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agentica-org/DeepScaleR-Preview-Dataset | Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$ . Compute the sum of all bases and exponents in the prime factorization of $A$ . For example, if $A=7\cdot 11^5$ , the answer would be $7+11+5=23$ . | 677 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x)=2\ln(3x)+8x$, find the value of $\lim_{\triangle x \to 0}\frac{f(1-2\triangle x)-f(1)}{\triangle x}$. | -20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Convert $\rm{A}03_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 in order. | 2563 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy] | 7 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the last two digits of \(\left[(\sqrt{29}+\sqrt{21})^{1984}\right]\). | 71 |
|
agentica-org/DeepScaleR-Preview-Dataset | For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute? | 4 |
|
agentica-org/DeepScaleR-Preview-Dataset | The maximum value of the function $y=\sin x \cos x + \sin x + \cos x$ is __________. | \frac{1}{2} + \sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$ . | 41 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that point $P$ moves on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\sqrt{5}=0$. | \frac{\sqrt{10}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset |
In the Cartesian coordinate system \( xOy \), find the area of the region defined by the inequalities
\[
y^{100}+\frac{1}{y^{100}} \leq x^{100}+\frac{1}{x^{100}}, \quad x^{2}+y^{2} \leq 100.
\] | 50 \pi |
|
agentica-org/DeepScaleR-Preview-Dataset | Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$. | 17 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate the value of $\frac{(2210-2137)^2 + (2137-2028)^2}{64}$. | 268.90625 |
|
agentica-org/DeepScaleR-Preview-Dataset | The ratio of the sums of the first \( n \) terms of two arithmetic sequences is \(\frac{9n+2}{n+7}\). Find the ratio of their 5th terms. | \frac{83}{16} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a trapezoid \(ABCD\). A point \(M\) is chosen on its lateral side \(CD\) such that \( \frac{CM}{MD} = \frac{4}{3} \). It turns out that segment \( BM \) divides the diagonal \( AC \) into two segments, the ratio of the lengths of which is also \( \frac{4}{3} \). What possible values can the ratio \( \frac{AD}{BC} \) take? If necessary, round your answer to 0.01 or write it as a common fraction. | 7/12 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $M = 2007 \div 3$, $N = M \div 3$, and $X = M - N$, then what is the value of $X$? | 446 |
|
agentica-org/DeepScaleR-Preview-Dataset | A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceiling such that the following conditions are satisfied: every point on the soccer field is illuminated by at least one floodlight; the height of the ceiling must be a multiple of 0.1 meters (e.g., 19.2 meters, 26 meters, 31.9 meters). | 27.1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find whole numbers $\heartsuit$ and $\clubsuit$ such that $\heartsuit \cdot \clubsuit = 48$ and $\heartsuit$ is even, then determine the largest possible value of $\heartsuit + \clubsuit$. | 26 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $m$ and $n$ be odd integers greater than $1.$ An $m\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099$. | 248 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $u,$ $v,$ and $w$ be the roots of the equation $x^3 - 18x^2 + 20x - 8 = 0.$ Find the value of $(2+u)(2+v)(2+w).$ | 128 |
|
agentica-org/DeepScaleR-Preview-Dataset | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | For a real number \( x \), \([x]\) denotes the greatest integer less than or equal to \( x \). Given a sequence of positive numbers \( \{a_n\} \) such that \( a_1 = 1 \) and \( S_n = \frac{1}{2} \left( a_n + \frac{1}{a_n} \right) \), where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \), then \(\left[ \frac{1}{S_1} + \frac{1}{S_2} + \cdots + \frac{1}{S_{100}} \right] = \, \). | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined. | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $x$, $y$, $z \in \mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$. | -3\sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given two lines $l_1: ax+2y+6=0$ and $l_2: x+(a-1)y+a^2-1=0$. When $a$ \_\_\_\_\_\_, $l_1$ intersects $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ is perpendicular to $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ coincides with $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ is parallel to $l_2$. | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Points $ A$ and $ B$ lie on a circle centered at $ O$ , and $ \angle AOB=60^\circ$ . A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle? | \frac{1}{9} |
|
agentica-org/DeepScaleR-Preview-Dataset | Determine the number of angles $\theta$ between 0 and $2 \pi$, other than integer multiples of $\pi / 2$, such that the quantities $\sin \theta, \cos \theta$, and $\tan \theta$ form a geometric sequence in some order. | 4 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called *good* if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number. | 667 |
|
agentica-org/DeepScaleR-Preview-Dataset | The inclination angle of the line $\sqrt{3}x+y-1=0$ is ____. | \frac{2\pi}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots which are numerically equal but of opposite signs, the value of $m$ must be: | \frac{a-b}{a+b} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists. | 64 |
|
agentica-org/DeepScaleR-Preview-Dataset | The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C_{}^{}$? | 144 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$. | 987 |
|
agentica-org/DeepScaleR-Preview-Dataset | In how many ways can 81 be written as the sum of three positive perfect squares if the order of the three perfect squares does not matter? | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle. | \frac{\pi-2}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$. | \frac{8\sqrt{2}}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that is not occupied by the cones. Express your answer in terms of $\pi$. | 5250\pi |
|
agentica-org/DeepScaleR-Preview-Dataset | Point $C$ is the midpoint of $\overline{AB}$, point $D$ is the midpoint of $\overline{AC}$, point $E$ is the midpoint of $\overline{AD}$, and point $F$ is the midpoint of $\overline{AE}$. If $AF=3$, what is the number of units in the length of $\overline{AB}$? | 48 |
|
agentica-org/DeepScaleR-Preview-Dataset | Calculate:
$$\frac{\left(1+\frac{1}{2}\right)^{2} \times\left(1+\frac{1}{3}\right)^{2} \times\left(1+\frac{1}{4}\right)^{2} \times\left(1+\frac{1}{5}\right)^{2} \times \cdots \times\left(1+\frac{1}{10}\right)^{2}}{\left(1-\frac{1}{2^{2}}\right) \times\left(1-\frac{1}{3^{2}}\right) \times\left(1-\frac{1}{4^{2}}\right) \times\left(1-\frac{1}{5^{2}}\right) \times \cdots \times\left(1-\frac{1}{10^{2}}\right)}$$ | 55 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the coefficient of $x^5$ in the expansion of $(1+2x-3x^2)^6$. | -168 |
|
agentica-org/DeepScaleR-Preview-Dataset | For a point $P = (a, a^2)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$ , $P_2 = (a_2, a_2^2)$ , $P_3 = (a_3, a_3^2)$ , such that the intersections of the lines $\ell(P_1)$ , $\ell(P_2)$ , $\ell(P_3)$ form an equilateral triangle $\triangle$ . Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles. | \[
\boxed{y = -\frac{1}{4}}
\] |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $α$ and $β$ are acute angles, and it is given that $\sin \alpha =\frac{3}{5}$ and $\cos (\alpha +\beta )=\frac{5}{13}$, calculate the value of $\cos \beta$. | \frac{56}{65} |
|
agentica-org/DeepScaleR-Preview-Dataset | The angle can be represented by the two uppercase letters on its sides and the vertex letter. The angle in the diagram $\angle A O B$ symbol ("$\angle$" represents angle) can also be represented by $\angle O$ (when there is only one angle). In the triangle $\mathrm{ABC}$ below, given $\angle B A O = \angle C A O$, $\angle C B O = \angle A B O$, $\angle A C O = \angle B C O$, and $\angle A O C = 110^{\circ}$, find $\angle C B O =$. | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $\mathbb Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\mathbb Z\rightarrow\mathbb Z$ and $g:\mathbb Z\rightarrow\mathbb Z$ satisfying \[f(g(x))=x+a\quad\text{and}\quad g(f(x))=x+b\] for all integers $x$ . | \[ |a| = |b| \] |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x)=\cos^4x+2\sin x\cos x-\sin^4x$
$(1)$ Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$. | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Points $P$ and $Q$ are on a circle of radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of the line segment $PR$. | \sqrt{98 - 14\sqrt{33}} |
|
agentica-org/DeepScaleR-Preview-Dataset | A magic square is an array of numbers in which the sum of the numbers in each row, in each column, and along the two main diagonals are equal. The numbers in the magic square shown are not written in base 10. For what base will this be a magic square?
[asy]
unitsize(0.75cm);
for (int i=0; i<4; ++i) {
draw((0,i)--(3,i),linewidth(0.7));
draw((i,0)--(i,3),linewidth(0.7));
}
label("1",(1.5,2),N);
label("2",(2.5,0),N);
label("3",(0.5,1),N);
label("4",(0.5,0),N);
label("10",(1.5,1),N);
label("11",(2.5,2),N);
label("12",(2.5,1),N);
label("13",(0.5,2),N);
label("14",(1.5,0),N);
[/asy] | 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | [i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\] | {a_n = 2^{n-1}} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6. | 936 |
|
agentica-org/DeepScaleR-Preview-Dataset | Six rhombi of side length 1 are arranged as shown. What is the perimeter of this figure? | 14 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the minimum value of the function \( f(x)=\cos 4x + 6\cos 3x + 17\cos 2x + 30\cos x \) for \( x \in \mathbb{R} \). | -18 |
|
agentica-org/DeepScaleR-Preview-Dataset | An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | 6 |
|
agentica-org/DeepScaleR-Preview-Dataset | Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color matches the color previously placed by Alex or Bonnie respectively. Calculate the probability that at least one box receives $3$ blocks all of the same color.
A) $\frac{27}{64}$
B) $\frac{29}{64}$
C) $\frac{37}{64}$
D) $\frac{55}{64}$
E) $\frac{63}{64}$ | \frac{37}{64} |
|
agentica-org/DeepScaleR-Preview-Dataset | A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established? | 68 |
|
agentica-org/DeepScaleR-Preview-Dataset | In triangle $ABC$, if median $\overline{AD}$ makes an angle of $30^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$ | 2\sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Robin bought a four-scoop ice cream cone having a scoop each of vanilla, chocolate, strawberry and cherry. In how many orders can the four scoops be stacked on the cone if they are stacked one on top of the other? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the point \( P(-2,5) \) lies on the circle \(\odot C: x^{2}+y^{2}-2x-2y-23=0\), and the line \( l: 3x+4y+8=0 \) intersects \(\odot C\) at points \( A \) and \( B \). Find \(\overrightarrow{AB} \cdot \overrightarrow{BC}\). | -32 |
|
agentica-org/DeepScaleR-Preview-Dataset | An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. | \frac{20}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A number in the set $\{50, 51, 52, 53, ... , 500\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction. | \frac{17}{451} |
|
agentica-org/DeepScaleR-Preview-Dataset | The sequence $3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, ...$ consists of $3$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. Calculate the sum of the first $1024$ terms of this sequence.
A) $4166$
B) $4248$
C) $4303$
D) $4401$ | 4248 |
|
agentica-org/DeepScaleR-Preview-Dataset | Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy] | 36 |
|
agentica-org/DeepScaleR-Preview-Dataset | In triangle $ABC$, $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3 \cos A = 1$. Find all possible values of $\angle C,$ in degrees. Enter all the possible values, separated by commas. | 30^\circ |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the polynomial $p(x),$ with real coefficients, such that
\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\]for all real numbers $x.$ | 6x^3 - 6 |
|
agentica-org/DeepScaleR-Preview-Dataset | A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$ | -1941 |
|
agentica-org/DeepScaleR-Preview-Dataset | In an isosceles triangle \(ABC\) (\(AB = BC\)), the angle bisectors \(AM\) and \(BK\) intersect at point \(O\). The areas of triangles \(BOM\) and \(COM\) are 25 and 30, respectively. Find the area of triangle \(ABC\). | 110 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate \(\left(d^d - d(d-2)^d\right)^d\) when \( d = 4 \). | 1358954496 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the event "Randomly select a point P on the side CD of rectangle ABCD, such that the longest side of ΔAPB is AB", with a probability of 1/3, determine the ratio of AD to AB. | \frac{\sqrt{5}}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is first seen 2 seconds after midnight, and the red signal is first seen 8 seconds after midnight. At what time will both signals be seen together for the first time? | 92 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many distinct four-digit numbers composed of the digits $1$, $2$, $3$, and $4$ are even? | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with point A on line $l$.
(1) Find the polar coordinate equation of curve $C_1$ and the rectangular coordinate equation of line $l$.
(2) After line $l$ is moved 6 units to the left to obtain $l'$, the intersection points of $l'$ and $C_1$ are M and N. Find the polar coordinate equation of $l'$ and the length of $|MN|$. | 2\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the product of $1011_2 \cdot 101_2$. Express your answer in base 2. | 110111 |
|
agentica-org/DeepScaleR-Preview-Dataset | Alexio has 120 cards numbered 1-120, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 3, 4, or 7? Express your answer as a common fraction. | \dfrac{69}{120} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $F_{1}$ and $F_{2}$ are two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\angle AF_{1}F_{2}=45^{\circ}$, calculate the area of triangle $AF_{1}F_{2}$. | \frac{7}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | The points $(-1,4)$ and $(2,-3)$ are adjacent vertices of a square. What is the area of the square? | 58 |
|
agentica-org/DeepScaleR-Preview-Dataset | (12 points in total) 4 students are sitting in a row to watch a movie, and there are 6 seats in the row.
(1) How many seating arrangements are there such that there is exactly one person between students A and B, and there are no empty seats between them?
(2) How many seating arrangements are there such that all empty seats are not adjacent? | 240 |
|
agentica-org/DeepScaleR-Preview-Dataset | Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters, and find the total number of different arrangements. | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \_\_\_\_\_\_\_\_. | 2.376 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the area of the portion of the circle defined by \(x^2 - 10x + y^2 = 9\) that lies above the \(x\)-axis and to the left of the line \(y = x-5\)? | 4.25\pi |
|
agentica-org/DeepScaleR-Preview-Dataset | A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th game for their mean number of points-per-game to exceed 22? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the area of the region enclosed by the graph of \( |x-75| + |y| = \left|\frac{x}{3}\right| \). | 703.125 |
|
agentica-org/DeepScaleR-Preview-Dataset | In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is | 34 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder. | 4\pi |
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agentica-org/DeepScaleR-Preview-Dataset | Let $P$, $Q$, and $R$ be points on a circle of radius $24$. If $\angle PRQ = 40^\circ$, what is the circumference of the minor arc $PQ$? Express your answer in terms of $\pi$. | \frac{32\pi}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | One of the factors of $x^4+4$ is: | $x^2-2x+2$ |
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agentica-org/DeepScaleR-Preview-Dataset | The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1. For example, $2^{4}=16$ is a perfect power, while $2\times3=6$ is not a perfect power)? Express your answer as a common fraction. | \frac{133}{150} |
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agentica-org/DeepScaleR-Preview-Dataset | Given that Let \\(S_{n}\\) and \\(T_{n}\\) be the sums of the first \\(n\\) terms of the arithmetic sequences \\(\{a_{n}\}\\) and \\(\{b_{n}\}\\), respectively, and \\( \frac {S_{n}}{T_{n}}= \frac {n}{2n+1} (n∈N^{*})\\), determine the value of \\( \frac {a_{6}}{b_{6}}\\). | \frac{11}{23} |
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agentica-org/DeepScaleR-Preview-Dataset | What is the result of adding 0.45 to 52.7 and then subtracting 0.25? | 52.9 |
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agentica-org/DeepScaleR-Preview-Dataset | 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 |
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agentica-org/DeepScaleR-Preview-Dataset | Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \). | -1 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and
\[ pq + pr + ps + qr + qs + rs = 20. \]
Find the largest possible value of $s$. | \frac{5 + \sqrt{105}}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice? | 5\pi |
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agentica-org/DeepScaleR-Preview-Dataset | If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? | 182 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the distance from Boguli to Bolifoyn? | 10 |
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agentica-org/DeepScaleR-Preview-Dataset | The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | 12 |
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agentica-org/DeepScaleR-Preview-Dataset | $100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?
(D. Fomin, Leningrad) | 101 |
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agentica-org/DeepScaleR-Preview-Dataset | What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped? | 1/24 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified. | 5 |
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agentica-org/DeepScaleR-Preview-Dataset | Find all real numbers $k$ such that $r^{4}+k r^{3}+r^{2}+4 k r+16=0$ is true for exactly one real number $r$. | \pm \frac{9}{4} |
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agentica-org/DeepScaleR-Preview-Dataset | Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$. | 5.2 |
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