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 |
|---|---|---|---|---|---|---|
The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers? | 26 | 0 | 1,621.6875 | -1 | 1,621.6875 | |
Someone claims that the first five decimal digits of the square root of 51 are the same as the first five significant digits of the square root of 2. Verify this claim. Which rational approximating fraction can we derive from this observation for \(\sqrt{2}\)? How many significant digits does this fraction share with t... | \frac{99}{70} | 0 | 7,475.6875 | -1 | 7,475.6875 | |
Find the quadratic polynomial, with real coefficients, which has $3 + i$ as a root, and where the coefficient of $x^2$ is 2. | 2x^2 - 12x + 20 | 1 | 1,842.375 | 1,842.375 | -1 | |
Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse: | \frac{12\sqrt{3}-9}{13} | 1. **Identify the triangle and its properties**: We are given a right triangle $\triangle ABC$ with legs $BC = 3$ and $AC = 4$. Since it is a right triangle, we can use the Pythagorean theorem to find the hypotenuse $AB$:
\[
AB = \sqrt{BC^2 + AC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.
\]
2. **U... | 0 | 7,921.125 | -1 | 7,921.125 |
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? | \sqrt{3} | 1. **Understanding the Path**: Crystal's path consists of four segments:
- North for 1 mile.
- Northeast for 1 mile.
- Southeast for 1 mile.
- A straight line back to the starting point.
2. **Breaking Down the Northeast and Southeast Runs**:
- **Northeast Run**: Running northeast for 1 mile forms a $45^... | 0.9375 | 3,702.6875 | 3,403.4 | 8,192 |
In a table containing $A$ columns and 100 rows, natural numbers from 1 to $100 \cdot A$ are written by rows in ascending order, starting from the first row. The number 31 is in the fifth row. In which row is the number 100? | 15 | 0.875 | 4,120.125 | 3,538.428571 | 8,192 | |
The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ b... | 34 | 0.4375 | 7,786.1875 | 7,264.428571 | 8,192 | |
Given the function $y= \sqrt {x^{2}-ax+4}$, find the set of all possible values of $a$ such that the function is monotonically decreasing on the interval $[1,2]$. | \{4\} | 0 | 7,163.9375 | -1 | 7,163.9375 | |
Let \[f(x) =
\begin{cases}
x^2+2 &\text{if } x<n, \\
2x+5 &\text{if }x\ge{n}.
\end{cases}
\]If the graph $y=f(x)$ is continuous, find the sum of all possible values of $n$. | 2 | 1 | 1,581.3125 | 1,581.3125 | -1 | |
A regular pentagon \(Q_1 Q_2 \dotsb Q_5\) is drawn in the coordinate plane with \(Q_1\) at \((1,0)\) and \(Q_3\) at \((5,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_5 + y_5 i).\] | 242 | 0 | 7,837.25 | -1 | 7,837.25 | |
Given the function $f(x)=ax^{2}+bx+c(a,b,c∈R)$, if there exists a real number $a∈[1,2]$, for any $x∈[1,2]$, such that $f(x)≤slant 1$, then the maximum value of $7b+5c$ is _____. | -6 | 0.125 | 8,142.875 | 7,799 | 8,192 | |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | \sqrt{3} | 0 | 3,703.75 | -1 | 3,703.75 | |
When two fair dice are thrown once each, what is the probability that the upward-facing numbers are different and that one of them shows a 3? | \frac{5}{18} | 0.875 | 4,485.625 | 3,956.142857 | 8,192 | |
In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order? | 36000 | 0 | 7,917.875 | -1 | 7,917.875 | |
$\Delta ABC$ is isosceles with $AC = BC$. If $m\angle C = 40^{\circ}$, what is the number of degrees in $m\angle CBD$? [asy] pair A,B,C,D,E;
C = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);
draw(B--C--A--E); dot(D);
label("$A$",A,S); label("$B$",B,S); label("$D$",D,S); label("$C$",C,N);
[/asy] | 110 | 0.0625 | 7,842.625 | 4,461 | 8,068.066667 | |
In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\). | 2177 | 0.3125 | 7,913.9375 | 7,302.2 | 8,192 | |
Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? | \frac{1}{2} | 1. **Identify the radii of the semicircles and the circle**:
- Assume the radius of each semicircle $POQ$ and $ROS$ is $r$. Since these semicircles pass through the center $O$ of the circle, and each semicircle's diameter is a chord of the circle that passes through $O$, the radius of the circle $O$ must be $r$ as ... | 0.875 | 5,298.875 | 5,344.071429 | 4,982.5 |
Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)?... | 50 | 0.25 | 7,767.625 | 6,494.5 | 8,192 | |
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? | \frac{1}{14} |
To solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations and calculate the probability.
#### Step 1: Understanding th... | 0 | 8,040.125 | -1 | 8,040.125 |
For every integers $ a,b,c$ whose greatest common divisor is $n$ , if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a}
{2x \plus{} y \minus{} 2z \equal{} b}
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$ ? | 28 | 0.0625 | 7,955.9375 | 8,192 | 7,940.2 | |
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from
left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos
n^2$. Evaluate
$\lim_{n\to\infty} d_n$. | 0 | The limit is $0$; we will show this by checking that $d_n = 0$ for all $n \geq 3$.
Starting from the given matrix, add the third column to the first column; this does not change the
determinant. However, thanks to the identity
$\cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$,
the resulting matrix has the for... | 0.125 | 7,995.5 | 6,620 | 8,192 |
Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$What is the value of $b_2 - b_1$? Express your answer as a common fraction. | \frac{1}{75} | 0.75 | 4,467.9375 | 3,226.583333 | 8,192 | |
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | 13 | 0.4375 | 7,154.625 | 5,820.857143 | 8,192 | |
Distribute 5 students into three groups: A, B, and C. Group A must have at least two students, while groups B and C must have at least one student each. Determine the number of different distribution schemes. | 80 | 0.0625 | 8,076.5 | 8,192 | 8,068.8 | |
Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is | 25 | 0.0625 | 8,192 | 8,192 | 8,192 | |
How many multiples of 4 are between 70 and 300? | 57 | 0.75 | 5,574 | 4,701.333333 | 8,192 | |
Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\... | \dfrac{1}{12} | 0.875 | 2,230.75 | 1,937.571429 | 4,283 | |
Find the circulation of the vector field
$$
\vec{a}=\frac{y}{3} \vec{i} 3-3 x \vec{j}+x \vec{k}
$$
along the closed contour $\Gamma$
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t - 2 \sin t
\end{array} \quad t \in [0,2\pi]
\right.
$$ | -\frac{52 \pi}{3} | 0.0625 | 8,005.75 | 7,564 | 8,035.2 | |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existi... | \frac{1}{8} | 0.0625 | 8,112.375 | 7,581 | 8,147.8 | |
Dave's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths contained cayenne, and one-sixths contained salted soy nuts. What is the largest possible number of pies that had none of these ingredients? | 9 | 0.3125 | 6,942.5625 | 5,360.2 | 7,661.818182 | |
The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 0$ and a vertical asymptote of $ x=-1 $. If $q(x)$ is quadratic, $p(2)=1$, and $q(2) = 3$, find $p(x) + q(x).$
[asy]
size(8cm);
import graph;
Label f;
f.p=fontsize(6);
real f(real x) {return (x-1)/((x-1)*(x+1)... | x^2 + x - 2 | 0.9375 | 4,460.375 | 4,211.6 | 8,192 | |
Given a circle with a diameter of $1$ unit, if a point $P$ on the circle starts from point $A$ representing $3$ on the number line and rolls one round to the left along the number line, find the real number represented by the point $B$ where $P$ arrives on the number line. | 3 - \pi | 0.125 | 7,939.125 | 6,169 | 8,192 | |
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... | 0.1875 | 7,719.4375 | 5,965.333333 | 8,124.230769 |
What is the sum of the integers from $-30$ to $50$, inclusive? | 810 | 0.9375 | 3,880.125 | 3,592.666667 | 8,192 | |
In the Cartesian coordinate system $xOy$, the equation of circle $C$ is $(x- \sqrt {3})^{2}+(y+1)^{2}=9$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of circle $C$;
$(2)$ The line $OP$: $\theta= \frac {\pi}{6}$ ($p\i... | 2 \sqrt {6} | 0 | 8,127.8125 | -1 | 8,127.8125 | |
There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \text{cm}^2$. What is the area (in $\text{cm}^2$) of the square inscribed in the same $\triangle ABC$ as shown in Figure 2 below?
[asy] draw((0,0)--... | 392 | 0 | 8,165.625 | -1 | 8,165.625 | |
A person has $440.55$ in their wallet. They purchase goods costing $122.25$. Calculate the remaining money in the wallet. After this, calculate the amount this person would have if they received interest annually at a rate of 3% on their remaining money over a period of 1 year. | 327.85 | 0.875 | 382.5 | 391.142857 | 322 | |
Given the function \( f(x) \) such that \( f(x+4) + f(x-4) = f(x) \) for all \( x \in \mathbb{R} \), determine the common minimum period of such functions. | 24 | 0.625 | 6,927.6875 | 6,575.5 | 7,514.666667 | |
Find the minimum value of
\[(\sin x + \csc x)^2 + (\cos x + \sec x)^2\]for $0 < x < \frac{\pi}{2}.$ | 9 | 0.9375 | 5,066.9375 | 4,858.6 | 8,192 | |
How many positive four-digit integers are divisible by $8$? | 1125 | 1 | 2,993.25 | 2,993.25 | -1 | |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 0 | 8,122.4375 | -1 | 8,122.4375 | |
Simplify first, then evaluate: $-2(-x^2y+xy^2)-[-3x^2y^2+3x^2y+(3x^2y^2-3xy^2)]$, where $x=-1$, $y=2$. | -6 | 0.625 | 4,753.4375 | 4,011.3 | 5,990.333333 | |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 48 | 0 | 7,730.125 | -1 | 7,730.125 | |
There are 5 students who signed up to participate in a two-day volunteer activity at the summer district science museum. Each day, two students are needed for the activity. The probability that exactly 1 person will participate in the volunteer activity for two consecutive days is ______. | \frac{3}{5} | 0.375 | 7,308 | 5,834.666667 | 8,192 | |
Randomly select a number $x$ in the interval $[0,4]$, the probability of the event "$-1 \leqslant \log_{\frac{1}{3}}(x+ \frac{1}{2}) \leqslant 1$" occurring is ______. | \frac{3}{8} | 0.0625 | 4,592.1875 | 3,521 | 4,663.6 | |
Calculate \(3^5 \cdot 6^5\). | 1,889,568 | 0.0625 | 3,793.75 | 551 | 4,009.933333 | |
In triangle \(ABC\), \(AB = 20\), \(BC = 21\), and \(CA = 29\). Point \(M\) is on side \(AB\) with \(\frac{AM}{MB}=\frac{3}{2}\), while point \(N\) is on side \(BC\) with \(\frac{CN}{NB}=2\). Points \(P\) and \(Q\) are on side \(AC\) such that line \(MP\) is parallel to \(BC\) and line \(NQ\) is parallel to \(AB\). Sup... | 224/15 | 0.625 | 6,961.4375 | 6,495.7 | 7,737.666667 | |
Given point $O$ inside $\triangle ABC$, and $\overrightarrow{OA}+\overrightarrow{OC}+2 \overrightarrow{OB}=0$, calculate the ratio of the area of $\triangle AOC$ to the area of $\triangle ABC$. | 1:2 | 0 | 7,992.8125 | -1 | 7,992.8125 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____. | \frac{5\pi }{12} | 0.375 | 4,332.8125 | 4,747.833333 | 4,083.8 | |
Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$. | 6.125 | 0 | 2,138.0625 | -1 | 2,138.0625 | |
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach? Express your answer in terms of $\pi$. | 3\pi | 0.6875 | 4,938.8125 | 4,058.727273 | 6,875 | |
A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.
A) $1.5\pi$
B) $2.25\pi$
C) $3\pi$
D) $4.5\pi$ | 2.25\pi | 0 | 8,091.625 | -1 | 8,091.625 | |
Given triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sqrt{3}\sin B + 2\cos^2\frac{B}{2} = 3$ and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A \sin B}{6\sin C}$, find the area of the circumcircle of $\triangle ABC$. | 16\pi | 0.375 | 6,979.3125 | 5,299.833333 | 7,987 | |
In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$. | \frac{40}{23} | 0 | 7,313.1875 | -1 | 7,313.1875 | |
Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass. | 42 | 0.5 | 6,781.875 | 6,093.625 | 7,470.125 | |
How many values of $x$, $-10 < x < 50$, satisfy $\cos^2 x + 3\sin^2 x = 1.5?$ (Note: $x$ is measured in radians.) | 18 | 0 | 7,997.75 | -1 | 7,997.75 | |
What is the greater of the solutions to the equation $x^2 + 15x -54=0$? | 3 | 1 | 1,893.75 | 1,893.75 | -1 | |
Olympus Corporation has released a new version of its popular Vowel Soup. In this version, each vowel (A, E, I, O, U) appears 7 times in each bowl. How many six-letter "words" can be formed from this Olympus Vowel Soup? Note: Words do not need to be actual words in the English language. | 15625 | 0.75 | 4,144 | 3,593.833333 | 5,794.5 | |
How many times does the letter "e" occur in all problem statements in this year's HMMT February competition? | 1661 | It is possible to arrive at a good estimate using Fermi estimation. See http: //en.wikipedia.org/wiki/Fermi_problem for more details. For example, there are 76 problems on the HMMT this year. You might guess that the average number of words in a problem is approximately 40, and the average number of letters in a word i... | 0 | 6,008.375 | -1 | 6,008.375 |
Given that $| \vec{e} | = 1$ and it satisfies $| \vec{a} + \vec{e} | = | \vec{a} - 2\vec{e} |$, find the projection of vector $\vec{a}$ in the direction of $\vec{e}$. | \frac{1}{2} | 0.9375 | 2,294.6875 | 2,318.2 | 1,942 | |
Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$ . Let $D$ be the point on ray $BC$ such that $CD=6$ . Let the intersection of $AD$ and $\omega$ be $E$ . Given that $AE=7$ , find $AC^2$ .
*Proposed by Ephram Chun and Euhan Kim* | 105 | 0.0625 | 8,067.3125 | 6,197 | 8,192 | |
Given that $\tan x = -\frac{12}{5}$ and $x \in (\frac{\pi}{2}, \pi)$, find the value of $\cos(-x + \frac{3\pi}{2})$. | -\frac{12}{13} | 0.9375 | 3,519.625 | 3,616.466667 | 2,067 | |
For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$
| 7 | 0 | 7,849.75 | -1 | 7,849.75 | |
On a $123 \times 123$ board, each cell is painted either purple or blue according to the following conditions:
- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.
... | 6724 | 0.3125 | 7,547.4375 | 6,129.4 | 8,192 | |
In the trapezoid \(MPQF\), the bases are \(MF = 24\) and \(PQ = 4\). The height of the trapezoid is 5. Point \(N\) divides the side into segments \(MN\) and \(NP\) such that \(MN = 3NP\).
Find the area of triangle \(NQF\). | 22.5 | 0 | 6,859.625 | -1 | 6,859.625 | |
If $x, y, k$ are positive reals such that $$3=k^{2}\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right)$$ find the maximum possible value of $k$. | (-1+\sqrt{7})/2 | We have $3=k^{2}(x^{2} / y^{2}+y^{2} / x^{2})+k(x / y+y / x) \geq 2 k^{2}+2 k$, hence $7 \geq 4 k^{2}+4 k+1=(2 k+1)^{2}$, hence $k \leq(\sqrt{7}-1) / 2$. Obviously $k$ can assume this value, if we let $x=y=1$. | 0 | 6,742.3125 | -1 | 6,742.3125 |
Find all irreducible positive fractions which increase threefold if both the numerator and the denominator are increased by 12. | \frac{2}{9} | 0.625 | 6,393.5 | 5,412.5 | 8,028.5 | |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of
\[\frac{a + b}{abc}.\] | \frac{16}{9} | 0.625 | 6,783.25 | 5,938 | 8,192 | |
Natasha has more than $\$1$ but less than $\$10$ worth of dimes. When she puts her dimes in stacks of 3, she has 1 left over. When she puts them in stacks of 4, she has 1 left over. When she puts them in stacks of 5, she also has 1 left over. How many dimes does Natasha have? | 61 | 0.9375 | 3,194.1875 | 2,861 | 8,192 | |
Given the function $f(x) = \left( \frac{1}{3}\right)^{ax^2-4x+3}$,
$(1)$ If $a=-1$, find the intervals of monotonicity for $f(x)$;
$(2)$ If $f(x)$ has a maximum value of $3$, find the value of $a$;
$(3)$ If the range of $f(x)$ is $(0,+\infty)$, find the range of values for $a$. | \{0\} | 0 | 6,926.8125 | -1 | 6,926.8125 | |
The average of 1, 3, and \( x \) is 3. What is the value of \( x \)? | 5 | Since the average of three numbers equals 3, then their sum is \( 3 \times 3 = 9 \). Therefore, \( 1+3+x=9 \) and so \( x=9-4=5 \). | 1 | 1,240.25 | 1,240.25 | -1 |
What is the value of $((2^2-2)-(3^2-3)+(4^2-4))$ | 8 | The problem requires us to evaluate the expression \[(2^2-2)-(3^2-3)+(4^2-4).\]
1. **Calculate each term separately:**
- For the first term, \(2^2 - 2 = 4 - 2 = 2\).
- For the second term, \(3^2 - 3 = 9 - 3 = 6\).
- For the third term, \(4^2 - 4 = 16 - 4 = 12\).
2. **Substitute the values into the expression... | 1 | 814.125 | 814.125 | -1 |
A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second time. How many minutes does it take for the... | 22 | 0 | 8,137.375 | -1 | 8,137.375 | |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 0 | 6,458.75 | -1 | 6,458.75 | |
Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle{ABP} = 30^{\circ}$. Given that $AP = 3$ and $CP = 1$, compute the area of triangle $ABC$. | \frac{12}{5} | 0 | 6,194.0625 | -1 | 6,194.0625 | |
Select two distinct integers, $m$ and $n$, randomly from the set $\{3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $3mn - m - n$ is a multiple of $5$? | \frac{2}{9} | 0 | 8,192 | -1 | 8,192 | |
A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{... | \sqrt{5 - \sqrt{5}} | 0 | 7,593.5625 | -1 | 7,593.5625 | |
Let \( x, y, z, w \) be different positive real numbers such that \( x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t \). Find \( t \). | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
How many four-digit numbers can be formed using three 1s, two 2s, and five 3s? | 71 | 0.5 | 6,082.3125 | 5,178.5 | 6,986.125 | |
If $(pq)(qr)(rp) = 16$, what is a possible value for $pqr$? | 4 | Since $(pq)(qr)(rp) = 16$, then $pqqrrp = 16$ or $p^2q^2r^2 = 16$. Thus, $(pqr)^2 = 16$ and so $pqr = \pm 4$. Using the given answers, $pqr$ is positive and so $pqr = 4$. | 0.75 | 3,281.6875 | 2,811.916667 | 4,691 |
At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$? | 2 | 1. **Determine the total number of quizzes Lisa needs to score an A on to meet her goal:**
Lisa's goal is to earn an A on at least 80% of her 50 quizzes. Therefore, the number of quizzes she needs to score an A on is:
\[
0.8 \times 50 = 40
\]
quizzes.
2. **Calculate the number of quizzes she has alrea... | 1 | 1,075.3125 | 1,075.3125 | -1 |
Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves? | 960 | 0.375 | 7,059.9375 | 5,803.166667 | 7,814 | |
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set \[
\{-7,-5,-3,-2,2,4,6,13\}.
\]What is the minimum possible value of \[
(a+b+c+d)^{2} + (e+f+g+h)^{2}?
\] | 34 | 0 | 8,192 | -1 | 8,192 | |
Two people, A and B, play a "guess the number" game using a fair six-sided die (the faces of the die are numbered $1, 2, \cdots, 6$). Each person independently thinks of a number on the die, denoted as $a$ and $b$. If $|a - b| \leqslant 1$, they are said to be "in sync." What is the probability that A and B are in sync... | 4/9 | 0.9375 | 4,974.5625 | 4,760.066667 | 8,192 | |
A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has one vertex tangent to the edge of the circle at point $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle b... | 9(\sqrt{3} - \pi) | 0 | 5,979.1875 | -1 | 5,979.1875 | |
What is the ratio of the legs in a right triangle, if the triangle formed by its altitudes as sides is also a right triangle? | \sqrt{\frac{-1 + \sqrt{5}}{2}} | 0 | 7,715.625 | -1 | 7,715.625 | |
Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$. | 512 | $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$ adding both we get \[\log_8{ab} +2\log_4{ab}=12\] then we see $\log_4 {ab}$ is 3/2 times $\log_8 {ab}$ putting $\log_8 {ab}$ as x we get x+3x=12 so x=3 and $\log_8 {ab}$ = 3 so ab= $8^3$=\[\boxed{512}\] ~ math_comb01 | 0.9375 | 3,640.1875 | 3,336.733333 | 8,192 |
A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set. | 2\sqrt{2} | 0.5 | 2,797.0625 | 3,492.75 | 2,101.375 | |
Draw five lines \( l_1, l_2, \cdots, l_5 \) on a plane such that no two lines are parallel and no three lines pass through the same point.
(1) How many intersection points are there in total among these five lines? How many intersection points are there on each line? How many line segments are there among these five li... | 10 | 0.25 | 8,036.8125 | 7,571.25 | 8,192 | |
Evaluate $\log_8 2$. | \frac{1}{3} | 1 | 1,841.375 | 1,841.375 | -1 | |
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$ . | 669 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Maria wants to purchase a book which costs \$35.50. She checks her purse and discovers she has two \$20 bills, and twelve quarters, and a bunch of nickels. Determine the minimum number of nickels that Maria needs with her to buy the book. | 150 | 0.25 | 4,486.5625 | 3,957.25 | 4,663 | |
Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is "divisible by $x^{2}+1$ modulo 3", or more precisely, either of the following equivalent conditions holds: there exist polynomials $P, Q$ with integer coefficients such that $(x+1)^{n}-1=\left(x^{2}+1\right) P(x)+3 Q(x)$; or more conceptua... | 8 | We have $(x+1)^{2}=x^{2}+2 x+1 \equiv 2 x,(x+1)^{4} \equiv(2 x)^{2} \equiv-4 \equiv-1$, and $(x+1)^{8} \equiv(-1)^{2}=1$. So the order $n$ divides 8, as $x+1$ and $x^{2}+1$ are relatively prime polynomials modulo 3 (or more conceptually, in $\mathbb{F}_{3}[x]$ ), but cannot be smaller by our computations of the 2 nd an... | 0.6875 | 5,776.5 | 4,678.545455 | 8,192 |
The first and thirteenth terms of an arithmetic sequence are $\frac79$ and $\frac45$, respectively. What is the seventh term? | \frac{71}{90} | 1 | 2,752.0625 | 2,752.0625 | -1 | |
The opposite of the real number $2023$ is | -2023 | 1 | 1,321.5625 | 1,321.5625 | -1 | |
Given that 50 students are selected from 2013 students using a two-step process, first eliminating 13 students through simple random sampling and then selecting 50 from the remaining 2000 using systematic sampling, determine the probability of each person being selected. | \frac{50}{2013} | 0.4375 | 6,357.75 | 5,613.285714 | 6,936.777778 | |
Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth. | 0.5 | 0.25 | 7,336.25 | 6,681 | 7,554.666667 | |
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all... | 76 | 1. **Identify the given information:**
- Mean score of the morning class, $M = 84$.
- Mean score of the afternoon class, $A = 70$.
- Ratio of the number of students in the morning class to the afternoon class, $\frac{m}{a} = \frac{3}{4}$.
2. **Express the number of students in the morning class in terms of th... | 1 | 1,792.8125 | 1,792.8125 | -1 |
A secret facility is a rectangle measuring $200 \times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along the outer perimeter (while the intruder rema... | 150 | 0.125 | 7,670.0625 | 4,016.5 | 8,192 | |
A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box ha... | 21 | 0.0625 | 8,022.5625 | 5,481 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.