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agentica-org/DeepScaleR-Preview-Dataset |
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$ | 6 \ln 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. | 1, 3, 18, 36 |
|
agentica-org/DeepScaleR-Preview-Dataset | In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the trees (along with their respective circles) and rotating each circle around its tree. In how many ways can the children be divided into the circles? | 50 |
|
agentica-org/DeepScaleR-Preview-Dataset | The amount of heat \( Q \) received by a certain substance when heated from 0 to \( T \) is determined by the formula \( Q = 0.1054t + 0.000002t^2 \) (\( Q \) is in joules, \( t \) is in kelvins). Find the heat capacity of this substance at \( 100 \) K. | 0.1058 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse. | n \neq 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces originally in the first box. | 131 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right triangle with sides of length $5$, $12$, and $13$ so that one side of the square lies on the hypotenuse of the triangle, find the value of $\frac{x}{y}$. | \frac{39}{51} |
|
agentica-org/DeepScaleR-Preview-Dataset | When $1 - i \sqrt{3}$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? | \frac{5\pi}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | On the Cartesian plane in which each unit is one foot, a dog is tied to a post on the point $(4,3)$ by a $10$ foot rope. What is the greatest distance the dog can be from the origin? | 15 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $\sin \left(x+ \frac {\pi}{3}\right)= \frac {1}{3}$, then the value of $\sin \left( \frac {5\pi}{3}-x\right)-\cos \left(2x- \frac {\pi}{3}\right)$ is \_\_\_\_\_\_. | \frac {4}{9} |
|
agentica-org/DeepScaleR-Preview-Dataset | In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 32805 |
|
agentica-org/DeepScaleR-Preview-Dataset | Calculate: $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^{2}=\_\_\_\_\_\_$. | \sqrt{3}-\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 20$ and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = xyz.\]Find $\frac{x^3 + y^3 + z^3}{xyz}.$ | 13 |
|
agentica-org/DeepScaleR-Preview-Dataset | The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd? | \frac{1}{23} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x)=2m\sin x-2\cos ^{2}x+ \frac{m^{2}}{2}-4m+3$, and the minimum value of the function $f(x)$ is $(-7)$, find the value of the real number $m$. | 10 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | 0.19 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - B$, then what is the value of $Y$? | 669 |
|
agentica-org/DeepScaleR-Preview-Dataset | David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? | 210 |
|
agentica-org/DeepScaleR-Preview-Dataset | Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell.
[/list]
At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$.
[i] | 2 \sum_{i=0}^{8} \binom{n}{i} - 1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is \_\_\_\_\_\_. | -3+2\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$, respectively. Find the ratio $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$. | \frac{1}{16} |
|
agentica-org/DeepScaleR-Preview-Dataset | The base of isosceles $\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides? | 13 |
|
agentica-org/DeepScaleR-Preview-Dataset | The function \( g(x) \) satisfies
\[ g(x) - 2 g \left( \frac{1}{x} \right) = 3^x + x \]
for all \( x \neq 0 \). Find \( g(2) \). | -4 - \frac{2\sqrt{3}}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent? | 20 \% |
|
agentica-org/DeepScaleR-Preview-Dataset | For a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right)\) has a Caesar sum of 1000, what is the Caesar sum of the 100-term sequence \(\left(1, p_{1}, p_{2}, \cdots, p_{99}\right)\)? | 991 |
|
agentica-org/DeepScaleR-Preview-Dataset | If a computer executes the following program:
1. Initial values: \( x = 3 \), \( S = 0 \).
2. \( x = x + 2 \).
3. \( S = S + x \).
4. If \( S \geq 10000 \), go to step 5; otherwise, go back to step 2.
5. Print \( x \).
6. Stop.
Then the value printed at step 5 is: | 201 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculate the value of $\sum _{k=1}^{22}f(k)$. | -24 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is: | 7 |
|
agentica-org/DeepScaleR-Preview-Dataset | The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas? | 103 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the diagram, \(ABCD\) is a square with a side length of \(8 \, \text{cm}\). Point \(E\) is on \(AB\) and point \(F\) is on \(DC\) so that \(\triangle AEF\) is right-angled at \(E\). If the area of \(\triangle AEF\) is \(30\%\) of the area of \(ABCD\), what is the length of \(AE\)? | 4.8 |
|
agentica-org/DeepScaleR-Preview-Dataset | Simplify the following expression:
$$5x + 6 - x + 12$$ | 4x + 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \). | -3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If there are 10 or more people, they can buy group tickets for 70 yuan per person. What is the minimum amount they should pay for the tickets? | 800 |
|
agentica-org/DeepScaleR-Preview-Dataset | A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the third prize is more than that for the second prize, and the total prize money for the second prize is more than that for the first prize, with the total number of winners not exceeding 20, then what is the minimum amount of the first prize? | 2700 |
|
agentica-org/DeepScaleR-Preview-Dataset | In triangle $ABC$, the altitude and the median from vertex $C$ each divide the angle $ACB$ into three equal parts. Determine the ratio of the sides of the triangle. | 2 : \sqrt{3} : 1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any additional seasoning? (Each day Sasha cooks the same portion of soup.) | 1.5 |
|
agentica-org/DeepScaleR-Preview-Dataset | On Monday at work, David produces $w$ widgets per hour, and works for $t$ hours. Exhausted by this work, on Tuesday, he decides to work for $2$ fewer hours, but manages to produce $4$ additional widgets per hour. If $w = 2t$, how many more widgets did David produce on Monday than on Tuesday? | 8 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $i^2=-1$, then $(i-i^{-1})^{-1}=$ | -\frac{i}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | If $x=3$, what is the value of $-(5x - 6x)$? | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Alice thinks of four positive integers $a \leq b \leq c \leq d$ satisfying $\{a b+c d, a c+b d, a d+b c\}=\{40,70,100\}$. What are all the possible tuples $(a, b, c, d)$ that Alice could be thinking of? | (1,4,6,16) |
|
agentica-org/DeepScaleR-Preview-Dataset | For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$. | 695 |
|
agentica-org/DeepScaleR-Preview-Dataset | A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist? | 10 |
|
agentica-org/DeepScaleR-Preview-Dataset | The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______. | 23 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let \( a, b, c, d \) be integers such that \( a > b > c > d \geq -2021 \) and
\[ \frac{a+b}{b+c} = \frac{c+d}{d+a} \]
(and \( b+c \neq 0 \neq d+a \)). What is the maximum possible value of \( a \cdot c \)? | 510050 |
|
agentica-org/DeepScaleR-Preview-Dataset | A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. The length, in centimeters, of the path followed by the dot is $c\pi$, where $c$ is a constant. What is $c$? | \dfrac{1+\sqrt{5}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | In our daily life, for a pair of new bicycle tires, the rear tire wears out faster than the front tire. Through testing, it is found that the front tire of a general bicycle is scrapped after traveling 11,000 kilometers, while the rear tire is scrapped after traveling 9,000 kilometers. It is evident that when the rear tire is scrapped after traveling 9,000 kilometers, the front tire can still be used, which inevitably leads to a certain waste. If the front and rear tires are swapped once, allowing the front and rear tires to be scrapped simultaneously, the bicycle can travel a longer distance. How many kilometers can the bicycle travel at most after swapping once? And after how many kilometers should the front and rear tires be swapped? | 4950 |
|
agentica-org/DeepScaleR-Preview-Dataset | Points \( D \) and \( E \) are located on side \( AC \) of triangle \( ABC \). Lines \( BD \) and \( BE \) divide the median \( AM \) of triangle \( ABC \) into three equal segments.
Find the area of triangle \( BDE \) if the area of triangle \( ABC \) is 1. | 0.3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the sum of all $x$ that satisfy the equation $\frac{-9x}{x^2-1} = \frac{2x}{x+1} - \frac{6}{x-1}.$ | -\frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute $\sin 510^\circ$. | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:
\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108} \qquad &\cdots\\ \frac 1{216} \qquad &\cdots\\ &\ddots \end{align*}
In general, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row. Then, each succeeding term in a row is $\frac 1p$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{2008}$ -array can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $2008$.
| 1 |
|
agentica-org/DeepScaleR-Preview-Dataset | The diagram shows three rectangles and three straight lines. What is the value of \( p + q + r \)?
A) 135
B) 180
C) 210
D) 225
E) 270 | 180 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the set S={1, 2, 3, ..., 40}, and a subset A⊆S containing three elements, find the number of such sets A that can form an arithmetic progression. | 380 |
|
agentica-org/DeepScaleR-Preview-Dataset | The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?
A) 41
B) 49
C) 51
D) 59
E) 61 | 59 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $a$ , $b$ , $c$ , $d$ , $e$ be positive reals satisfying \begin{align*} a + b &= c a + b + c &= d a + b + c + d &= e.\end{align*} If $c=5$ , compute $a+b+c+d+e$ .
*Proposed by Evan Chen* | 40 |
|
agentica-org/DeepScaleR-Preview-Dataset | Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 515 |
|
agentica-org/DeepScaleR-Preview-Dataset | The equation
$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$
is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective. | 2016 |
|
agentica-org/DeepScaleR-Preview-Dataset | Positive real numbers $r,s$ satisfy the equations $r^2 + s^2 = 1$ and $r^4 + s^4= \frac{7}{8}$. Find $rs$. | \frac{1}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | Zachary paid for a $\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used? | 17 |
|
agentica-org/DeepScaleR-Preview-Dataset | John has 12 marbles of different colors, including one red, one green, and one blue marble. In how many ways can he choose 4 marbles, if exactly one of the chosen marbles is red, green, or blue? | 252 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$ and $0 < \alpha < \pi$, find the value of $\tan(\alpha - \frac{\pi}{4})$. | 2\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $1+2i$. Gracie chooses $-1+i$. How far apart are Gracie and Joe's points? | \sqrt{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the smallest positive integer \( n \) such that \( n(n+1)(n+2) \) is divisible by 247. | 37 |
|
agentica-org/DeepScaleR-Preview-Dataset | From Moscow to city \( N \), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \( N \) in 10 hours, including the waiting time. By how many times is the plane’s speed greater than the train’s speed, given that the plane will be above this train 8/9 hours after departure from the airport and will have traveled the same number of kilometers as the train by that time? | 10 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$? | 1120 |
|
agentica-org/DeepScaleR-Preview-Dataset | During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \(4: 7: 14\). On the twelfth day, more people participated in the survey, which changed the ratio of the response frequencies to \(6: 9: 16\). What is the minimum number of people who must have responded to the survey on the twelfth day? | 75 |
|
agentica-org/DeepScaleR-Preview-Dataset | Coach Randall is preparing a 6-person starting lineup for her soccer team, the Rangers, which has 15 players. Among the players, three are league All-Stars (Tom, Jerry, and Spike), and they are guaranteed to be in the starting lineup. Additionally, the lineup must include at least one goalkeeper, and there is only one goalkeeper available among the remaining players. How many different starting lineups are possible? | 55 |
|
agentica-org/DeepScaleR-Preview-Dataset | Class A and Class B each send 2 students to participate in the grade math competition. The probability of each participating student passing the competition is 0.6, and the performance of the participating students does not affect each other. Find:
(1) The probability that there is exactly one student from each of Class A and Class B who passes the competition;
(2) The probability that at least one student from Class A and Class B passes the competition. | 0.9744 |
|
agentica-org/DeepScaleR-Preview-Dataset | The line with equation $y = x$ is an axis of symmetry of the curve with equation
\[y = \frac{px + q}{rx + s},\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?
(A) $p + q = 0$
(B) $p + r = 0$
(C) $p + s = 0$
(D) $q + r = 0$
(E) $q + s = 0$
(F) $r + s = 0$ | \text{(C)} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$ | 1/288 |
|
agentica-org/DeepScaleR-Preview-Dataset | If point $P$ is the golden section point of segment $AB$, and $AP < BP$, $BP=10$, then $AP=\_\_\_\_\_\_$. | 5\sqrt{5} - 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her? | 54 |
|
agentica-org/DeepScaleR-Preview-Dataset | Xiao Wang loves mathematics and chose the six numbers $6$, $1$, $8$, $3$, $3$, $9$ to set as his phone's startup password. If the two $3$s are not adjacent, calculate the number of different passwords Xiao Wang can set. | 240 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the sum of the digits of the decimal representation of $2^{2005} \times 5^{2007} \times 3$? | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $A B$ be a segment of length 2 with midpoint $M$. Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $A M$ and $B M$ and internally tangent to the circle with diameter $A B$. Determine the value of $r$. | \frac{1}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, calculate the diameter of the circumcircle of $\triangle ABC$. | 5\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given \(0 \leqslant x \leqslant 2\), the function \(y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5\) reaches its minimum value at? | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the number of subsets $S$ of $\{1,2, \ldots, 48\}$ satisfying both of the following properties: - For each integer $1 \leq k \leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$. - There are exactly nine integers $1 \leq m \leq 47$ so that both $m$ and $m+1$ are in $S$. | 177100 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the right triangular prism $ABC - A_1B_1C_1$, $\angle ACB = 90^\circ$, $AC = 2BC$, and $A_1B \perp B_1C$. Find the sine of the angle between $B_1C$ and the lateral face $A_1ABB_1$. | \frac{\sqrt{10}}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that
$$
\begin{array}{l}
a + b + c = 5, \\
a^2 + b^2 + c^2 = 15, \\
a^3 + b^3 + c^3 = 47.
\end{array}
$$
Find the value of \((a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)\). | 625 |
|
agentica-org/DeepScaleR-Preview-Dataset | Suppose that $x, y, z$ are real numbers such that $x=y+z+2$, $y=z+x+1$, and $z=x+y+4$. Compute $x+y+z$. | -7 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the number of ordered triples of positive integers $(a, b, c)$ such that $6a+10b+15c=3000$. | 4851 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $a, b$, and $c$ are complex numbers satisfying $$\begin{aligned} a^{2}+a b+b^{2} & =1+i \\ b^{2}+b c+c^{2} & =-2 \\ c^{2}+c a+a^{2} & =1 \end{aligned}$$ compute $(a b+b c+c a)^{2}$. (Here, $\left.i=\sqrt{-1}.\right)$ | \frac{-11-4 i}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | Sue owns 11 pairs of shoes: six identical black pairs, three identical brown pairs and two identical gray pairs. If she picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction. | \frac{7}{33} |
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agentica-org/DeepScaleR-Preview-Dataset | Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice? | \frac{19}{40} |
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agentica-org/DeepScaleR-Preview-Dataset | Using systematic sampling method to select 32 people from 960 for a questionnaire survey, they are randomly numbered from 1 to 960. After grouping, the number drawn by simple random sampling in the first group is 9. Among the 32 people drawn, those with numbers in the interval [1,450] will fill out questionnaire A, those in the interval [451,750] will fill out questionnaire B, and the rest will fill out questionnaire C. How many of the drawn people will fill out questionnaire B? | 10 |
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agentica-org/DeepScaleR-Preview-Dataset | If $x=1$ is a solution of the equation $x^{2} + ax + 1 = 0$, what is the value of $a$? | -2 |
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agentica-org/DeepScaleR-Preview-Dataset | A sequence $a_1$, $a_2$, $\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible? | 324 |
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agentica-org/DeepScaleR-Preview-Dataset | Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. | 100 |
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agentica-org/DeepScaleR-Preview-Dataset | In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?
A) $\frac{24}{5}$
B) $\frac{12}{5}$
C) $\frac{17}{4}$
D) $\frac{15}{3}$ | \frac{24}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | The coefficients of the polynomial \(P(x)\) are nonnegative integers, each less than 100. Given that \(P(10)=331633\) and \(P(-10)=273373\), compute \(P(1)\). | 100 |
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agentica-org/DeepScaleR-Preview-Dataset | A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. | 81\sqrt{3} |
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agentica-org/DeepScaleR-Preview-Dataset | Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
*(Proposed by Walther Janous)* | 16 |
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agentica-org/DeepScaleR-Preview-Dataset | 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 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial. | 8\sqrt{6} |
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agentica-org/DeepScaleR-Preview-Dataset | Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? | 0.4 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $a_{1}=1$, and let $a_{n}=\left\lfloor n^{3} / a_{n-1}\right\rfloor$ for $n>1$. Determine the value of $a_{999}$. | 999 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the difference between the sum of the numbers $3$, $-4$, and $-5$ and the sum of their absolute values. | -18 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $\frac {35x - 29}{x^2 - 3x + 2} = \frac {N_1}{x - 1} + \frac {N_2}{x - 2}$ be an identity in $x$. The numerical value of $N_1N_2$ is: | -246 |
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