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 |
|---|---|---|---|---|---|---|
An investor put $\$12,\!000$ into a three-month savings certificate that paid a simple annual interest rate of $8\%$. After three months, she reinvested the resulting amount in another three-month certificate. The investment value after another three months was $\$12,\!435$. If the annual interest rate of the second ce... | 6.38\% | 0 | 8,038.625 | -1 | 8,038.625 | |
On a rectangular sheet of paper, a picture in the shape of a "cross" made of two rectangles $ABCD$ and $EFGH$ was drawn, with sides parallel to the edges of the sheet. It is known that $AB = 9$, $BC = 5$, $EF = 3$, $FG = 10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 0 | 6,787.25 | -1 | 6,787.25 | |
Let $O$ be the origin. There exists a scalar $m$ such that for any points $E,$ $F,$ $G,$ and $H$ satisfying the vector equation:
\[4 \overrightarrow{OE} - 3 \overrightarrow{OF} + 2 \overrightarrow{OG} + m \overrightarrow{OH} = \mathbf{0},\]
the four points $E,$ $F,$ $G,$ and $H$ are coplanar. Find the value of $m.$ | -3 | 0.6875 | 5,583.375 | 4,397.636364 | 8,192 | |
Find the center of the circle with equation $9x^2-18x+9y^2+36y+44=0.$ | (1,-2) | 0.9375 | 2,632.625 | 2,262 | 8,192 | |
Given a line $l$ with an inclination angle of $\theta$, if $\cos\theta= \frac {4}{5}$, calculate the slope of this line. | \frac{3}{4} | 1 | 2,024.75 | 2,024.75 | -1 | |
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$ | 9 | 1. **Identify the relationship between angles**: Given that $\angle B = \angle C = 4\angle A$, we can use the fact that the sum of angles in a triangle is $180^\circ$. Therefore, we write:
\[
\angle B + \angle C + \angle A = 180^\circ
\]
Substituting $\angle B = 4\angle A$ and $\angle C = 4\angle A$, we get... | 1 | 2,219.875 | 2,219.875 | -1 |
The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of $x$? | 11 | The sum of the odd numbers from 5 to 21 is $5+7+9+11+13+15+17+19+21=117$. Therefore, the sum of the numbers in any row is one-third of this total, or 39. This means as well that the sum of the numbers in any column or diagonal is also 39. Since the numbers in the middle row add to 39, then the number in the centre squa... | 0 | 8,192 | -1 | 8,192 |
A natural number is called a square if it can be written as the product of two identical numbers. For example, 9 is a square because \(9 = 3 \times 3\). The first squares are 1, 4, 9, 16, 25, ... A natural number is called a cube if it can be written as the product of three identical numbers. For example, 8 is a cube b... | 2067 | 0.375 | 6,896.375 | 5,762 | 7,577 | |
Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the v... | 182 | 0.375 | 7,663.8125 | 6,783.5 | 8,192 | |
There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat ne... | 7/13 | 0 | 8,002.125 | -1 | 8,002.125 | |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \frac{52}{221} | 0 | 6,848.75 | -1 | 6,848.75 | |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is | 13 | 0.0625 | 6,721.1875 | 7,797 | 6,649.466667 | |
In the following diagram, \(ABCD\) is a square, \(BD \parallel CE\) and \(BE = BD\). Let \(\angle E = x^{\circ}\). Find \(x\). | 30 | 0.75 | 6,320.0625 | 5,696.083333 | 8,192 | |
Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate... | \frac{n+1}{3} | \textbf{First solution:} By the linearity of expectation, the average number of local maxima is equal to the sum of the probability of having a local maximum at $k$ over $k=1,\dots, n$. For $k=1$, this probability is 1/2: given the pair $\{\pi(1), \pi(2)\}$, it is equally likely that $\pi(1)$ or $\pi(2)$ is bigger. Sim... | 0.25 | 7,454 | 5,240 | 8,192 |
What real number is equal to the expression $3 + \frac{5}{2 + \frac{5}{3 + \frac{5}{2 + \cdots}}}$, where the $2$s and $3$s alternate? | \frac{5}{3} | 0 | 6,419.75 | -1 | 6,419.75 | |
If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 16? | \frac{1}{36} | 0.6875 | 6,594.4375 | 5,868.272727 | 8,192 | |
For all values of $x$ for which it is defined, let $g(x) = \cot \frac{x}{2} - \cot 2x$. This expression can be written as
\[g(x) = \frac{\sin kx}{\sin \frac{x}{2} \sin 2x}.\]
Find the value of $k$. | \frac{3}{2} | 0.8125 | 3,901.75 | 2,911.692308 | 8,192 | |
After eating lunch together, Jay and Paul start walking in opposite directions. Jay walks 0.75 miles every 15 minutes and Paul power walks 2.5 miles every 30 minutes. In miles, how far apart are they after 1.5 hours? | 12 | 1 | 1,449.375 | 1,449.375 | -1 | |
Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form... | \frac{7}{18} | 0.125 | 7,697.6875 | 6,145.5 | 7,919.428571 | |
A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?
[asy]
fill((0,0)--(21,0)--(21,21)--(0,21)--cycle,black);
fill((9,4)--(9,13)--(18,13)... | 40 | 1 | 2,094.5 | 2,094.5 | -1 | |
When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S. | 375 | 0.4375 | 5,335.5 | 3,801.714286 | 6,528.444444 | |
Let \( p, q, r, s, t, u \) be positive real numbers such that \( p + q + r + s + t + u = 10 \). Find the minimum value of
\[ \frac{1}{p} + \frac{9}{q} + \frac{4}{r} + \frac{16}{s} + \frac{25}{t} + \frac{36}{u}. \] | 44.1 | 0 | 4,947.0625 | -1 | 4,947.0625 | |
A chessboard has its squares labeled according to the rule $\frac{1}{c_i + r_j}$, where $c_i$ is the column number and $r_j$ is the row number. Eight squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of these eight chosen squares. | \frac{8}{9} | 0.1875 | 7,862.25 | 6,433.333333 | 8,192 | |
Given that $a+b=3$ and $a^3+b^3=81$, find $ab$. | -6 | 1 | 2,016.5625 | 2,016.5625 | -1 | |
If $z$ is a complex number such that
\[
z + z^{-1} = 2\sqrt{2},
\]
what is the value of
\[
z^{100} + z^{-100} \, ?
\] | -2 | 0 | 8,192 | -1 | 8,192 | |
The numbers \(a, b, c, d\) belong to the interval \([-6.5 ; 6.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 182 | 0.1875 | 7,768 | 5,930.666667 | 8,192 | |
Simplify $(3-2i)^2$. (Your answer should be of the form $a+bi$.) | 5-12i | 1 | 1,758.3125 | 1,758.3125 | -1 | |
(1) How many natural numbers are there that are less than 5000 and have no repeated digits?
(2) From the digits 1 to 9, five digits are selected each time to form a 5-digit number without repeated digits.
(i) If only odd digits can be placed in odd positions, how many such 5-digit numbers can be formed?
(ii) If o... | 1800 | 0 | 7,768.4375 | -1 | 7,768.4375 | |
Find the number of 3-digit positive integers whose digits multiply to 30. | 12 | 0.625 | 6,955.75 | 6,474.1 | 7,758.5 | |
Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. | -2 | 1 | 4,848.1875 | 4,848.1875 | -1 | |
An isosceles trapezoid has sides labeled as follows: \(AB = 25\) units, \(BC = 12\) units, \(CD = 11\) units, and \(DA = 12\) units. Compute the length of the diagonal \( AC \). | \sqrt{419} | 0.8125 | 5,105.375 | 4,885 | 6,060.333333 | |
Find the phase shift of the graph of $y = 2 \sin \left( 2x + \frac{\pi}{3} \right).$ | -\frac{\pi}{6} | 1 | 3,194.5 | 3,194.5 | -1 | |
Calculate the arc length of the curve defined by the equation in the rectangular coordinate system.
\[ y = \ln 7 - \ln x, \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \frac{1}{2} \ln \frac{3}{2} | 0.0625 | 6,760.1875 | 6,417 | 6,783.066667 | |
The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one? | 162 | 0 | 8,118 | -1 | 8,118 | |
What is the nearest integer to $(3 + 2)^6$? | 9794 | 0 | 2,685.8125 | -1 | 2,685.8125 | |
In a rectangular prism $A^{\prime}C$, with $AB=5$, $BC=4$, and $B^{\prime}B=6$, $E$ is the midpoint of $AA^{\prime}$. Find the distance between the skew lines $BE$ and $A^{\prime}C^{\prime}$. | \frac{60}{\sqrt{769}} | 0 | 7,872 | -1 | 7,872 | |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are denoted as $S_n$ and $T_n$ respectively. If for any natural number $n$, it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, calculate the value of $\dfrac{a_3+a_{15}}{2(b_3+b_9)}+ \dfrac{a_3}{b_2+b_{10}}$. | \dfrac{19}{41} | 0.4375 | 5,638.5 | 4,893.285714 | 6,218.111111 | |
A certain lottery has tickets labeled with the numbers $1,2,3, \ldots, 1000$. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ... | 1/501 | Notice that the outcome is the same as if the lottery instead draws all the tickets, in random order, and awards a prize to the holder of the odd ticket drawn earliest and each even ticket drawn before it. Thus, the probability of your winning is the probability that, in a random ordering of the tickets, your ticket pr... | 0.0625 | 7,144.0625 | 6,821 | 7,165.6 |
What is the total number of digits used when the first 2500 positive even integers are written? | 9448 | 0.6875 | 6,244.9375 | 5,571.545455 | 7,726.4 | |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 | 1. **Given Information and Assumptions**:
- Convex quadrilateral $ABCD$ with $AB = 9$, $CD = 12$, and diagonals $AC = 14$.
- Diagonals $AC$ and $BD$ intersect at $E$.
- $\triangle AED$ and $\triangle BEC$ have equal areas.
2. **Using Equal Areas to Infer Similarity**:
- Since $\triangle AED$ and $\triangle... | 0.375 | 7,046.5 | 5,214.166667 | 8,145.9 |
From post office $A$, a car leaves heading towards post office $B$. After 20 minutes, a motorcyclist departs in pursuit of the car, traveling at a speed of 60 km/h. Upon catching up with the car, the motorcyclist delivers a package to the driver's cab and immediately turns back. The car reaches $B$ at the moment when t... | 45 | 0.125 | 7,696.1875 | 4,762 | 8,115.357143 | |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( ... | 1/2 | 0 | 8,192 | -1 | 8,192 | |
The sum of Jim's weight and Bob's weight is 180 pounds. If you subtract Jim's weight from Bob's weight, you get half of Bob's weight. How many pounds does Bob weigh? | 120 | 1 | 1,853.4375 | 1,853.4375 | -1 | |
Given that the function $f(x) = \frac{1}{2}\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) is an even function, and points P and Q are the highest and lowest points respectively on the graph of $y = f(x)$ such that $| \overrightarrow{PQ} | = \sqrt{2}$,
(1) Find the explicit formula of the function $f(x)$;... | \frac{\pi}{12} | 0 | 7,542.875 | -1 | 7,542.875 | |
In trapezoid \(ABCD\), the sides \(AB\) and \(CD\) are parallel and \(CD = 2AB\). Points \(P\) and \(Q\) are chosen on sides \(AD\) and \(BC\), respectively, such that \(DP : PA = 2\) and \(BQ : QC = 3 : 4\). Find the ratio of the areas of quadrilaterals \(ABQP\) and \(CDPQ\). | 19/44 | 0.0625 | 7,570.6875 | 6,711 | 7,628 | |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | -80 | 1 | 2,788.1875 | 2,788.1875 | -1 | |
(Elective 4-4: Coordinate Systems and Parametric Equations)
In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta =4... | \sqrt{3}+2 | 0 | 6,495.25 | -1 | 6,495.25 | |
Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\circ}$ and $85^{\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\circ}$ more than twice $C$. Find the measure of the largest angle. | 205^\circ | 1 | 2,247.75 | 2,247.75 | -1 | |
Two real numbers are selected independently at random from the interval $[-15, 15]$. The product of those numbers is considered only if both numbers are outside the interval $[-5, 5]$. What is the probability that the product of those numbers, when considered, is greater than zero? | \frac{2}{9} | 0.125 | 5,976.5625 | 7,761.5 | 5,721.571429 | |
The probability that a light bulb lasts more than 1000 hours is 0.2. Determine the probability that 1 out of 3 light bulbs fails after 1000 hours of use. | 0.096 | 0.875 | 3,204.875 | 2,803.642857 | 6,013.5 | |
What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$? | 75 | 1 | 1,816.1875 | 1,816.1875 | -1 | |
Given vectors $\overrightarrow {a}=(\sqrt{3}\sin x,\sin x)$ and $\overrightarrow {b}=(\cos x,\sin x)$, where $x \in (0, \frac{\pi}{2})$,
1. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x$;
2. Let the function $f(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{b}$,... | \frac{5}{2} | 0.9375 | 4,221.125 | 3,956.4 | 8,192 | |
Let \( a_{n} = 1 + 2 + \cdots + n \), where \( n \in \mathbf{N}_{+} \), and \( S_{m} = a_{1} + a_{2} + \cdots + a_{m} \), \( m = 1, 2, \cdots, m \). Find the number of values among \( S_{1}, S_{2}, \cdots, S_{2017} \) that are divisible by 2 but not by 4. | 252 | 0.0625 | 7,968.5625 | 6,943 | 8,036.933333 | |
If 25$\%$ of a number is the same as 20$\%$ of 30, what is the number? | 24 | 1 | 1,094.4375 | 1,094.4375 | -1 | |
What is the greatest common factor of all two-digit palindromes? (Note: A palindrome is a number that reads the same forwards as backwards.) | 11 | 1 | 1,878.4375 | 1,878.4375 | -1 | |
Solve
\[\sqrt{1 + \sqrt{2 + \sqrt{x}}} = \sqrt[3]{1 + \sqrt{x}}.\] | 49 | 0.8125 | 4,614.0625 | 3,788.384615 | 8,192 | |
On the board, the product of the numbers $\overline{\text{IKS}}$ and $\overline{\text{KSI}}$ is written, where the letters correspond to different non-zero decimal digits. This product is a six-digit number and ends with S. Vasya erased all the zeros from the board, after which only IKS remained. What was written on th... | 100602 | 0 | 8,192 | -1 | 8,192 | |
Given points O(0,0) and A(1,1), the slope angle of line OA is $\boxed{\text{answer}}$. | \frac{\pi}{4} | 0.125 | 1,165.5 | 1,267.5 | 1,150.928571 | |
There is a unique quadruple of positive integers $(a, b, c, k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^{4}-20 x^{3}+108 x^{2}-k x+9$. Compute $c$. | 7 | The four roots are $a \pm \sqrt{b \pm \sqrt{c}}$, so the sum of roots is 20 , so $a=5$. Next, we compute the sum of squares of roots: $$(a+\sqrt{b \pm \sqrt{c}})^{2}+(a-\sqrt{b \pm \sqrt{c}})^{2}=2 a^{2}+2 b \pm 2 \sqrt{c}$$ so the sum of squares of roots is $4 a^{2}+4 b$. However, from Vieta, it is $20^{2}-2 \cdot 108... | 0.3125 | 7,339.3125 | 5,463.4 | 8,192 |
In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equati... | \frac{3 \sqrt{5}}{10} | 0 | 8,017.6875 | -1 | 8,017.6875 | |
Let $Z$ be as in problem 15. Let $X$ be the greatest integer such that $|X Z| \leq 5$. Find $X$. | 2 | Problems 13-15 go together. See below. | 0.3125 | 5,824.1875 | 4,249.2 | 6,540.090909 |
We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone? | 8\pi | 0.1875 | 7,478.875 | 5,476 | 7,941.076923 | |
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors? | 44 | 0 | 8,192 | -1 | 8,192 | |
In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona is sequentially taking out one apple at a time from the basket. If at any point the number of green apples she has taken out is less than the number of yellow apples, and the number of yellow apples is less than the number of red apples, then she... | 39 | 0 | 8,173.125 | -1 | 8,173.125 | |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the higher total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5,... | 13 | 0.25 | 7,038.125 | 5,505.5 | 7,549 | |
Point $A$ is at $(0, 0)$ and point $B$ is on the line $y = 4$. The slope of segment $AB$ is $\frac{2}{3}$. What is the sum of the $x$- and $y$-coordinates of point $B$? | 10 | 1 | 1,409.4375 | 1,409.4375 | -1 | |
Given a trapezoid \(ABCD\). A point \(M\) is chosen on its lateral side \(CD\) such that \( \frac{CM}{MD} = \frac{4}{3} \). It turns out that segment \( BM \) divides the diagonal \( AC \) into two segments, the ratio of the lengths of which is also \( \frac{4}{3} \). What possible values can the ratio \( \frac{AD}{BC}... | 7/12 | 0.0625 | 7,922.9375 | 5,735 | 8,068.8 | |
Jason is trying to remember the five digit combination to his safe. He knows that he only used digits 1 through 5 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try? | 180 | 0.8125 | 4,084.875 | 3,137.076923 | 8,192 | |
Let $z$ be a complex number such that $|z| = 3.$ Find the largest possible distance between $(1 + 2i)z^3$ and $z^4$ when plotted in the complex plane. | 216 | 0 | 6,975.3125 | -1 | 6,975.3125 | |
How many positive integers $n \leq 2009$ have the property that $\left\lfloor\log _{2}(n)\right\rfloor$ is odd? | 682 | We wish to find $n$ such that there is some natural number $k$ for which $2 k-1 \leq \log _{2} n<$ $2 k$. Since $n \leq 2009$ we must have $k \leq 5$. This is equivalent to finding the number of positive integers $n \leq 2009$ satisfying $2^{2 k-1} \leq n<2^{2 k}$ for some $k \leq 5$, so the number of such integers is ... | 0.75 | 6,289.5 | 5,934.833333 | 7,353.5 |
Given that \(a\), \(b\), \(c\), and \(d\) are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers, find the smallest possible value of \(a + b + c + d\). Note that the four numbers \(a\), \(b\), \(c\), and \(d\) are not necessarily distin... | 28 | 0.0625 | 8,079.5625 | 6,393 | 8,192 | |
A coin is altered so that the probability that it lands on heads is less than $\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\frac{1}{6}$. What is the probability that the coin lands on heads? | \frac{3-\sqrt{3}}{6} | 1. **Define the probability of heads**: Let $x$ be the probability that the coin lands on heads. Consequently, the probability that the coin lands on tails is $1 - x$.
2. **Set up the probability equation for 2 heads and 2 tails in 4 flips**: The number of ways to choose 2 heads out of 4 flips is given by the binomial... | 0 | 3,082.6875 | -1 | 3,082.6875 |
Comprehensive exploration: When two algebraic expressions containing square roots are multiplied together and the product does not contain square roots, we call these two expressions rationalizing factors of each other. For example, $\sqrt{2}+1$ and $\sqrt{2}-1$, $2\sqrt{3}+3\sqrt{5}$ and $2\sqrt{3}-3\sqrt{5}$ are all ... | 1011 | 0.25 | 4,826.3125 | 3,857.75 | 5,149.166667 | |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 37 | 0.875 | 4,266.1875 | 3,705.357143 | 8,192 | |
In an opaque bag, there are three balls, each labeled with the numbers $-1$, $0$, and $\frac{1}{3}$, respectively. These balls are identical except for the numbers on them. Now, a ball is randomly drawn from the bag, and the number on it is denoted as $m$. After putting the ball back and mixing them, another ball is dr... | \frac{5}{9} | 0.25 | 6,651 | 4,951.5 | 7,217.5 | |
Given $6$ cards labeled $1$, $2$, $3$, $4$, $5$, and $6$ are drawn without replacement, calculate the probability that the product of the numbers of the $2$ cards is a multiple of $4$. | \frac{2}{5} | 0.5625 | 6,767.5625 | 5,659.666667 | 8,192 | |
Given the function $f(x)=\cos (2x- \frac {π}{6})\sin 2x- \frac{1}{4}(x∈R)$
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) Find the maximum and minimum values of the function $f(x)$ on $\[- \frac {π}{4},0\]$. | -\frac {1}{2} | 0.8125 | 6,335.5625 | 6,216.076923 | 6,853.333333 | |
The denominator of a fraction is 7 less than 3 times the numerator. If the fraction is equivalent to $2/5$, what is the numerator of the fraction? | 14 | 1 | 1,599.6875 | 1,599.6875 | -1 | |
Find all three-digit numbers \( \overline{\mathrm{MGU}} \) consisting of distinct digits \( M, \Gamma, \) and \( U \) for which the equality \( \overline{\mathrm{MGU}} = (M + \Gamma + U) \times (M + \Gamma + U - 2) \) holds. | 195 | 0.3125 | 8,021.375 | 7,646 | 8,192 | |
Given that the polar coordinate equation of curve $C$ is $ρ=2\cos θ$, and the polar coordinate equation of line $l$ is $ρ\sin (θ+ \frac {π}{6})=m$. If line $l$ and curve $C$ have exactly one common point, find the value of the real number $m$. | \frac{3}{2} | 0.3125 | 7,693.1875 | 7,898.8 | 7,599.727273 | |
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | 6 | To find the number of ways $345$ can be expressed as the sum of two or more consecutive positive integers, we start by considering the general form of such a sequence. Let the sequence start with $n$ and have $k$ terms. The sum of the sequence can be expressed as:
\[ n + (n+1) + (n+2) + \ldots + (n+k-1) \]
This sum ca... | 0.0625 | 6,353.6875 | 7,074 | 6,305.666667 |
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | \frac{63}{400} | 1 | 3,160.6875 | 3,160.6875 | -1 | |
A function $f(x, y, z)$ is linear in $x, y$, and $z$ such that $f(x, y, z)=\frac{1}{x y z}$ for $x, y, z \in\{3,4\}$. What is $f(5,5,5)$? | \frac{1}{216} | We use a similar method to the previous problem. Notice that $f(x, y, 5)=2 f(x, y, 4)-f(x, y, 3)$. Let $f_{2}$ denote the function from the previous problem and $f_{3}$ the function from this problem. Since $3 f_{3}(x, y, 3)$ is linear in $x$ and $y$, and $3 f_{3}(x, y, 3)=\frac{1}{x y}$ for all $x, y \in\{3,4\}$, the ... | 0 | 8,192 | -1 | 8,192 |
What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$? | 26 | 0 | 2,273.375 | -1 | 2,273.375 | |
Factor $(x^2 + 3x + 2)(x^2 + 7x + 12) + (x^2 + 5x - 6)$ as the product of two non-constant polynomials. | (x^2 + 5x + 2)(x^2 + 5x + 9) | 0.875 | 4,099.375 | 3,867.571429 | 5,722 | |
Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 25$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $200$ square units, what is the area of triangle $FYG$? | 46.875 | 0 | 8,192 | -1 | 8,192 | |
Lily is taking a 30-question, multiple-choice Biology quiz. Each question offers four possible answers. Lily guesses on the last six questions. What is the probability that she will get at least two of these last six questions wrong? | \frac{4077}{4096} | 0.1875 | 5,989.0625 | 4,195 | 6,403.076923 | |
Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\triangle PF_{1}F_{2}$ is $30^{\circ}$, then the eccentricity of $C$ is ______. | \sqrt{3} | 0.625 | 6,743.3125 | 5,924.7 | 8,107.666667 | |
A student reads a book, reading 35 pages on the first day and then 5 more pages each subsequent day, until only 35 pages are left on the last day. The second time he reads it, he reads 45 pages on the first day and then 5 more pages each subsequent day, until only 40 pages are left on the last day. How many pages does ... | 385 | 0.0625 | 5,956.5 | 1,902 | 6,226.8 | |
Given that $A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$ | 40 | Though the problem may appear to be quite daunting, it is actually not that difficult. $\frac {k(k-1)}2$ always evaluates to an integer (triangular number), and the cosine of $n\pi$ where $n \in \mathbb{Z}$ is 1 if $n$ is even and -1 if $n$ is odd. $\frac {k(k-1)}2$ will be even if $4|k$ or $4|k-1$, and odd otherwise.
... | 0.0625 | 7,875.4375 | 3,127 | 8,192 |
How many integers are there from 1 to 1,000,000 that are neither perfect squares, nor perfect cubes, nor fourth powers? | 998910 | 0.375 | 7,414.1875 | 6,795.666667 | 7,785.3 | |
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. ... | 051 | Denote by $N_{i,j}$ the optimal expected number of cards that Alice guesses correctly, where the number of cards are $i$ and $j \ge i.$
If $i = 0$ then Alice guesses correctly all cards, so $N_{0,j} = j.$
If $j = i$ then Alice guesses next card with probability $\frac {1}{2} \implies N_{i,i} = \frac {1}{2} + N_{i-1,i... | 0 | 7,902.5625 | -1 | 7,902.5625 |
A group consists of 4 male students and 3 female students. From this group, 4 people are selected to complete three different tasks, with the condition that at least two of the selected individuals must be female, and each task must have at least one person assigned to it. The number of different ways to select and ass... | 792 | 0.625 | 5,252.5 | 4,772 | 6,053.333333 | |
The quotient of two positive integers is $\frac{5}{2}$ and their product is 160. What is the value of the larger of the two integers? | 20 | 1 | 1,989.0625 | 1,989.0625 | -1 | |
What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$ | 18 | 1. **Convert the given equation to a simpler form by completing the square:**
\[
|x^2 - 12x + 34| = 2
\]
Completing the square for the quadratic expression:
\[
x^2 - 12x + 34 = (x^2 - 12x + 36) - 2 = (x-6)^2 - 2
\]
Thus, the equation becomes:
\[
|(x-6)^2 - 2| = 2
\]
2. **Solve the abso... | 0.875 | 2,924 | 2,492.571429 | 5,944 |
The numbers \( a, b, c, d \) belong to the interval \([-5, 5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 110 | 0.125 | 7,686.9375 | 6,022 | 7,924.785714 | |
An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?
[asy]
import markers; defaultpen(linewidth(0.8));
pair A,B,C,D,E,F,G;
draw(unitcircle)... | 75^\circ | 0 | 8,192 | -1 | 8,192 | |
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(... | 130 | Let $QP = PR = x$. Extend the line containing the centers of the two circles to meet $R$, and to meet the other side of the large circle at a point $S$.
The part of this line from $R$ to the point nearest to $R$ where it intersects the larger circle has length $6+(12-8)=10$. The length of the diameter of the larger ci... | 0.25 | 7,667.4375 | 6,837.75 | 7,944 |
Jeff's five assignment scores are 89, 92, 88, 95 and 91. What is the arithmetic mean of these five scores? | 91 | 1 | 2,110.3125 | 2,110.3125 | -1 | |
In triangle $\triangle JKL$ shown, $\tan K = \frac{3}{2}$. What is $KL$?
[asy]
pair J,K,L;
L = (0,0);
J = (0,3);
K = (2,3);
draw(L--J--K--L);
draw(rightanglemark(L,J,K,7));
label("$L$",L,SW);
label("$J$",J,NW);
label("$K$",K,NE);
label("$2$",(J+K)/2,N);
[/asy] | \sqrt{13} | 0.3125 | 777.5625 | 852.4 | 743.545455 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.