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agentica-org/DeepScaleR-Preview-Dataset | Let $ABC$ be a triangle with $BC = a$ , $CA = b$ , and $AB = c$ . The $A$ -excircle is tangent to $\overline{BC}$ at $A_1$ ; points $B_1$ and $C_1$ are similarly defined.
Determine the number of ways to select positive integers $a$ , $b$ , $c$ such that
- the numbers $-a+b+c$ , $a-b+c$ , and $a+b-c$ are even integers at most 100, and
- the circle through the midpoints of $\overline{AA_1}$ , $\overline{BB_1}$ , and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$ .
| 125000 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $y= \frac{x^2+2x+8}{x-4}$, at what value of $x$ will there be a vertical asymptote? | 4 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions? | 160 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the magnitude of vector $\overrightarrow {a}$ is 1, the magnitude of vector $\overrightarrow {b}$ is 2, and the magnitude of $\overrightarrow {a}+ \overrightarrow {b}$ is $\sqrt {7}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac {\pi}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $z = \cos \frac{4 \pi}{7} + i \sin \frac{4 \pi}{7}.$ Compute
\[\frac{z}{1 + z^2} + \frac{z^2}{1 + z^4} + \frac{z^3}{1 + z^6}.\] | -2 |
|
agentica-org/DeepScaleR-Preview-Dataset | Define a function $h(x),$ for positive integer values of $x,$ by \[h(x) = \left\{\begin{aligned} \log_2 x & \quad \text{ if } \log_2 x \text{ is an integer} \\ 1 + h(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $h(100).$ | 35 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the function $f(x)=\sin x+a\cos x$ has a symmetry axis on $x=\frac{5π}{3}$, determine the maximum value of the function $g(x)=a\sin x+\cos x$. | \frac {2\sqrt {3}}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,100] = \gcd(n,100)+450~?$$ | 250 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest? | $c$ |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the tower function $T(n)$ defined by $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n \geq 1$, calculate the largest integer $k$ for which $\underbrace{\log_3\log_3\log_3\ldots\log_3B}_{k\text{ times}}$ is defined, where $B = (T(2005))^A$ and $A = (T(2005))^{T(2005)}$. | 2005 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus. | \frac{90\sqrt{261}}{261} |
|
agentica-org/DeepScaleR-Preview-Dataset | For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way. | n \geq 14 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $n$ be given, $n \geq 4$, and suppose that $P_1, P_2, \dots, P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i$. What is the probability that at least one of the vertex angles of this polygon is acute? | n(n-2) 2^{-n+1} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find all solutions to
\[\sqrt{x + 3 - 4 \sqrt{x - 1}} + \sqrt{x + 8 - 6 \sqrt{x - 1}} = 1.\] | [5,10] |
|
agentica-org/DeepScaleR-Preview-Dataset | You want to arrange the numbers $1,2,3, \ldots, 25$ in a sequence with the following property: if $n$ is divisible by $m$, then the $n$th number is divisible by the $m$ th number. How many such sequences are there? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.
[asy] pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("$A$",A,dir(0)); label("$B$",B,dir(75)); label("$C$",C,dir(90)); label("$D$",D,dir(105)); label("$E$",E,dir(180)); label("$F$",F,dir(225)); label("$G$",G,dir(260)); label("$H$",H,dir(280)); label("$I$",I,dir(315)); [/asy] | 58 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given vectors $\overrightarrow {a}=( \sqrt {3}\sin x, m+\cos x)$ and $\overrightarrow {b}=(\cos x, -m+\cos x)$, and a function $f(x)= \overrightarrow {a}\cdot \overrightarrow {b}$
(1) Find the analytical expression of function $f(x)$;
(2) When $x\in[- \frac {\pi}{6}, \frac {\pi}{3}]$, the minimum value of $f(x)$ is $-4$. Find the maximum value of the function $f(x)$ and the corresponding $x$ value. | \frac {\pi}{6} |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the sum of the first ten positive multiples of $13$? | 715 |
|
agentica-org/DeepScaleR-Preview-Dataset | The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a "wealth number." Among the numbers from 1 to 100, how many "wealth numbers" are there? | 88 |
|
agentica-org/DeepScaleR-Preview-Dataset | Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? | 10 |
|
agentica-org/DeepScaleR-Preview-Dataset | $ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\angle{AB'C'} \cong \angle{B'EA}$ and $\angle{B'C'A} = 90^\circ$. If $AB' = 3$ and $BE = 12$, compute the area of rectangle $ABCD$ in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Compute $a + b + c$. | 57 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $a$ and $b$ be even integers such that $ab = 144$. Find the minimum value of $a + b$. | -74 |
|
agentica-org/DeepScaleR-Preview-Dataset | Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3,\ldots, 1000\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimension $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | In triangle \( \triangle ABC \), \(\angle A\) is the smallest angle, \(\angle B\) is the largest angle, and \(2 \angle B = 5 \angle A\). If the maximum value of \(\angle B\) is \(m^{\circ}\) and the minimum value of \(\angle B\) is \(n^{\circ}\), then find \(m + n\). | 175 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many six-digit numbers are there in which only the middle two digits are the same? | 90000 |
|
agentica-org/DeepScaleR-Preview-Dataset | A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters? | 2880 |
|
agentica-org/DeepScaleR-Preview-Dataset | A rotating disc is divided into five equal sectors labeled $A$, $B$, $C$, $D$, and $E$. The probability of the marker stopping on sector $A$ is $\frac{1}{5}$, the probability of it stopping in $B$ is $\frac{1}{5}$, and the probability of it stopping in sector $C$ is equal to the probability of it stopping in sectors $D$ and $E$. What is the probability of the marker stopping in sector $C$? Express your answer as a common fraction. | \frac{1}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O. | 48\pi |
|
agentica-org/DeepScaleR-Preview-Dataset | A group of children, numbering between 50 and 70, attended a spring math camp. To celebrate Pi Day (March 14), they decided to give each other squares if they were just acquaintances and circles if they were friends. Andrey noted that each boy received 3 circles and 8 squares, and each girl received 2 squares and 9 circles. Katya discovered that the total number of circles and squares given out was the same. How many children attended the camp? | 60 |
|
agentica-org/DeepScaleR-Preview-Dataset | Three positive reals \( x \), \( y \), and \( z \) are such that
\[
\begin{array}{l}
x^{2}+2(y-1)(z-1)=85 \\
y^{2}+2(z-1)(x-1)=84 \\
z^{2}+2(x-1)(y-1)=89
\end{array}
\]
Compute \( x + y + z \). | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?
| 465 |
|
agentica-org/DeepScaleR-Preview-Dataset | The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. | 12.8470588235 |
|
agentica-org/DeepScaleR-Preview-Dataset | Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive? | 9 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a,\frac{b}{a},1\}=\{a^{2},a+b,0\}$, determine the value of $a^{2023}+b^{2024}$. | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let set $A=\{x|\left(\frac{1}{2}\right)^{x^2-4}>1\}$, $B=\{x|2<\frac{4}{x+3}\}$
(1) Find $A\cap B$
(2) If the solution set of the inequality $2x^2+ax+b<0$ is $B$, find the values of $a$ and $b$. | -6 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $ABC$ be a triangle with $\angle BAC=117^\circ$ . The angle bisector of $\angle ABC$ intersects side $AC$ at $D$ . Suppose $\triangle ABD\sim\triangle ACB$ . Compute the measure of $\angle ABC$ , in degrees. | 42 |
|
agentica-org/DeepScaleR-Preview-Dataset | The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000. | 371 |
|
agentica-org/DeepScaleR-Preview-Dataset | The matrices
\[\begin{pmatrix} 3 & -8 \\ a & 11 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 11 & b \\ 4 & 3 \end{pmatrix}\]are inverses. Enter the ordered pair $(a,b).$ | (-4,8) |
|
agentica-org/DeepScaleR-Preview-Dataset | A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix},$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = -7.$ | \begin{pmatrix} -17 \\ 76 \end{pmatrix} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
[asy]
defaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);
dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));
} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));
[/asy] | 300 |
|
agentica-org/DeepScaleR-Preview-Dataset | For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses. | 409 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked? | 32 |
|
agentica-org/DeepScaleR-Preview-Dataset | A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \neq 0$ and $A \neq B$. (The hundreds digit of the product is zero.) For example, the integers in the set $\{3,5,7,13,33\}$ have a product of 45045. In total, how many different sets of five different odd positive integers have these properties? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | 33 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ have left and right foci $F_{1}$, $F_{2}$, and line $l$ passing through point $F_{1}$. If the point $P$, which is the symmetric point of $F_{2}$ with respect to line $l$, lies exactly on ellipse $C$, and $\overrightarrow{F_1P} \cdot \overrightarrow{F_1F_2} = \frac{1}{2}a^2$, find the eccentricity of $C$. | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | In a bus, there are single and double seats. In the morning, 13 people were sitting in the bus, and there were 9 completely free seats. In the evening, 10 people were sitting in the bus, and there were 6 completely free seats. How many seats are there in the bus? | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | 12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form such a team? | 36 |
|
agentica-org/DeepScaleR-Preview-Dataset | Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first? | 1500 |
|
agentica-org/DeepScaleR-Preview-Dataset | For all $m, n$ satisfying $1 \leqslant n \leqslant m \leqslant 5$, the number of distinct hyperbolas represented by the polar equation $\rho = \frac{1}{1 - c_{m}^{n} \cos \theta}$ is: | 15 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the Cartesian coordinate system $xOy$, let $D$ be the region represented by the inequality $|x| + |y| \leq 1$, and $E$ be the region consisting of points whose distance from the origin is no greater than 1. If a point is randomly thrown into $E$, the probability that this point falls into $D$ is. | \frac{4}{\pi} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$ , for integers $i,j$ with $0\leq i,j\leq n$ , such that:
1. for all $0\leq i,j\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and
2. $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$ .
Contents 1 Solution 1 2 Solution 2 2.1 Lemma 2.2 Filling in the rest of the grid 2.3 Finishing off 3 See also | \[
(2n)! \cdot 2^{n^2}
\] |
|
agentica-org/DeepScaleR-Preview-Dataset | I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color? | \frac{4}{9} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that an ellipse has the equation $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $e = \frac {\sqrt {6}}{3}$. The distance from the origin to the line that passes through points $A(0,-b)$ and $B(a,0)$ is $\frac {\sqrt {3}}{2}$.
$(1)$ Find the equation of the ellipse.
$(2)$ Given the fixed point $E(-1,0)$, if the line $y = kx + 2 \ (k \neq 0)$ intersects the ellipse at points $C$ and $D$, is there a value of $k$ such that the circle with diameter $CD$ passes through point $E$? Please provide an explanation. | \frac {7}{6} |
|
agentica-org/DeepScaleR-Preview-Dataset | Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board? | 13 |
|
agentica-org/DeepScaleR-Preview-Dataset | If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number). | 243 |
|
agentica-org/DeepScaleR-Preview-Dataset | Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly one of the triplets is in the starting lineup? | 1386 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \ldots + a_{12}$. | 364 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the expected value of the number formed by rolling a fair 6-sided die with faces numbered 1, 2, 3, 5, 7, 9 infinitely many times. | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m. | \frac{3}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4\) cube. How many different possibilities are there for this total? | 49 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ are $(S_n)$ and $(T_n)$, respectively. If for any positive integer $n$, $\frac{S_n}{T_n}=\frac{2n-5}{3n-5}$, determine the value of $\frac{a_7}{b_2+b_8}+\frac{a_3}{b_4+b_6}$. | \frac{13}{22} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the line $l: 4x+3y-8=0$ passes through the center of the circle $C: x^2+y^2-ax=0$ and intersects circle $C$ at points A and B, with O as the origin.
(I) Find the equation of circle $C$.
(II) Find the equation of the tangent to circle $C$ at point $P(1, \sqrt {3})$.
(III) Find the area of $\triangle OAB$. | \frac{16}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that
\[
2^{-\frac{5}{3} + \sin 2\theta} + 2 = 2^{\frac{1}{3} + \sin \theta},
\]
compute \(\cos 2\theta.\) | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the largest real number $c$ such that $$ \sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i $$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$ | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Shift the graph of the function $f(x)=2\sin(2x+\frac{\pi}{6})$ to the left by $\frac{\pi}{12}$ units, and then shift it upwards by 1 unit to obtain the graph of $g(x)$. If $g(x_1)g(x_2)=9$, and $x_1, x_2 \in [-2\pi, 2\pi]$, then find the maximum value of $2x_1-x_2$. | \frac {49\pi}{12} |
|
agentica-org/DeepScaleR-Preview-Dataset | Solve for $x$: $2^{x-3}=4^{x+1}$ | -5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$ , $x_2y_1-x_1y_2=5$ , and $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of $y_1^2+5y_2^2$ | 23 |
|
agentica-org/DeepScaleR-Preview-Dataset | A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$. | 2009 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6},$ the middle part is: | 17\frac{1}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | In the geometric sequence $\{a_n\}$, $a_2a_3=5$ and $a_5a_6=10$. Calculate the value of $a_8a_9$. | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Express $\sqrt{x} \div\sqrt{y}$ as a common fraction, given:
$\frac{ {\left( \frac{1}{2} \right)}^2 + {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 + {\left( \frac{1}{5} \right)}^2} = \frac{13x}{41y} $ | \frac{10}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the greatest common divisor of $654321$ and $543210$? | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle. | 1.5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $x$ and $y$ be positive real numbers. Find the maximum value of
\[\frac{(x + y)^2}{x^2 + y^2}.\] | 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 70 |
|
agentica-org/DeepScaleR-Preview-Dataset | Vinny wrote down all the single-digit base-$b$ numbers and added them in base $b$, getting $34_b$.
What is $b$? | 8 |
|
agentica-org/DeepScaleR-Preview-Dataset | Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? | 32/17 |
|
agentica-org/DeepScaleR-Preview-Dataset | Cátia leaves school every day at the same time and returns home by bicycle. When she pedals at $20 \mathrm{~km/h}$, she arrives home at $4:30$ PM. If she pedals at $10 \mathrm{~km/h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM? | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? | \frac{1}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the parametric equations of curve $C$ are
$$
\begin{cases}
x=3+ \sqrt {5}\cos \alpha \\
y=1+ \sqrt {5}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}),
$$
with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system.
$(1)$ Find the polar equation of curve $C$;
$(2)$ If the polar equation of a line is $\sin \theta-\cos \theta= \frac {1}{\rho }$, find the length of the chord cut from curve $C$ by the line. | \sqrt {2} |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{4}{3}x - \frac{2y}{5}\right)^8$? | -\frac{114688}{84375} |
|
agentica-org/DeepScaleR-Preview-Dataset | Triangle $ABC$ with vertices of $A(6,2)$, $B(2,5)$, and $C(2,2)$ is reflected over the x-axis to triangle $A'B'C'$. This triangle is reflected over the y-axis to triangle $A''B''C''$. What are the coordinates of point $C''$? | (-2, -2) |
|
agentica-org/DeepScaleR-Preview-Dataset | If \( a \) and \( b \) are given real numbers, and \( 1 < a < b \), then the absolute value of the difference between the average and the median of the four numbers \( 1, a+1, 2a+b, a+b+1 \) is ______. | \frac{1}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | In a hypothetical math competition, contestants are given the problem to find three distinct positive integers $X$, $Y$, and $Z$ such that their product $X \cdot Y \cdot Z = 399$. What is the largest possible value of the sum $X+Y+Z$? | 29 |
|
agentica-org/DeepScaleR-Preview-Dataset | A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the distance in meters the train has traveled during this period. | 405 |
|
agentica-org/DeepScaleR-Preview-Dataset | The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss. | 56 |
|
agentica-org/DeepScaleR-Preview-Dataset | A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 173 |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute $\sin(-30^\circ)$ and verify by finding $\cos(-30^\circ)$, noticing the relationship, and confirming with the unit circle properties. | \frac{\sqrt{3}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | To be able to walk to the center $C$ of a circular fountain, a repair crew places a 16-foot plank from $A$ to $B$ and then a 10-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$ . What is the area of the circular base of the fountain? Express your answer in terms of $\pi$. [asy]
size(250); import olympiad; import geometry; defaultpen(linewidth(0.8));
draw((-10,0)..(-5,0.8)..(0,1)..(5,0.8)..(10,0)^^(10,0)..(5,-0.8)..(0,-1)..(-5,-0.8)..(-10,0));
draw((-10,0)--(-10,-2)^^(10,-2)..(5,-2.8)..(0,-3)..(-5,-2.8)..(-10,-2)^^(10,-2)--(10,0));
draw(origin..(-1,5)..(-4,8));
draw(origin..(1,5)..(4,8));
draw(origin..(-0.5,5)..(-2,8));
draw(origin..(0.5,5)..(2,8));
draw(origin..(-0.2,6)..(-1,10));
draw(origin..(0.2,6)..(1,10));
label("Side View",(0,-2),3*S);
pair C = (25,8);
draw(Circle(C,10));
pair A = C + 10*dir(80);
pair B = C + 10*dir(20);
pair D = midpoint(A--B);
draw(A--B);
draw(C--D);
dot(Label("$A$",align=SW),A);
dot(Label("$B$",align=SE),B);
dot(Label("$C$",align=S),C);
dot(Label("$D$",align=S),D);
for(int i = 0; i < 5; ++i){
draw(C--(C + 5*dir(72*i)));
}
label("Top View",(25,-2),3*S);
[/asy] | 164 \pi \mbox{ square feet} |
|
agentica-org/DeepScaleR-Preview-Dataset | The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values? | 15,450 |
|
agentica-org/DeepScaleR-Preview-Dataset | Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed? | 7 |
|
agentica-org/DeepScaleR-Preview-Dataset | Mary and Sally were once the same height. Since then, Sally grew \( 20\% \) taller and Mary's height increased by half as many centimetres as Sally's height increased. Sally is now 180 cm tall. How tall, in cm, is Mary now? | 165 |
|
agentica-org/DeepScaleR-Preview-Dataset | Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, d\right)$. | \frac{5}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | How many of the positive divisors of 3240 are multiples of 3? | 32 |
|
agentica-org/DeepScaleR-Preview-Dataset | The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ (on the number line) is | \frac{4}{15} |
|
agentica-org/DeepScaleR-Preview-Dataset | In a plane, there are 10 lines, among which 4 lines are parallel to each other. Then, these 10 lines can divide the plane into at most how many parts? | 50 |
|
agentica-org/DeepScaleR-Preview-Dataset | Ali Baba and the thief are dividing a hoard consisting of 100 gold coins, distributed in 10 piles of 10 coins each. Ali Baba chooses 4 piles, places a cup near each of them, and puts a certain number of coins (at least one, but not the entire pile) into each cup. The thief must then rearrange the cups, changing their initial positions, after which the coins are poured from the cups back into the piles they are now next to. Ali Baba then chooses 4 piles from the 10 again, places cups near them, and so on. At any moment, Ali Baba can leave, taking with him any three piles of his choice. The remaining coins go to the thief. What is the maximum number of coins Ali Baba can take with him if the thief also tries to get as many coins as possible? | 72 |
|
agentica-org/DeepScaleR-Preview-Dataset | There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac{1}{2}$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked. | \frac{32}{7} |
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