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 |
|---|---|---|---|---|---|---|
Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$:
(1) If $a = -8$, find the value of $\tan C$;
(2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$. | \frac{4}{3} | 0.4375 | 5,103.25 | 4,827.428571 | 5,317.777778 | |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | 2 | 1 | 1,586.75 | 1,586.75 | -1 | |
Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$ . Let $A$ , $B$ , and $C$ be random points chosen independently from $S$ , with each of the 81 points being equally likely. (The points $A$ , $B$ , and $C$ do not have to be different.) Let $K$ be ... | \frac{200}{3} | 0.0625 | 7,804.375 | 8,138 | 7,782.133333 | |
The area of the large square \(ABCD\) in the diagram is 1, and the other points are the midpoints of the sides.
Question: What is the area of the shaded triangle? | \frac{3}{32} | 0 | 7,827.9375 | -1 | 7,827.9375 | |
In Mr. Jacob's music class, 18 of the 27 students participated in the annual school musical. Mr. Jacob's ratio of participating to not participating students is applied to Mr. Steve's class for the regional music competition. If Mr. Steve has 45 students in total, how many students from Mr. Steve's class are expected t... | 30 | 0.75 | 503.4375 | 553.5 | 353.25 | |
Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method) | -3-\sqrt{14} | 1 | 2,667.3125 | 2,667.3125 | -1 | |
In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at point $Q$ ($A$ is between $Q$ and $B$). Find the ratio $QA:AB$. | 2:1 | 0.5 | 5,652.9375 | 4,355.75 | 6,950.125 | |
Captain Zarnin of Planet Hvan has four job openings for his battle station: Assistant Engineer, Weapons Maintenance, Field Technician, and Radio Specialist. Zarnin has received 24 resumes and found himself less than thrilled about half of them and does not hire them. The rest, he thinks, could fill any of the open post... | 11,\!880 | 0.25 | 821.375 | 571.5 | 904.666667 | |
How many integers $n$ satisfy the inequality $-\frac{9\pi}{2} \leq n \leq 12\pi$? | 53 | 0 | 2,767.375 | -1 | 2,767.375 | |
Consider the polynomial $Q(x) = 3x^3 + dx^2 + ex + f$. The polynomial has the property that the reciprocal of the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. Find the value of $e$. | -16 | 0.75 | 2,928.6875 | 2,069.5 | 5,506.25 | |
Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? | 16 | 0.625 | 574 | 549.1 | 615.5 | |
A positive integer \( A \) divided by \( 3! \) gives a result where the number of factors is \(\frac{1}{3}\) of the original number of factors. What is the smallest such \( A \)? | 12 | 0.5625 | 7,188.6875 | 6,408.333333 | 8,192 | |
Evaluate $\log_464$. | 3 | 1 | 1,684.9375 | 1,684.9375 | -1 | |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | \frac{5\pi}{8} | 0.25 | 7,395.5 | 5,871.75 | 7,903.416667 | |
The function $f$ satisfies $f(x)+f(2 x+y)+5 x y=f(3 x-y)+2 x^{2}+1$ for all real numbers $x, y$. Determine the value of $f(10)$. | -49 | Setting $x=10$ and $y=5$ gives $f(10)+f(25)+250=f(25)+200+1$, from which we get $f(10)=-49$. Remark: By setting $y=\frac{x}{2}$, we see that the function is $f(x)=-\frac{1}{2} x^{2}+1$, and it can be checked that this function indeed satisfies the given equation. | 1 | 3,219 | 3,219 | -1 |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$ | 196 | Define $x_i = 2y_i - 1$. Then $2\left(\sum_{i = 1}^4 y_i\right) - 4 = 98$, so $\sum_{i = 1}^4 y_i = 51$.
So we want to find four natural numbers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50;... | 1 | 3,056.9375 | 3,056.9375 | -1 |
Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO. | \frac{5}{16} | 0 | 7,367.8125 | -1 | 7,367.8125 | |
Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position. | 504 | 0 | 7,535.1875 | -1 | 7,535.1875 | |
Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB =... | 21 | 0.3125 | 7,562.125 | 6,176.4 | 8,192 | |
Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\] | 4044 | 0.1875 | 7,116.5625 | 6,166.333333 | 7,335.846154 | |
How many odd integers are there between $ rac{17}{4}$ and $ rac{35}{2}$? | 7 | We note that $ rac{17}{4}=4 rac{1}{4}$ and $ rac{35}{2}=17 rac{1}{2}$. Therefore, the integers between these two numbers are the integers from 5 to 17, inclusive. The odd integers in this range are $5,7,9,11,13,15$, and 17, of which there are 7. | 1 | 2,678.75 | 2,678.75 | -1 |
Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers cont... | 71 | 0.3125 | 8,040.625 | 7,707.6 | 8,192 | |
Find the greatest prime that divides $$ 1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2. $$ | 11 | 0.8125 | 5,797 | 5,244.307692 | 8,192 | |
There are three candidates standing for one position as student president and 130 students are voting. Sally has 24 votes so far, while Katie has 29 and Alan has 37. How many more votes does Alan need to be certain he will finish with the most votes? | 17 | 0.1875 | 1,849.25 | 2,432 | 1,714.769231 | |
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = $ | .3 | 1. **Simplify each fraction**:
- The fraction $\frac{1}{10}$ is already in its simplest form.
- The fraction $\frac{2}{20}$ can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 2. Thus, $\frac{2}{20} = \frac{1}{10}$.
- Similarly, the fraction $\frac{3}{30}... | 0 | 2,773.5 | -1 | 2,773.5 |
Find the product of all positive integral values of $m$ such that $m^2 - 40m + 399 = q$ for some prime number $q$. Note that there is at least one such $m$. | 396 | 0.875 | 5,231.6875 | 4,808.785714 | 8,192 | |
Exhibit a $13$ -digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$ s and $9$ s. | 8888888888888 | 0 | 8,192 | -1 | 8,192 | |
Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$ , $b$ , $c$ . Define $Q(x)$ to be the monic cubic polynomial with roots $ab$ , $bc$ , $ca$ . Compute $|Q(1)|$ .
*Proposed by Nathan Xiong* | 75 | 0.875 | 4,168.875 | 3,594.142857 | 8,192 | |
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai +bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \begin{bmatrix} 6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \... | (-1)^{\lceil n/2 \rceil-1} 2 \lceil \frac{n}{2} \rceil | The determinant equals $(-1)^{\lceil n/2 \rceil-1} 2 \lceil \frac{n}{2} \rceil$. To begin with, we read off the following features of $S$. \begin{itemize} \item $S$ is symmetric: $S_{ij} = S_{ji}$ for all $i,j$, corresponding to $(a,b) \mapsto (b,a)$). \item $S_{11} = n+1$, corresponding to $(a,b) = (0,n),(1,n-1),\dots... | 0 | 8,192 | -1 | 8,192 |
Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parallel to the sides of the rectangle they touc... | 648 | 0 | 6,430.5 | -1 | 6,430.5 | |
In the Cartesian coordinate system $xOy$, it is known that the circle $C: x^{2} + y^{2} + 8x - m + 1 = 0$ intersects with the line $x + \sqrt{2}y + 1 = 0$ at points $A$ and $B$. If $\triangle ABC$ is an equilateral triangle, then the value of the real number $m$ is. | -11 | 0.6875 | 6,350.6875 | 5,513.727273 | 8,192 | |
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. Calculate the total number of minutes Walter spent on the bus. | 60 | 0.875 | 3,282.5 | 2,961.928571 | 5,526.5 | |
Suppose $A$ is a set with $n$ elements, and $k$ is a divisor of $n$. Find the number of consistent $k$-configurations of $A$ of order 1. | \[
\frac{n!}{(n / k)!(k!)^{n / k}}
\] | Given such a \( k \)-configuration, we can write out all the elements of one of the \( k \)-element subsets, then all the elements of another subset, and so forth, eventually obtaining an ordering of all \( n \) elements of \( A \). Conversely, given any ordering of the elements of \( A \), we can construct a consisten... | 0 | 5,981.8125 | -1 | 5,981.8125 |
Given that point $P$ is a moving point on the circle $x^2+y^2=18$, and $PQ \perp x$-axis at point $Q$, if the moving point $M$ satisfies $\overrightarrow{OM}=\frac{1}{3}\overrightarrow{OP}+\frac{2}{3}\overrightarrow{OQ}$.
(Ⅰ) Find the equation of the trajectory $C$ of the moving point $M$;
(Ⅱ) The line passing throug... | \frac{\sqrt{2}}{3} | 0 | 6,903.0625 | -1 | 6,903.0625 | |
Player A and Player B play a number guessing game. First, Player A thinks of a number denoted as $a$, then Player B guesses the number that Player A is thinking of, and denotes this guessed number as $b$. Both $a$ and $b$ belong to the set $\{1,2,3,4,5,6\}$. If $|a-b| \leqslant 1$, it is said that "Player A and Player ... | \frac{4}{9} | 1 | 3,565.875 | 3,565.875 | -1 | |
A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. What is the $y$-intercept of the line? | 2\sqrt{2} | 0.4375 | 6,875.75 | 5,183.428571 | 8,192 | |
How many two-digit prime numbers can be formed by choosing two different digits from the set $\{2, 7, 8, 9\}$ to be used as the tens digit and units digit? | 4 | 0.8125 | 4,275.5 | 3,371.692308 | 8,192 | |
The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island and that among any 7 islands, there are always two that are connected by a bridge. What is the maximum possible value of $N$? | 36 | 0.5 | 5,947.1875 | 4,234 | 7,660.375 | |
The perimeter of a rectangle is 24 inches. What is the number of square inches in the maximum possible area for this rectangle? | 36 | 1 | 1,924.5 | 1,924.5 | -1 | |
Natural numbers are written in sequence on the blackboard, skipping over any perfect squares. The sequence looks like this:
$$
2,3,5,6,7,8,10,11, \cdots
$$
The first number is 2, the fourth number is 6, the eighth number is 11, and so on. Following this pattern, what is the 1992nd number written on the blackboard?
(Hig... | 2036 | 0.0625 | 6,006.6875 | 8,192 | 5,861 | |
Let $a,$ $b,$ $c,$ $d$ be positive integers such that
\[\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 18 & 12 \\ -20 & -13 \end{pmatrix}.\]Find the smallest possible value of $a + b + c + d.$ | 16 | 0.875 | 4,641.0625 | 4,133.785714 | 8,192 | |
Find the value of $1006 \sin \frac{\pi}{1006}$. Approximating directly by $\pi=3.1415 \ldots$ is worth only 3 points. | 3.1415875473 | The answer is $1006 \sin \frac{\pi}{1006}$. Using the third-degree Taylor polynomial for sin we can approximate $\sin x \approx x-\frac{x^{3}}{6}$. This gives an answer of 3.1415875473 worth full points. If during the calculation we use the approximation $\pi^{3} \approx 30$, this gives an answer worth 9 points. | 0 | 8,192 | -1 | 8,192 |
In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, three edges with vertex $A$ as an endpoint are all of length $2$, and their angles with each other are all $60^{\circ}$. Determine the cosine value of the angle between the line $BD_{1}$ and the line $AC$. | \frac{\sqrt{6}}{6} | 0 | 5,941.6875 | -1 | 5,941.6875 | |
Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the ... | 10 | 0.5625 | 6,526.25 | 5,230.666667 | 8,192 | |
A two-inch cube ($2\times2\times2$) of silver weighs 3 pounds and is worth $\$200$. How much is a three-inch cube of silver worth? Round your answer to the nearest dollar. | \$675 | 0.9375 | 2,365.5 | 1,977.066667 | 8,192 | |
Given that the two distinct square roots of a positive number $a$ are $2x-2$ and $6-3x$, find the cube root of $a$. | \sqrt[3]{36} | 0.9375 | 4,205.125 | 3,939.333333 | 8,192 | |
Solve for $x$ and $y$ given:
1. $\frac{4x - 2}{5x - 5} = \frac{3}{4}$
2. $x + y = 3$ | 10 | 0.9375 | 1,352.75 | 1,265.4 | 2,663 | |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 1$. | \frac1{216} | 1 | 1,976.1875 | 1,976.1875 | -1 | |
The probability that the distance between any two points selected from the four vertices and the center of a square is not less than the side length of the square is $\boxed{\text{answer}}$. | \frac{3}{5} | 0.875 | 5,780.0625 | 5,435.5 | 8,192 | |
Let $q(x) = x^{2007} + x^{2006} + \cdots + x + 1$, and let $s(x)$ be the polynomial remainder when $q(x)$ is divided by $x^3 + 2x^2 + x + 1$. Find the remainder when $|s(2007)|$ is divided by 1000. | 49 | 0 | 8,192 | -1 | 8,192 | |
If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation | r^2+r+1=0 | 1. **Identify the terms of the geometric progression**: Given that $x(y-z)$, $y(z-x)$, and $z(x-y)$ form a geometric progression, we denote these terms as $a$, $ar$, and $ar^2$ respectively. Thus, we have:
\[ a = x(y-z), \quad ar = y(z-x), \quad ar^2 = z(x-y). \]
2. **Sum the terms of the geometric progression**: A... | 0.0625 | 8,120.875 | 7,054 | 8,192 |
What is the smallest number that can be written as a sum of $2$ squares in $3$ ways? | 325 | 0.25 | 6,848.1875 | 6,538.75 | 6,951.333333 | |
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]{2}+\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.) | x^{3}-3x^{2}+9x-9 | Note that $(1-\sqrt[3]{2}+\sqrt[3]{4})(1+\sqrt[3]{2})=3$, so $1-\sqrt[3]{2}+\sqrt[3]{4}=\frac{3}{1+\sqrt[3]{2}}$. Now, if $f(x)=x^{3}-2$, we have $f(\sqrt[3]{2})=0$, so if we let $g(x)=f(x-1)=(x-1)^{3}-2=x^{3}-3x^{2}+3x-3$, then $g(1+\sqrt[3]{2})=f(\sqrt[3]{2})=0$. Finally, we let $h(x)=g\left(\frac{3}{x}\right)=\frac{... | 0 | 6,993.5625 | -1 | 6,993.5625 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is
\[
\begin{cases}
x=4t^2 \\
y=4t
\end{cases}
\]
where $t$ is the parameter. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit length. The pol... | 16 | 0.8125 | 5,880 | 5,346.461538 | 8,192 | |
Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the sam... | 48 | 0.75 | 4,295.25 | 2,996.333333 | 8,192 | |
A geometric sequence of positive integers is formed for which the first term is 3 and the fourth term is 192. What is the third term of the sequence? | 48 | 1 | 1,102.75 | 1,102.75 | -1 | |
Eight consecutive three-digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of these eight integers? | 13 | 0.1875 | 7,666.0625 | 5,387 | 8,192 | |
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$ | 63 | 0.1875 | 7,999.25 | 7,164 | 8,192 | |
Given the function $f(x)= \begin{cases} ax^{2}-2x-1, & x\geqslant 0\\ x^{2}+bx+c, & x < 0\end{cases}$ is an even function, and the line $y=t$ intersects the graph of $y=f(x)$ from left to right at four distinct points $A$, $B$, $C$, $D$. If $AB=BC$, then the value of the real number $t$ is \_\_\_\_\_\_. | - \dfrac {7}{4} | 0.5 | 6,823.4375 | 5,590.25 | 8,056.625 | |
Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$? | \frac{35}{132} | 0.8125 | 5,586.375 | 4,985.076923 | 8,192 | |
Compute $-8\cdot 4-(-6\cdot -3)+(-10\cdot -5)$. | 0 | 0.9375 | 4,289.25 | 4,029.066667 | 8,192 | |
Let $C(k)$ denotes the sum of all different prime divisors of a positive integer $k$. For example, $C(1)=0$, $C(2)=2, C(45)=8$. Find all positive integers $n$ such that $C(2^{n}+1)=C(n)$ | n=3 | Let $P(t)$ be the largest prime divisor of a positive integer $t>1$. Let $m$ be the largest odd divisor of $n: n=2^{k} m$. Then $2^{n}+1=2^{2^{k} m}+1=a^{m}+1$, where $a=2^{2^{k}}$. If $k>0$, that is, $n$ is even, then $C(n)=C(m)+2$ and $C(2^{n}+1)=C(a^{m}+1)$. We need the following two lemmas. Lemma 1. For every prime... | 0 | 8,192 | -1 | 8,192 |
Find the smallest \( n > 4 \) for which we can find a graph on \( n \) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them. | 16 | 0.0625 | 7,865.5 | 6,123 | 7,981.666667 | |
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients. | 136 | 1 | 2,825.1875 | 2,825.1875 | -1 | |
The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$? | 1120 | 0 | 8,073.3125 | -1 | 8,073.3125 | |
In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the ... | \frac{7}{92} | 0.25 | 7,704.1875 | 7,067 | 7,916.583333 | |
The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\) | 3999999 | 0.1875 | 7,412.25 | 6,585 | 7,603.153846 | |
What is the perimeter of the shaded region in a \( 3 \times 3 \) grid where some \( 1 \times 1 \) squares are shaded? | 10 | The top, left and bottom unit squares each contribute 3 sides of length 1 to the perimeter. The remaining square contributes 1 side of length 1 to the perimeter. Therefore, the perimeter is \( 3 \times 3 + 1 \times 1 = 10 \). | 0.0625 | 6,148.6875 | 5,905 | 6,164.933333 |
Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$ | 375 | As noted in the previous solutions, $a+b+c = 0$. Let $a = a_1+a_2 i$, $b = b_1+b_2 i$, $c = c_1+c_2 i$ and we have $a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = 0$. Then the given $|a|^2 + |b|^2 + |c|^2 = 250$ translates to $\sum_{} ( {a_1}^2 + {a_2}^2 ) = 250.$ Note that in a right triangle, the sum of the squares of the three... | 0.1875 | 7,631.4375 | 6,078 | 7,989.923077 |
Two fair six-sided dice are rolled. What is the probability that their sum is at least 10? | \frac{1}{6} | There are $3,2,1$ outcomes with sum $10,11,12$, so the probability is $\frac{3+2+1}{6^{2}}=\frac{1}{6}$. | 0.875 | 4,169.5625 | 3,594.928571 | 8,192 |
$S$ is a set of complex numbers such that if $u, v \in S$, then $u v \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most 1 is finite. What is the largest possible value of $N$ ? | 13 | First, if $S$ contained some $u \neq 0$ with absolute value $<1$, then (by the first condition) every power of $u$ would be in $S$, and $S$ would contain infinitely many different numbers of absolute value $<1$. This is a contradiction. Now suppose $S$ contains some number $u$ of absolute value 1 and argument $\theta$.... | 0 | 8,192 | -1 | 8,192 |
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
| 125 | 1 | 2,663.375 | 2,663.375 | -1 | |
The factory's planned output value for this year is $a$ million yuan, which is a 10% increase from last year. If the actual output value this year can exceed the plan by 1%, calculate the increase in the actual output value compared to last year. | 11.1\% | 0.125 | 6,025.5625 | 551.5 | 6,807.571429 | |
Given the vector $$\overrightarrow {a_{k}} = (\cos \frac {k\pi}{6}, \sin \frac {k\pi}{6} + \cos \frac {k\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\sum\limits_{k=0}^{11} (\overrightarrow {a_{k}} \cdot \overrightarrow {a_{k+1}})$$. | 9\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the ... | 10 | 0 | 8,192 | -1 | 8,192 | |
The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is | 2 | 1. **Given Information and Equations:**
- The equations of lines $L_1$ and $L_2$ are $y = mx$ and $y = nx$ respectively.
- $L_1$ makes twice as large of an angle with the horizontal as does $L_2$.
- $L_1$ has 4 times the slope of $L_2$.
- $L_1$ is not horizontal.
2. **Relating Slopes to Angles:**
- Let ... | 0.9375 | 3,481.5625 | 3,167.533333 | 8,192 |
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil? | 4 | 1. **Convert the total payments to cents**:
- Seventh graders paid $1.43$ dollars, which is $143$ cents.
- Sixth graders paid $1.95$ dollars, which is $195$ cents.
2. **Determine the cost of one pencil**:
- The cost of a pencil must be a common factor of both $143$ and $195$ since each group paid a whole num... | 0.9375 | 2,215.3125 | 2,241.866667 | 1,817 |
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer. | $(p,n)=(p,p),(2,4)$ |
To solve the problem of finding all pairs \((p, n)\) of a prime number \(p\) and a positive integer \(n\) for which \(\frac{n^p + 1}{p^n + 1}\) is an integer, we start by analyzing the expression:
\[
\frac{n^p + 1}{p^n + 1}.
\]
**Step 1: Initial observation**
We need to determine when this ratio is an integer. Clea... | 0 | 8,059.6875 | -1 | 8,059.6875 |
A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \frac{1}{3} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair. | (18,6) | 0.8125 | 5,582.125 | 4,979.846154 | 8,192 | |
Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \). | -1/2 | 0 | 8,192 | -1 | 8,192 | |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(-2, -1, -1) \)
\( A_{2}(0, 3, 2) \)
\( A_{3}(3, 1, -4) \)
\( A_{4}(-4, 7, 3) \) | \frac{140}{\sqrt{1021}} | 0 | 6,543.625 | -1 | 6,543.625 | |
A math field day competition is held in a room with many tables, and there are 6 stools at each table. Each stool has 3 legs, and each table has 4 legs. If there is a total of 484 legs on all the tables and stools in the room, how many tables are in the room? | 22 | 1 | 1,472.4375 | 1,472.4375 | -1 | |
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between t... | 18 | 0 | 7,657.375 | -1 | 7,657.375 | |
Dave's sister Amy baked $4$ dozen pies. Among these:
- $5/8$ of them contained chocolate.
- $3/4$ of them contained marshmallows.
- $2/3$ of them contained cayenne.
- $1/4$ of them contained salted soy nuts.
Additionally, all pies with salted soy nuts also contained marshmallows. How many pies at most did not contain a... | 16 | 0 | 8,179.6875 | -1 | 8,179.6875 | |
Let $w, x, y, z$ be real numbers such that $w+x+y+z =5$, $2 w+4 x+8 y+16 z =7$, $3 w+9 x+27 y+81 z =11$, $4 w+16 x+64 y+256 z =1$. What is the value of $5 w+25 x+125 y+625 z ?$ | -60 | We note this system of equations is equivalent to evaluating the polynomial (in $a$ ) $P(a)=w a+x a^{2}+y a^{3}+z a^{4}$ at $1,2,3$, and 4 . We know that $P(0)=0, P(1)=5, P(2)=7, P(3)=11$, $P(4)=1$. The finite difference of a polynomial $f$ is $f(n+1)-f(n)$, which is a polynomial with degree one less than the degree of... | 0.1875 | 7,362.5625 | 6,687.666667 | 7,518.307692 |
Given real numbers \( x, y, z, w \) satisfying \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \). | \frac{3}{2} | 0 | 8,192 | -1 | 8,192 | |
John surveyed a group of people about their knowledge of rats. To the nearest tenth of a percent, he found that $86.8\%$ of the people surveyed thought rats carried diseases. Of the people who thought rats carried diseases, $45.7\%$ said that rats frequently carried rabies. Since rats do not frequently carry rabies, th... | 53 | 0.875 | 5,777.5625 | 5,432.642857 | 8,192 | |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, forming a pyramid $DABC$ with all triangular faces.
Suppose every edge of $DABC$ has a length either $25$ or $60$, and no face of $DABC$ is equilateral. Determine the total surface area of $DABC$. | 3600\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Given $3\sin \left(-3\pi +\theta \right)+\cos \left(\pi -\theta \right)=0$, then the value of $\frac{sinθcosθ}{cos2θ}$ is ____. | -\frac{3}{8} | 0.9375 | 5,254.25 | 5,328.733333 | 4,137 | |
Given the function $f(x)= \begin{cases} 2x,& x > 0 \\ f(x+1),& x\leqslant 0 \\ \end{cases}$, find $f(- \frac {4}{3})=$\_\_\_\_\_\_ and the maximum value of the real number $x_{0}$ that satisfies $f(f(x_{0}))=2$. | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end ... | \frac{2}{9} | 0.9375 | 3,861.8125 | 3,573.133333 | 8,192 | |
Find the number of triangulations of a general convex 7-gon into 5 triangles by 4 diagonals that do not intersect in their interiors. | 42 | Define the Catalan numbers by $C(n)=\frac{1}{n+1}\binom{2 n}{n}$. The current solution is the $C$ (number of triangles) $=C(5)=42$. | 0.9375 | 3,489.9375 | 3,176.466667 | 8,192 |
Given the cubic equation $10x^3 - 25x^2 + 8x - 1 = 0$, whose roots are $p$, $q$, and $s$, all of which are positive and less than 1. Calculate the sum of
\[\frac{1}{1-p} + \frac{1}{1-q} + \frac{1}{1-s}.\] | 0.5 | 0 | 5,904.4375 | -1 | 5,904.4375 | |
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line? | 1000000 | 0 | 665.3125 | -1 | 665.3125 | |
Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB? | 90 | 0 | 8,192 | -1 | 8,192 | |
Find the constant $c$ such that the remainder when $2x+7$ divides $2x^3+cx^2-11x+39$ is $4$. | 1 | 0.875 | 4,113.125 | 3,530.428571 | 8,192 | |
Simplify the expression: $({1-\frac{1}{{x+3}}})÷\frac{{{x^2}-9}}{{{x^2}+6x+9}}$, then choose a suitable number from $-3$, $2$, $3$ to substitute and evaluate. | -4 | 0.875 | 2,645.8125 | 2,730.642857 | 2,052 | |
Each of the numbers \(1, 2, 3, 4, 5, 6\) is to be placed in the cells of a \(2 \times 3\) table, with one number in each cell. In how many ways can this be done so that in each row and in each column the sum of the numbers is divisible by 3? | 48 | 0 | 8,192 | -1 | 8,192 | |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also find the le... | \frac{5 \sqrt{3}}{3} | 0 | 5,134.3125 | -1 | 5,134.3125 | |
Let $x = (2 + \sqrt{3})^{1000},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[x(1 - f).\] | 1 | 0.8125 | 3,800.5 | 2,787.076923 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.