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agentica-org/DeepScaleR-Preview-Dataset | If $a, b, c$ are positive integers less than $10$, then $(10a + b)(10a + c) = 100a(a + 1) + bc$ if: | $b+c=10$ |
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agentica-org/DeepScaleR-Preview-Dataset | A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | 100 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[\frac{4z}{2x + y} + \frac{4x}{y + 2z} + \frac{y}{x + z}.\] | 3 |
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agentica-org/DeepScaleR-Preview-Dataset | Calculate $[x]$, where $x = -3.7 + 1.5$. | -3 |
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agentica-org/DeepScaleR-Preview-Dataset | Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]
import graph;
filldraw(circle((0,0),10), lightgray, black+linewidth(1));
filldraw(circle((0,0),6), white, black+linewidth(1));
dot((0,0));
label("$C$",(0,0),NE);
[/asy] | 64\pi |
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agentica-org/DeepScaleR-Preview-Dataset | Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. What is the smallest number of moves that he can make to get to the point \((1056,1007)\)? | 2111 |
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agentica-org/DeepScaleR-Preview-Dataset | In the Cartesian coordinate system, it is known that the terminal side of an angle $\alpha$ with the origin as the vertex and the non-negative half-axis of the $x$-axis as the initial side passes through the point $(-3,-4)$.
$(1)$ Find the value of $\frac{sin\alpha}{tan\alpha}$;
$(2)$ Find the value of $\frac{sin(\alpha+\frac{\pi}{2})\cdot cos(\frac{9\pi}{2}-\alpha)\cdot tan(2\pi-\alpha)\cdot cos(-\frac{3\pi}{2}+\alpha)}{sin(2\pi-\alpha)\cdot tan(-\alpha-\pi)\cdot sin(\pi+\alpha)}$. | \frac{3}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Find the probability $p$ that the object reaches $(3,1)$ exactly in four steps. | \frac{1}{32} |
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agentica-org/DeepScaleR-Preview-Dataset | A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection. | 185 |
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agentica-org/DeepScaleR-Preview-Dataset | 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 |
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agentica-org/DeepScaleR-Preview-Dataset | A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week. | 36 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the value of $x$ that satisfies the equation $25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.$ | 3 |
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agentica-org/DeepScaleR-Preview-Dataset | In an equilateral triangle \(ABC\), a point \(P\) is chosen such that \(AP = 10\), \(BP = 8\), and \(CP = 6\). Find the area of this triangle. | 36 + 25\sqrt{3} |
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agentica-org/DeepScaleR-Preview-Dataset | In the sequence $\{a_n\}$, $a_n$ is the closest positive integer to $\sqrt{n}$ ($n \in \mathbb{N}^*$). Compute the sum $\sum_{i=1}^{100}\frac{1}{a_i} = \_\_\_\_\_\_\_\_$. | 19 |
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agentica-org/DeepScaleR-Preview-Dataset | There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. | 548 |
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agentica-org/DeepScaleR-Preview-Dataset | Given the number
\[e^{11\pi i/40} + e^{21\pi i/40} + e^{31 \pi i/40} + e^{41\pi i /40} + e^{51 \pi i /40},\]
express it in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. Find $\theta$. | \frac{11\pi}{20} |
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agentica-org/DeepScaleR-Preview-Dataset | What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? | 23 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$ . Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ | $\pi/2$ |
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agentica-org/DeepScaleR-Preview-Dataset | Find the number of positive integers \( x \), where \( x \neq 9 \), such that
\[
\log _{\frac{x}{9}}\left(\frac{x^{2}}{3}\right)<6+\log _{3}\left(\frac{9}{x}\right) .
\] | 223 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$. | \frac{4}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | The regular hexagon \(ABCDEF\) has diagonals \(AC\) and \(CE\). The internal points \(M\) and \(N\) divide these diagonals such that \(AM: AC = CN: CE = r\). Determine \(r\) if it is known that points \(B\), \(M\), and \(N\) are collinear. | \frac{1}{\sqrt{3}} |
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agentica-org/DeepScaleR-Preview-Dataset | Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = 4$ when $\beta = 9$, find $\alpha$ when $\beta = -72$. | -\frac{1}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | In the complex plane, the graph of \( |z - 5| = 3|z + 5| \) intersects the graph of \( |z| = k \) in exactly one point. Find all possible values of \( k \). | 12.5 |
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agentica-org/DeepScaleR-Preview-Dataset | What is $\frac{1}{4}\%$ of 120? Express your answer as a decimal. | .3 |
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agentica-org/DeepScaleR-Preview-Dataset | Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? | 51 |
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agentica-org/DeepScaleR-Preview-Dataset | In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \sqrt{2}$. Find the value of $x$. | -6 |
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agentica-org/DeepScaleR-Preview-Dataset | Let the first term of a geometric sequence be $\frac{3}{4}$, and let the second term be $15$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by one million? | 7 |
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agentica-org/DeepScaleR-Preview-Dataset | The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)? | 44 |
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agentica-org/DeepScaleR-Preview-Dataset | The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$, $8\text{ in}^{2}$, and $6\text{ in}^{2}$ respectively is: | $24\text{ in}^{3}$ |
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agentica-org/DeepScaleR-Preview-Dataset | Unit circle $\Omega$ has points $X, Y, Z$ on its circumference so that $X Y Z$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $W Y Z$ is also equilateral. Determine the area of the region inside triangle $W Y Z$ that lies outside circle $\Omega$. | $\frac{3 \sqrt{3}-\pi}{3}$ |
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agentica-org/DeepScaleR-Preview-Dataset | Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that $|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.$ What is the smallest possible value of $n$? | 20 |
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agentica-org/DeepScaleR-Preview-Dataset | Given that the domains of functions f(x) and g(x) are both 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, determine the value of \sum _{k=1}^{22}f(k). | -24 |
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agentica-org/DeepScaleR-Preview-Dataset | The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are: | \frac{3}{5} < x \le 2 |
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agentica-org/DeepScaleR-Preview-Dataset | If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\). | \sqrt{10} |
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agentica-org/DeepScaleR-Preview-Dataset | A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$. | 32 |
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agentica-org/DeepScaleR-Preview-Dataset | Calculate the definite integral:
$$
\int_{0}^{\frac{2\pi}{3}} \frac{\cos^2 x \, dx}{(1 + \cos x + \sin x)^2}
$$ | \frac{\sqrt{3}}{2} - \ln 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum? | \frac{500}{1001} |
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agentica-org/DeepScaleR-Preview-Dataset | Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$. What is the probability that, among those selected, the second smallest is $3$?
$\textbf{(A)}\ \frac{1}{60}\qquad \textbf{(B)}\ \frac{1}{6}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{1}{2}\qquad \textbf{(E)}\ \text{none of these}$
| \frac{1}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle? | 1.5 |
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agentica-org/DeepScaleR-Preview-Dataset | Given positive numbers \(a, b, c, x, y, z\) that satisfy the equations \(cy + bz = a\), \(az + cx = b\), and \(bx + ay = c\), find the minimum value of the function
\[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \] | 1/2 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the maximum number of consecutive positive integers starting from 10 that can be added together before the sum exceeds 500? | 23 |
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agentica-org/DeepScaleR-Preview-Dataset | In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \). | 21 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$ | \frac{\sqrt{7}}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | If $A$ is an angle such that $\tan A + \sec A = 2,$ enter all possible values of $\cos A,$ separated by commas. | \frac{4}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:
\[
\sum_{i = 0}^{100} \frac{q_i}{r_i} - \sum_{i = 0}^{99} \frac{p_i}{q_i}.
\] | \frac{1}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Given that three numbers are randomly selected from the set {1, 2, 3, 4, 5}, find the probability that the sum of the remaining two numbers is odd. | 0.6 |
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agentica-org/DeepScaleR-Preview-Dataset | Square \(ABCD\) has sides of length 14. A circle is drawn through \(A\) and \(D\) so that it is tangent to \(BC\). What is the radius of the circle? | 8.75 |
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agentica-org/DeepScaleR-Preview-Dataset | In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$. | 177 |
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agentica-org/DeepScaleR-Preview-Dataset | In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\). | \frac{\sqrt{7}}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Given $\tan (\alpha +\beta )=7$ and $\tan (\alpha -\beta )=1$, find the value of $\tan 2\alpha$. | -\dfrac{4}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | John is thinking of a number. He gives the following 3 clues. ``My number has 125 as a factor. My number is a multiple of 30. My number is between 800 and 2000.'' What is John's number? | 1500 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $S$ be the set of points $(a, b)$ with $0 \leq a, b \leq 1$ such that the equation $x^{4}+a x^{3}-b x^{2}+a x+1=0$ has at least one real root. Determine the area of the graph of $S$. | \frac{1}{4} |
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agentica-org/DeepScaleR-Preview-Dataset | The repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$? | 11 |
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agentica-org/DeepScaleR-Preview-Dataset | A graduating high school class has $45$ people. Each student writes a graduation message to every other student, with each pair writing only one message between them. How many graduation messages are written in total? (Answer with a number) | 1980 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=0$ and $\tan \angle PF_{1}F_{2}= \frac{1}{2}$. | \frac{\sqrt{5}}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $ \begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases} $ (where $\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta- \frac {\pi}{4})= \sqrt {2}$.
$(1)$ Find the implicit equation for $C$ and the inclination angle of $l$.
$(2)$ Let point $P(0,2)$ be given, and line $l$ intersects curve $C$ at points $A$ and $B$. Find the $|PA|+|PB|$. | \frac {18 \sqrt {2}}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | Add 78.621 to 34.0568 and round to the nearest thousandth. | 112.678 |
|
agentica-org/DeepScaleR-Preview-Dataset | In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\). | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | Determine the length of the interval of solutions of the inequality $a \le 3x + 6 \le b$ where the length of the interval is $15$. | 45 |
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agentica-org/DeepScaleR-Preview-Dataset | Given $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\cos 2C+2c\cos A\cos C+a+b=0$.
$(1)$ Find the size of angle $C$;
$(2)$ If $b=4\sin B$, find the maximum value of the area $S$ of $\triangle ABC$. | \sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Observe the following equations:<br/>$\frac{1}{1×2}=1-\frac{1}{2}=\frac{1}{2}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}=\frac{2}{3}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=\frac{3}{4}$;<br/>$\ldots $<br/>Based on the pattern you discovered, answer the following questions:<br/>$(1)\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}×\frac{1}{4×5}=$______;<br/>$(2)$If $n$ is a positive integer, then $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\frac{1}{4×5}+…+\frac{1}{n(n+1)}=$______;<br/>$(3)$Calculate $1-\frac{1}{1×2}-\frac{1}{2×3}-\frac{1}{3×4}-…-\frac{1}{99×100}$. | \frac{1}{100} |
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agentica-org/DeepScaleR-Preview-Dataset | Given the circle with radius $6\sqrt{2}$, diameter $\overline{AB}$, and chord $\overline{CD}$ intersecting $\overline{AB}$ at point $E$, where $BE = 3\sqrt{2}$ and $\angle AEC = 60^{\circ}$, calculate $CE^2+DE^2$. | 216 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate the expression \[ (a^2+b)^2 - (a^2-b)^2, \]if $a=4$ and $b=1$. | 64 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $S=1!+2!+3!+\cdots +99!$, then the units' digit in the value of S is: | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute: $9 \cdot \frac{1}{13} \cdot 26.$ | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. | 4104 |
|
agentica-org/DeepScaleR-Preview-Dataset | (15) Given the following propositions:
(1) "If $x > 2$, then $x > 0$" - the negation of the proposition
(2) "For all $a \in (0, +\infty)$, the function $y = a^x$ is strictly increasing on its domain" - the negation
(3) "$π$ is a period of the function $y = \sin x$" or "$2π$ is a period of the function $y = \sin 2x$"
(4) "$x^2 + y^2 = 0$" is a necessary condition for "$xy = 0$"
The sequence number(s) of the true proposition(s) is/are _______. | (2)(3) |
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agentica-org/DeepScaleR-Preview-Dataset | The repeating decimal \( 0.\dot{x}y\dot{3} = \frac{a}{27} \), where \( x \) and \( y \) are distinct digits. Find the integer \( a \). | 19 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let \( z \) be a complex number that satisfies
\[ |z - 2| + |z - 7i| = 10. \]
Find the minimum value of \( |z| \). | 1.4 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $PQRST$ be a convex pentagon with $PQ \parallel RT, QR \parallel PS, QS \parallel PT, \angle PQR=100^\circ, PQ=4, QR=7,$ and $PT = 21.$ Given that the ratio between the area of triangle $PQR$ and the area of triangle $RST$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 232 |
|
agentica-org/DeepScaleR-Preview-Dataset | Simplify the expression \[\sqrt{45 - 28\sqrt{2}}.\] | 5 - 3\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | $\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=$ | $16$ |
|
agentica-org/DeepScaleR-Preview-Dataset | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$ | 683 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given an arithmetic sequence $\{a_n\}$ with common difference $d \neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\{a_n^2\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\{b_n\}$ with a common ratio $q$ that is a positive rational number less than $1$. The first term of this geometric sequence is $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. Find the possible value(s) of $q$ such that $\frac{S_3}{T_3}$ is a positive integer. | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | If $z=1+i$, then $|{iz+3\overline{z}}|=\_\_\_\_\_\_$. | 2\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Calculate the probability that athlete A cannot run the first leg and athlete B cannot run the last leg in a 4x100 meter relay race selection from 6 short-distance runners, including athletes A and B, to form a team of 4 runners. | \frac{7}{10} |
|
agentica-org/DeepScaleR-Preview-Dataset | Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than they disassemble one. When they arrived at the workshop in the morning, several grinders were already assembled.
At 7:00 AM, Hubert started assembling and Robert started disassembling. Exactly at 12:00 PM, Hubert finished assembling a grinder and Robert finished disassembling another one. In total, 70 grinders were added during this shift.
At 1:00 PM, Robert started assembling and Hubert started disassembling. Exactly at 10:00 PM, Robert finished assembling the last grinder and Hubert finished disassembling another one. In total, 36 grinders were added during this shift.
How long would it take for Robert and Hubert to assemble 360 grinders if both of them worked together assembling? | 15 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the diagram, if $\angle PQR = 48^\circ$, what is the measure of $\angle PMN$? [asy]
size(6cm);
pair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * dir(-24);
label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);
draw(m--q--r--n--cycle);
add(pathticks(m--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
add(pathticks(r--q, 2, spacing=0.9, s=4));
[/asy] | 66^\circ |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$. | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate $(-27)^{5/3}$. | -243 |
|
agentica-org/DeepScaleR-Preview-Dataset | The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$ . Find the volume of the tetrahedron. | 1/8 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$. | 148 |
|
agentica-org/DeepScaleR-Preview-Dataset | If 20$\%$ of 10$\%$ of a number is 12, what is 10$\%$ of 20$\%$ of the same number? | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | Simplify $\frac{{x}^{2}-4x+4}{{x}^{2}-1}÷\frac{{x}^{2}-2x}{x+1}+\frac{1}{x-1}$ first, then choose a suitable integer from $-2\leqslant x\leqslant 2$ as the value of $x$ to evaluate. | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m + n + p.$ | 578 |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute the following expression:
\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \] | 1398100 |
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agentica-org/DeepScaleR-Preview-Dataset | How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
| 2148 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $x_{1}, \ldots, x_{100}$ be defined so that for each $i, x_{i}$ is a (uniformly) random integer between 1 and 6 inclusive. Find the expected number of integers in the set $\{x_{1}, x_{1}+x_{2}, \ldots, x_{1}+x_{2}+\ldots+x_{100}\}$ that are multiples of 6. | \frac{50}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x)=\frac{1}{2}x^{2}-a\ln x+b$ where $a\in R$.
(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.
(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real number $a$.
(III) If $-2\leqslant a < 0$, for any $x_{1}$, $x_{2}\in(0,2]$, the inequality $|f(x_{1})-f(x_{2})|\leqslant m| \frac{1}{x_{1}}- \frac{1}{x_{2}}|$ always holds. Find the minimum value of $m$. | 12 |
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agentica-org/DeepScaleR-Preview-Dataset | Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$? | \frac{1}{16} |
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agentica-org/DeepScaleR-Preview-Dataset | Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$. | -6 |
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agentica-org/DeepScaleR-Preview-Dataset | Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be? | 14 |
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agentica-org/DeepScaleR-Preview-Dataset | How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 32 |
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agentica-org/DeepScaleR-Preview-Dataset | The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black? Express your answer as a common fraction. | \frac{650}{51} |
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agentica-org/DeepScaleR-Preview-Dataset | The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle. | \frac{\sqrt{2}}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Given \( P \) is the product of \( 3,659,893,456,789,325,678 \) and \( 342,973,489,379,256 \), find the number of digits of \( P \). | 34 |
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agentica-org/DeepScaleR-Preview-Dataset | How many solutions of the equation $\tan x = \tan (\tan x)$ are on the interval $0 \le x \le \tan^{-1} 942$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.)
Note: You can assume the result that $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}.$ | 300 |
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agentica-org/DeepScaleR-Preview-Dataset | An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times. | \frac{11}{2048} |
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agentica-org/DeepScaleR-Preview-Dataset | Given a sequence $\{a_n\}$ that satisfies: $a_{n+2}=\begin{cases} 2a_{n}+1 & (n=2k-1, k\in \mathbb{N}^*) \\ (-1)^{\frac{n}{2}} \cdot n & (n=2k, k\in \mathbb{N}^*) \end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \leqslant 2046$. | 19 |
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agentica-org/DeepScaleR-Preview-Dataset | Some squares of a $n \times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers. | 3n^{2}-5n+2 |
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