 8,192 |
Let $P(z)=z^3+az^2+bz+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$. | 136 | Set $w=x+yi$, so $x_1 = x+(y+3)i$, $x_2 = x+(y+9)i$, $x_3 = 2x-4+2yi$.
Since $a,b,c\in{R}$, the imaginary part of $a,b,c$ must be $0$.
Start with a, since it's the easiest one to do: $y+3+y+9+2y=0, y=-3$,
and therefore: $x_1 = x$, $x_2 = x+6i$, $x_3 = 2x-4-6i$.
Now, do the part where the imaginary part of c is 0 si... | 0.4375 | 6,992.8125 | 6,042.142857 | 7,732.222222 |
Compute $\dbinom{505}{505}$. | 1 | 1 | 1,269.4375 | 1,269.4375 | -1 | |
Find all real solutions to $x^4+(2-x)^4=34$. Enter all the solutions, separated by commas. | 1 + \sqrt{2}, 1 - \sqrt{2} | 0 | 4,424.1875 | -1 | 4,424.1875 | |
The number of inches in the perimeter of a square is equal to the number of square inches in its area. Find the length, in inches, of a side of the square. | 4 | 1 | 1,378.5 | 1,378.5 | -1 | |
A regular tetrahedron is inscribed in a cylinder such that two opposite edges of the tetrahedron are the diameters of the cylinder's bases. Find the ratio of the volume of the cylinder to the volume of the tetrahedron. | \frac{3 \pi}{2} | 0.3125 | 7,669.75 | 6,581 | 8,164.636364 | |
Given $e^{i \theta} = \frac{3 + i \sqrt{8}}{5}$, find $\cos 4 \theta$. | -\frac{287}{625} | 0 | 8,011.6875 | -1 | 8,011.6875 | |
Two people, A and B, alternately pick distinct numbers from the set \(\{0, 1, \cdots, 81\}\). A picks first, and each person picks one number per turn. After all 82 numbers are picked, let \(S_A\) and \(S_B\) be the sums of the numbers chosen by A and B, respectively. During the selection process, A wants to maximize t... | 41 | 0.1875 | 7,946.1875 | 6,881 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $45^\circ$ between them, and $|\overrightarrow{a}|=1$, $|2\overrightarrow{a}-\overrightarrow{b}|=\sqrt{10}$, find the magnitude of vector $\overrightarrow{b}$. | 3\sqrt{2} | 1 | 2,842.3125 | 2,842.3125 | -1 | |
What fraction of $\frac{1}{2}$ is $\frac{1}{7}$? | \frac{2}{7} | 1 | 2,014.5 | 2,014.5 | -1 | |
How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? | 1002 | To solve this problem, we need to find the number of integers between $1$ and $2005$ that are multiples of $3$ or $4$ but not multiples of $12$. We will use the principle of inclusion-exclusion and then subtract the multiples of $12$.
1. **Count multiples of $3$:**
The multiples of $3$ from $1$ to $2005$ are given ... | 0 | 3,973.4375 | -1 | 3,973.4375 |
Calculate both the product and the sum of the least common multiple (LCM) and the greatest common divisor (GCD) of $12$ and $15$. | 63 | 0.625 | 1,268.125 | 1,652.8 | 627 | |
Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+2n}$. | \frac{3}{4} | We know that $\frac{1}{n^{2}+2n}=\frac{1}{n(n+2)}=\frac{\frac{1}{n}-\frac{1}{n+2}}{2}$. So, if we sum this from 1 to $\infty$, all terms except for $\frac{1}{2}+\frac{\frac{1}{2}}{2}$ will cancel out (a 'telescoping' series). Therefore, the sum will be $\frac{3}{4}$. | 0.875 | 4,745.875 | 4,253.571429 | 8,192 |
Let the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin(2x) + 2a$.
(1) Find the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{4}]$, the minimum value of $f(x)$ is 0. Find the maximum value of $f(x)$. | \frac{1}{2} | 0.4375 | 6,880.0625 | 6,844.142857 | 6,908 | |
A classroom has 10 chairs arranged in a row. Tom and Jerry choose their seats at random, but they are not allowed to sit on the first and last chairs. What is the probability that they don't sit next to each other? | \frac{3}{4} | 0.5 | 6,146.5 | 4,687.75 | 7,605.25 | |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
\[\textbf{(A)} \ 0 \qquad\textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad\textbf{(D)} \ 3\qquad \textbf{(E)} \ \text{More than three, but finite}\]
| 1 | 0 | 6,686.5 | -1 | 6,686.5 | |
The first $20$ numbers of an arrangement are shown below. What would be the value of the $40^{\mathrm{th}}$ number if the arrangement were continued?
$\bullet$ Row 1: $2,$ $2$
$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$
$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$
$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$ | 12 | 0.75 | 3,480.5625 | 3,375.75 | 3,795 | |
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$? | 10 | 1 | 1,798.5 | 1,798.5 | -1 | |
How many real \( x \) satisfy the equation \( x = \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{5} \right\rfloor \)? | 30 | 0 | 8,146.375 | -1 | 8,146.375 | |
Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$? | 1200 | 0.6875 | 4,894.3125 | 3,875.090909 | 7,136.6 | |
Let
\[\mathbf{A} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.\]Compute $\mathbf{A}^{100}.$ | \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} | 0.8125 | 3,486.0625 | 3,411 | 3,811.333333 | |
Solve for $x$:
\[\arcsin 3x - \arccos (2x) = \frac{\pi}{6}.\] | -\frac{1}{\sqrt{7}} | 0 | 7,824.5 | -1 | 7,824.5 | |
A jar contains $29\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number. | 14\frac{6}{7} | 1 | 2,488.75 | 2,488.75 | -1 | |
The expression $3y^2-y-24$ can be written as $(3y + a)(y + b),$ where $a$ and $b$ are integers. What is $a - b$? | 11 | 1 | 3,016.25 | 3,016.25 | -1 | |
Find the least positive integer $x$ that satisfies $x+4609 \equiv 2104 \pmod{12}$. | 3 | 1 | 2,237.1875 | 2,237.1875 | -1 | |
Given the function $f(x) = \begin{cases} \log_{10} x, & x > 0 \\ x^{-2}, & x < 0 \end{cases}$, if $f(x\_0) = 1$, find the value of $x\_0$. | 10 | 0.3125 | 6,679.5625 | 6,674.8 | 6,681.727273 | |
Suppose $A>B>0$ and A is $x$% greater than $B$. What is $x$? | 100\left(\frac{A-B}{B}\right) | 1. **Understanding the Problem Statement:**
Given that $A > B > 0$ and $A$ is $x\%$ greater than $B$, we need to find the expression for $x$.
2. **Expressing $A$ in terms of $B$ and $x$:**
Since $A$ is $x\%$ greater than $B$, we can write:
\[
A = B + \frac{x}{100} \cdot B
\]
Simplifying, we get:
\... | 0.3125 | 2,363.6875 | 1,957 | 2,548.545455 |
The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: | -\frac{1}{3} | To solve the equation $\frac{64^{x-1}}{4^{x-1}} = 256^{2x}$, we start by expressing all terms with base 2:
1. **Expressing each term with base 2:**
- $64 = 2^6$, so $64^{x-1} = (2^6)^{x-1} = 2^{6(x-1)}$
- $4 = 2^2$, so $4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)}$
- $256 = 2^8$, so $256^{2x} = (2^8)^{2x} = 2^{16x}$
2... | 1 | 2,400.3125 | 2,400.3125 | -1 |
A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y=2$. Then $x$ is | 1 | 1. **Identify the Triangle and its Properties**:
The problem involves a triangle formed by the x-axis, y-axis, and the line $x+y=2$. The vertices of this triangle are at the origin $O(0,0)$, the y-intercept $A(0,2)$, and the x-intercept $B(2,0)$.
2. **Understanding the Incenter**:
The point that is equidistant f... | 0 | 8,142.5 | -1 | 8,142.5 |
Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___. | -1 | 0.625 | 7,150.75 | 6,931.8 | 7,515.666667 | |
Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points? | 16 | 0 | 8,145.25 | -1 | 8,145.25 | |
What is the largest three-digit integer $n$ that satisfies $$55n\equiv 165\pmod{260}~?$$ | 991 | 0.9375 | 4,466.4375 | 4,218.066667 | 8,192 | |
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest point on the wall is 400 meters, and from ... | 1803 | 0.3125 | 7,174.0625 | 4,934.6 | 8,192 | |
Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$ . Compute the area of region $R$ . Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$ . | 4 - 2\sqrt{2} | 0.4375 | 7,464.625 | 6,850.857143 | 7,942 | |
Let \begin{align*}
f(x) &= 2x + 5 \\
g(x) &= \sqrt{f(x)} - 2 \\
h(x) &= f(g(x))
\end{align*} What is $h(2)$? | 7 | 1 | 1,676.125 | 1,676.125 | -1 | |
The number of students in Carlos' graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Carlos' graduating class? | 182 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. Ahead of Peter, there were 50 people; ahead of Nicholas, there were 100; ahead of Denis, there were 170. When the command "About-face!" was given, everyone turned to face the oppo... | 211 | 0.1875 | 7,751.3125 | 5,894 | 8,179.923077 | |
The store bought a batch of New Year cards at 0.21 yuan each and sold them for a total of 14.57 yuan. If each card was sold at the same price and did not exceed twice the purchase price, how much profit did the store make? | 4.7 | 0 | 617.75 | -1 | 617.75 | |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 39 | 0.25 | 7,583.1875 | 6,508.5 | 7,941.416667 | |
Given the real numbers $a$, $b$, $c$, $d$ that satisfy $$\frac {a-2e^{a}}{b}= \frac {2-c}{d-1}=1$$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^2+(b-d)^2$. | \frac{25}{2} | 0.9375 | 5,810.5 | 5,651.733333 | 8,192 | |
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is | \frac{2}{x+1} | Let's consider a string of unit length (length = 1) for simplicity. We cut the string at a point $C$ chosen uniformly at random along the string. Let $C$ be the length of the shorter piece after the cut, and $1 - C$ be the length of the longer piece. We need to find the probability that the longer piece is at least $x$... | 0.375 | 7,702 | 6,885.333333 | 8,192 |
One bag contains 2 red marbles and 2 blue marbles. A second bag contains 2 red marbles, 2 blue marbles, and $g$ green marbles, with $g>0$. For each bag, Maria calculates the probability of randomly drawing two marbles of the same colour in two draws from that bag, without replacement. If these two probabilities are equ... | 5 | First, we consider the first bag, which contains a total of $2+2=4$ marbles. There are 4 possible marbles that can be drawn first, leaving 3 possible marbles that can be drawn second. This gives a total of $4 \times 3=12$ ways of drawing two marbles. For both marbles to be red, there are 2 possible marbles (either red ... | 1 | 2,634.375 | 2,634.375 | -1 |
In triangle $ABC$, points $A$, $B$, and $C$ are located such that $AB = 35$ units, $BC = 40$ units, and $CA = 45$ units. A point $X$ lies on side $AB$ such that $CX$ bisects $\angle ACB$. Given that $BX = 21$ units, find the length of segment $AX$. | 14 | 0 | 7,961.9375 | -1 | 7,961.9375 | |
Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average? | \frac{43}{14} | Note that there are 13 prime numbers amongst the cards. We may view these as separating the remaining 29 cards into 14 groups of nonprimes - those appearing before the first prime, between the first and second, etc. Each of these groups is equally likely to appear first, so 29/14 nonprimes are removed on average. We ar... | 0 | 8,192 | -1 | 8,192 |
In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$ .
 | 1 + \sqrt{2} | 0 | 8,029.375 | -1 | 8,029.375 | |
Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 40^\circ$, and $\angle ADC = 140^\circ$. What is the degree measure of $\angle
BAD$? | 50^{\circ} | 0.625 | 6,575.4375 | 5,605.5 | 8,192 | |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{3}{5}$ and a minor axis length of $8$,
(1) Find the standard equation of the ellipse $C$;
(2) Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$, respectively. A line $l$ passing through $F... | \frac{5}{3} | 0 | 8,192 | -1 | 8,192 | |
A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.
How many additional rods will the farmer need? | 21 | 0.1875 | 4,297.5625 | 4,161 | 4,329.076923 | |
Given a parallelogram with an acute angle of \(60^{\circ}\). Find the ratio of the sides of the parallelogram if the ratio of the squares of the diagonals is \(\frac{1}{3}\). | 1:1 | 0.4375 | 7,482.25 | 6,569.714286 | 8,192 | |
In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to th... | 36 | 0 | 8,192 | -1 | 8,192 | |
The sequence $(x_n)$ is defined by $x_1 = 25$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dotsb.\] | \frac{1}{25} | 0.25 | 6,944.8125 | 3,203.25 | 8,192 | |
If $2010a + 2014b = 2018$ and $2012a + 2016b = 2020$, what is the value of $a - b$ ? | -3 | 0.8125 | 5,474.75 | 4,847.692308 | 8,192 | |
The coefficient sum of the expansion of the binomial ${{\left(\frac{1}{x}-2x^2\right)}^9}$, excluding the constant term, is $671$. | 671 | 0.875 | 3,899.0625 | 3,285.785714 | 8,192 | |
How many nonempty subsets of $\{1,2,3, \ldots, 12\}$ have the property that the sum of the largest element and the smallest element is 13? | 1365 | If $a$ is the smallest element of such a set, then $13-a$ is the largest element, and for the remaining elements we may choose any (or none) of the $12-2 a$ elements $a+1, a+2, \ldots,(13-a)-1$. Thus there are $2^{12-2 a}$ such sets whose smallest element is $a$. Also, $13-a \geq a$ clearly implies $a<7$. Summing over ... | 0.1875 | 7,720.1875 | 5,988.333333 | 8,119.846154 |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 0.125 | 7,680.375 | 6,876.5 | 7,795.214286 | |
How many distinct numbers can you get by multiplying two or more distinct members of the set $\{1, 2, 3, 7, 13\}$, or taking any number to the power of another (excluding power 1) from the same set? | 23 | 0.125 | 8,152.75 | 8,191 | 8,147.285714 | |
Given two distinct geometric progressions with first terms both equal to 1, and the sum of their common ratios equal to 3, find the sum of the fifth terms of these progressions if the sum of the sixth terms is 573. If the answer is ambiguous, provide the sum of all possible values of the required quantity. | 161 | 0.1875 | 4,758.5625 | 5,227.333333 | 4,650.384615 | |
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$? | 52.5 | 1. **Identify the properties of the triangles**: Given that $AB = BC = CD = CE$, we can conclude that $\triangle ABC$ and $\triangle CDE$ are both isosceles. Additionally, it is given that $\angle A = \frac{5}{2} \angle B$.
2. **Analyze $\triangle ABC$**:
- Since $\triangle ABC$ is isosceles with $AB = BC$, we have... | 0.0625 | 7,995.75 | 7,600 | 8,022.133333 |
Given the side lengths of four squares with side lengths \( 3\sqrt{2} \) , \( 4\sqrt{2} \) , \( 5\sqrt{2} \) , and \( 6\sqrt{2} \) units, determine the area of the fifth square with an unknown side length. | 36 | 0 | 5,715.4375 | -1 | 5,715.4375 | |
How many whole numbers are between $\sqrt[3]{10}$ and $\sqrt[3]{200}$? | 3 | 0.875 | 6,066.6875 | 5,763.071429 | 8,192 | |
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
| 108 | 0.0625 | 8,181.75 | 8,028 | 8,192 | |
Let \mathcal{V} be the volume enclosed by the graph $x^{2016}+y^{2016}+z^{2}=2016$. Find \mathcal{V} rounded to the nearest multiple of ten. | 360 | Let $R$ be the region in question. Then we have $$[-1,1]^{2} \times[-\sqrt{2014}, \sqrt{2014}] \subset R \subset[-\sqrt[2016]{2016}, \sqrt[2016]{2016}]^{2} \times[\sqrt{2016}, \sqrt{2016}]$$ We find some bounds: we have $$\sqrt{2016}<\sqrt{2025}=45$$ By concavity of $\sqrt{\cdot}$, we have the bound $$\sqrt{2014} \leq ... | 0 | 8,192 | -1 | 8,192 |
Given that Lucas's odometer showed 27372 miles, which is a palindrome, and 3 hours later it showed another palindrome, calculate Lucas's average speed, in miles per hour, during this 3-hour period. | 33.33 | 0 | 972.8125 | -1 | 972.8125 | |
A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is: | 3r^2 | To find the area of a regular dodecagon inscribed in a circle, we can break the dodecagon into 12 congruent isosceles triangles, each having a vertex at the center of the circle.
1. **Calculate the central angle of each triangle:**
A regular dodecagon has 12 sides, so the central angle for each of the 12 segments (... | 1 | 3,249.125 | 3,249.125 | -1 |
Given an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\sqrt{3}$, and the ellipse passes through the point $(\sqrt{3}, \frac{1}{2})$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) Through point P$(-2, 0)$, draw two lines with slopes $k_1$ and $k_2$ respectively. These tw... | \frac{1}{4} | 0.4375 | 7,096.25 | 5,917.571429 | 8,013 | |
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$. | 45^\circ |
Given a square \(ABCD\) with side length \(1\), points \(P\) and \(Q\) are on sides \(AB\) and \(AD\) respectively. We are to find the angle \( \angle PCQ \) given that the perimeter of \( \triangle APQ \) is \(2\).
Let \( AP = x \) and \( AQ = y \). Then, \( PB = 1 - x \) and \( QD = 1 - y \). We need to find \( \ta... | 0.625 | 6,296.3125 | 5,158.9 | 8,192 |
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]
for all integers $x$. | $\left | a \right | = \left | b \right |$ |
We are tasked with finding all pairs of integers \((a, b)\) such that there exist functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) and \( g: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying the conditions:
\[ f(g(x)) = x + a \quad \text{and} \quad g(f(x)) = x + b \]
for all integers \( x \).
To solve this proble... | 0 | 7,324.375 | -1 | 7,324.375 |
We define $|\begin{array}{l}{a}&{c}\\{b}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{3}\\{2}&{4}\end{array}|=1\times 4-2\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \lt |\begin{array}{l}{2}&{y}\\{x}&{3}\end{array}| \lt 3$, then the minimum value of $x+y$ is ____. | -5 | 0.75 | 3,018.875 | 3,015.75 | 3,028.25 | |
The side length of the regular hexagon is 10 cm. What is the number of square centimeters in the area of the shaded region? Express your answer in simplest radical form.
[asy]
size(100);
pair A,B,C,D,E,F;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300);
fill(B--C--E--F--cycle,heavycya... | 100\sqrt{3} | 0.875 | 5,189.5 | 4,760.571429 | 8,192 | |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | 600 | 1 | 1,349.25 | 1,349.25 | -1 | |
Find the area of triangle $ABC$ below.
[asy]
unitsize(1inch);
pair A, B, C;
A = (0,0);
B= (sqrt(2),0);
C = (0,sqrt(2));
draw (A--B--C--A, linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$8$",(B+C)/2,NE);
label("$45^\circ$",(0,0.7),E);
[/asy] | 32 | 0 | 3,333.6875 | -1 | 3,333.6875 | |
In $\triangle ABC$, it is known that $\cos A= \frac{1}{7}$, $\cos (A-B)= \frac{13}{14}$, and $0 < B < A < \frac{\pi}{2}$. Find the measure of angle $B$. | \frac{\pi}{3} | 0.75 | 5,199.1875 | 4,915.833333 | 6,049.25 | |
Given any two positive real numbers $x$ and $y$, then $x \, \Diamond \, y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \, \Diamond \, y$ satisfies the equations $(xy) \, \Diamond \, y=x(y \, \Diamond \, y)$ and $(x \, \Diamond \, 1) \, \Diamond \, x = x \, \Di... | 19 | 0.625 | 5,846.4375 | 4,439.1 | 8,192 | |
Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\] | 1.381 | 0 | 7,986.125 | -1 | 7,986.125 | |
Let \( n = 2^{25} 3^{17} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | 424 | 0.0625 | 6,830.875 | 6,635 | 6,843.933333 |
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