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agentica-org/DeepScaleR-Preview-Dataset | What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$? | 46656 |
|
agentica-org/DeepScaleR-Preview-Dataset | On a sphere, there are four points A, B, C, and D satisfying $AB=1$, $BC=\sqrt{3}$, $AC=2$. If the maximum volume of tetrahedron D-ABC is $\frac{\sqrt{3}}{2}$, then the surface area of this sphere is _______. | \frac{100\pi}{9} |
|
agentica-org/DeepScaleR-Preview-Dataset | A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenant decides to arrange the recruits again but cannot remember how many rows there were initially. So, he orders them to form as many rows as his age. It turns out that each row again has the same number of soldiers, but in each row, there are 4 more soldiers than there were in the original arrangement. How old is the lieutenant? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone and that hitting the bullseye is worth 315 points. How many points is hitting the blue (second to last) zone worth? | 45 |
|
agentica-org/DeepScaleR-Preview-Dataset | A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ? | 4:26.8 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate $(2 + 1)(2^2 + 1^2)(2^4 + 1^4)$. | 255 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate $\lfloor-5.77\rfloor+\lceil-3.26\rceil+\lfloor15.93\rfloor+\lceil32.10\rceil$. | 39 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$ . | 23 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a sequence \( x_{n} \), satisfying \( (n+1) x_{n+1}=x_{n}+n \), and \( x_{1}=2 \), find \( x_{2009} \). | \frac{2009! + 1}{2009!} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals. | \[
f(n) = \left\lfloor \frac{(n-1)^2}{2} \right\rfloor
\] |
|
agentica-org/DeepScaleR-Preview-Dataset | Given \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \). | \frac{1}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.
(Ⅰ) Find the equation of the ellipse $C$.
(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y = -\frac{1}{m}x + \frac{1}{2}$, find the maximum value of the area of $\triangle OAB$ ($O$ is the origin). | \frac{\sqrt{2}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? | 120^\circ |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the product of the numerator and the denominator when $0.\overline{009}$ is expressed as a fraction in lowest terms? | 111 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $13^{3n}=\left(\frac{1}{13}\right)^{n-24}$, find $n$. | 6 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the remainder when $9^{2048}$ is divided by $50$? | 21 |
|
agentica-org/DeepScaleR-Preview-Dataset | After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 handshakes at the event, what is the fewest number of handshakes the coaches could have participated in? | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\] | a = 4/9 |
|
agentica-org/DeepScaleR-Preview-Dataset | Dots are placed two units apart both horizontally and vertically on a coordinate grid. Calculate the number of square units enclosed by the polygon formed by connecting these dots:
[asy]
size(90);
pair a=(0,0), b=(20,0), c=(20,20), d=(40,20), e=(40,40), f=(20,40), g=(0,40), h=(0,20);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
dot(g);
dot(h);
draw(a--b--c--d--e--f--g--h--cycle);
[/asy] | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the limits:
1) \(\lim_{x \to 3}\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)\)
2) \(\lim_{x \to \infty}\left(\sqrt{x^2 + 1}-x\right)\)
3) \(\lim_{n \to \infty} 2^n \sin \frac{x}{2^n}\)
4) \(\lim_{x \to 1}(1-x) \tan \frac{\pi x}{2}\). | \frac{2}{\pi} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the matrix that corresponds to rotating about the origin by an angle of $120^\circ$ counter-clockwise. | \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix} |
|
agentica-org/DeepScaleR-Preview-Dataset | A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded? | 33.33\% |
|
agentica-org/DeepScaleR-Preview-Dataset | How many solutions does the equation $\tan x = \tan (\tan x + \frac{\pi}{4})$ have in the interval $0 \leq x \leq \tan^{-1} 1884$? | 600 |
|
agentica-org/DeepScaleR-Preview-Dataset | Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | \frac{335}{2011} |
|
agentica-org/DeepScaleR-Preview-Dataset | The numbers 2, 3, 5, 7, 11, 13, 17, 19 are arranged in a multiplication table, with four along the top and the other four down the left. The multiplication table is completed and the sum of the sixteen entries is tabulated. What is the largest possible sum of the sixteen entries?
\[
\begin{array}{c||c|c|c|c|}
\times & a & b & c & d \\ \hline \hline
e & & & & \\ \hline
f & & & & \\ \hline
g & & & & \\ \hline
h & & & & \\ \hline
\end{array}
\] | 1482 |
|
agentica-org/DeepScaleR-Preview-Dataset | An up-right path from $(a, b) \in \mathbb{R}^{2}$ to $(c, d) \in \mathbb{R}^{2}$ is a finite sequence $(x_{1}, y_{1}), \ldots,(x_{k}, y_{k})$ of points in $\mathbb{R}^{2}$ such that $(a, b)=(x_{1}, y_{1}),(c, d)=(x_{k}, y_{k})$, and for each $1 \leq i<k$ we have that either $(x_{i+1}, y_{i+1})=(x_{i}+1, y_{i})$ or $(x_{i+1}, y_{i+1})=(x_{i}, y_{i}+1)$. Let $S$ be the set of all up-right paths from $(-400,-400)$ to $(400,400)$. What fraction of the paths in $S$ do not contain any point $(x, y)$ such that $|x|,|y| \leq 10$? Express your answer as a decimal number between 0 and 1. | 0.2937156494680644 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations. | 126 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$ with eccentricity $e$, and the parabola $x=2py^{2}$ with focus at $(e,0)$, find the value of the real number $p$. | \frac{1}{16} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $x > 0$, $y > 0$, and the inequality $2\log_{\frac{1}{2}}[(a-1)x+ay] \leq 1 + \log_{\frac{1}{2}}(xy)$ always holds, find the minimum value of $4a$. | \sqrt{6}+\sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | The product $$ \left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right) $$ can be written as $\frac{r}{s2^t}$ where $r$ , $s$ , and $t$ are positive integers and $r$ and $s$ are odd and relatively prime. Find $r+s+t$ . | 3769 |
|
agentica-org/DeepScaleR-Preview-Dataset | The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$. | \frac{125}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. Find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\] | \frac{49}{58} |
|
agentica-org/DeepScaleR-Preview-Dataset | The sum of two numbers is $30$. Their difference is $4$. What is the larger of the two numbers? | 17 |
|
agentica-org/DeepScaleR-Preview-Dataset | 9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained? | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. | 70 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. | 14 |
|
agentica-org/DeepScaleR-Preview-Dataset | There are 306 different cards with numbers \(3, 19, 3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}\) (each card has exactly one number, and each number appears exactly once). How many ways can you choose 2 cards such that the product of the numbers on the selected cards is a perfect square? | 17328 |
|
agentica-org/DeepScaleR-Preview-Dataset | Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
[asy]
pair A,B,C,D,E,F,G,H,J;
A=(20,20(2+sqrt(2)));
B=(20(1+sqrt(2)),20(2+sqrt(2)));
C=(20(2+sqrt(2)),20(1+sqrt(2)));
D=(20(2+sqrt(2)),20);
E=(20(1+sqrt(2)),0);
F=(20,0);
G=(0,20);
H=(0,20(1+sqrt(2)));
J=(10(2+sqrt(2)),10(2+sqrt(2)));
draw(A--B);
draw(B--C);
draw(C--D);
draw(D--E);
draw(E--F);
draw(F--G);
draw(G--H);
draw(H--A);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(J);
label("$A$",A,NNW);
label("$B$",B,NNE);
label("$C$",C,ENE);
label("$D$",D,ESE);
label("$E$",E,SSE);
label("$F$",F,SSW);
label("$G$",G,WSW);
label("$H$",H,WNW);
label("$J$",J,SE);
size(4cm);
[/asy] | 1152 |
|
agentica-org/DeepScaleR-Preview-Dataset | The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. | 3\sqrt{13} |
|
agentica-org/DeepScaleR-Preview-Dataset | The base of the quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is a rhombus \( A B C D \) with \( B D = 12 \) and \( \angle B A C = 60^{\circ} \). A sphere passes through the vertices \( D, A, B, B_{1}, C_{1}, D_{1} \).
a) Find the area of the circle obtained in the cross section of the sphere by the plane passing through points \( A_{1}, B_{1} \), and \( C_{1} \).
b) Find the angle \( A_{1} C B \).
c) Given that the radius of the sphere is 8, find the volume of the prism. | 192\sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Convert $BD4_{16}$ to base 4. | 233110_4 |
|
agentica-org/DeepScaleR-Preview-Dataset | Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$. | 120^\circ |
|
agentica-org/DeepScaleR-Preview-Dataset | If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]what is the value of
\[
z^{2010} + z^{-2010} \, ?
\] | -2 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles? | 0.4667 |
|
agentica-org/DeepScaleR-Preview-Dataset | Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability of $\frac{2}{3}$ of winning each game in project $A$, and a probability of $\frac{1}{2}$ of winning each game in project $B$, with no influence between games.
$(1)$ Find the probability of player A winning in project $A$ and project $B$ respectively.
$(2)$ Let $X$ be the number of projects player A wins. Find the distribution and mathematical expectation of $X$. | \frac{209}{162} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$ , where $a,\,b,\,c,\,d$ are positive integers. | 11 |
|
agentica-org/DeepScaleR-Preview-Dataset | The area of a triangle \(ABC\) is \(\displaystyle 40 \text{ cm}^2\). Points \(D, E\) and \(F\) are on sides \(AB, BC\) and \(CA\) respectively. If \(AD = 3 \text{ cm}, DB = 5 \text{ cm}\), and the area of triangle \(ABE\) is equal to the area of quadrilateral \(DBEF\), find the area of triangle \(AEC\) in \(\text{cm}^2\). | 15 |
|
agentica-org/DeepScaleR-Preview-Dataset | Gustave has 15 steel bars of masses $1 \mathrm{~kg}, 2 \mathrm{~kg}, 3 \mathrm{~kg}, \ldots, 14 \mathrm{~kg}, 15 \mathrm{~kg}$. He also has 3 bags labelled $A, B, C$. He places two steel bars in each bag so that the total mass in each bag is equal to $M \mathrm{~kg}$. How many different values of $M$ are possible? | 19 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $F$ be the set of functions from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$ such that $f(3x) \geq f(f(2x)) + x$. Maximize $\alpha$ such that $\forall x \geq 0, \forall f \in F, f(x) \geq \alpha x$. | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initially, the organisation had twenty people total. How many total people will be in the organization six years from now? | 10895 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the smallest positive integer with exactly 10 positive integer divisors? | 48 |
|
agentica-org/DeepScaleR-Preview-Dataset | Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_. | 60 |
|
agentica-org/DeepScaleR-Preview-Dataset | $f(x)$ is a monic polynomial such that $f(0)=4$ and $f(1)=10$. If $f(x)$ has degree $2$, what is $f(x)$? Express your answer in the form $ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers. | x^2+5x+4 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. | 2 - \sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A workshop produces products of types $A$ and $B$. Producing one unit of product $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while producing one unit of product $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling one unit of product $A$ is 80 thousand rubles, and for product $B$, it's 100 thousand rubles. The daily steel reserve is 700 kg, and the reserve of non-ferrous metals is 642 kg. How many units of products $A$ and $B$ should be produced per shift to maximize profit without exceeding the available resources? Record the maximum profit (in thousands of rubles) that can be obtained under these conditions as an integer without indicating the unit. | 2180 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that point $A(1,1)$ is a point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\_1$, $F\_2$ are the two foci of the ellipse such that $|AF\_1|+|AF\_2|=4$.
(I) Find the standard equation of the ellipse;
(II) Find the equation of the tangent line to the ellipse that passes through $A(1,1)$;
(III) Let points $C$ and $D$ be two points on the ellipse such that the slopes of lines $AC$ and $AD$ are complementary. Determine whether the slope of line $CD$ is a constant value. If it is, find the value; if not, explain the reason. | \frac{1}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $\sin \theta$ and $\cos \theta$ are the two roots of the equation $4x^{2}-4mx+2m-1=0$, and $\frac {3\pi}{2} < \theta < 2\pi$, find the angle $\theta$. | \frac {5\pi}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $f(x) = a\cos x - \left(x - \frac{\pi}{2}\right)\sin x$, where $x \in \left[0, \frac{\pi}{2}\right]$.
$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$.
$(2)$ If $f(x) \leq 0$ always holds, find the maximum value of the real number $a$. | -1 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the next 3 days, a meteorological station forecasts the weather with an accuracy rate of 0.8. The probability that the forecast is accurate for at least two consecutive days is ___. | 0.768 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $m\in R$, determine the condition for the lines $l_{1}$: $mx+2y-1=0$ and $l_{2}$: $3x+\left(m+1\right)y+1=0$ to be perpendicular. | -\frac{2}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 200 or 600 when divided by 900. Under this proposal, how many leap years will there be that end in double zeroes between 1996 and 4096? | 5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\) | \frac{\sqrt{3}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Olga Ivanovna, the homeroom teacher of class 5B, is staging a "Mathematical Ballet". She wants to arrange the boys and girls so that every girl has exactly 2 boys at a distance of 5 meters from her. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | A $ 4\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ 16$ cells were black. Find all possible values of $ n$. | 6, 8, 10, 12, 14, 16, \ldots |
|
agentica-org/DeepScaleR-Preview-Dataset | When simplified, what is the value of $\sqrt{3} \times 3^{\frac{1}{2}} + 12 \div 3 \times 2 - 4^{\frac{3}{2}}$? | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given 6 persons, with the restriction that person A and person B cannot visit Paris, calculate the total number of distinct selection plans for selecting 4 persons to visit Paris, London, Sydney, and Moscow, where each person visits only one city. | 240 |
|
agentica-org/DeepScaleR-Preview-Dataset | On a section of the map, three roads form a right triangle. When motorcyclists were asked about the distance between $A$ and $B$, one of them responded that after traveling from $A$ to $B$, then to $C$, and back to $A$, his odometer showed 60 km. The second motorcyclist added that he knew by chance that $C$ was 12 km from the road connecting $A$ and $B$, i.e., from point $D$. Then the questioner, making a very simple mental calculation, said:
- It's clear, from $A$ to $B \ldots$
Can the reader quickly determine this distance as well? | 22.5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $ | (0, 0, 2), (1, 1, 2), (2, 2, 3) |
|
agentica-org/DeepScaleR-Preview-Dataset | A typesetter scattered part of a set - a set of a five-digit number that is a perfect square, written with the digits $1, 2, 5, 5,$ and $6$. Find all such five-digit numbers. | 15625 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the probability that a randomly drawn positive factor of $60$ is less than $7$? | \frac{1}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | The distance from city $A$ to city $B$ is $999$ km. Along the highway leading from $A$ to $B$, there are kilometer markers indicating the distances from the marker to $A$ and $B$ as shown:
How many of these markers use only two different digits to indicate both distances? | 40 |
|
agentica-org/DeepScaleR-Preview-Dataset | A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome? | \frac{11}{30} |
|
agentica-org/DeepScaleR-Preview-Dataset | Determine the smallest positive integer $m$ such that $11m-3$ and $8m + 5$ have a common factor greater than $1$. | 108 |
|
agentica-org/DeepScaleR-Preview-Dataset | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? | 250 |
|
agentica-org/DeepScaleR-Preview-Dataset | For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) represent the number of \(P\)-type and \(Q\)-type numbers respectively. Find the value of \(N(P) - N(Q)\). | 285 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let the set \( T \) consist of integers between 1 and \( 2^{30} \) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \( T \), what is the probability that it is divisible by 9? | 5/29 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,200 \pi).$ | 200 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | \frac{3\sqrt{2}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given \( |z|=2 \) and \( u=\left|z^{2}-z+1\right| \), find the minimum value of \( u \) where \( z \in \mathbf{C} \). | \frac{3}{2} \sqrt{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | For a sample of size \( n = 41 \), a biased estimate \( D_{\text{в}} = 3 \) of the population variance is found. Find the unbiased estimate of the population variance. | 3.075 |
|
agentica-org/DeepScaleR-Preview-Dataset | Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\pi$ inches per minute, the other crawling along the smaller circle at $2.5\pi$ inches per minute. How many minutes is it before their next meeting at point P? | 40 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles? | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | The equation of curve $C$ is $\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$, where $m$ and $n$ are the numbers obtained by rolling a die twice in succession. Let event $A$ be "The equation $\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$ represents an ellipse with foci on the $x$-axis". Then, the probability of event $A$ occurring, $P(A)=$ . | \frac{5}{12} |
|
agentica-org/DeepScaleR-Preview-Dataset | The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit | 0 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the distance between the planes $x - 3y + 3z = 8$ and $2x - 6y + 6z = 2.$ | \frac{7 \sqrt{19}}{19} |
|
agentica-org/DeepScaleR-Preview-Dataset | So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother. | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\). | 35 |
|
agentica-org/DeepScaleR-Preview-Dataset | There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that
\[
\cos^7 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta + a_6 \cos 6 \theta + a_7 \cos 7 \theta
\]
for all angles $\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2 + a_7^2.$ | \frac{1716}{4096} |
|
agentica-org/DeepScaleR-Preview-Dataset | A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | 21 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 225 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the value of $n$ if $2^{n}=8^{20}$? | 60 |
|
agentica-org/DeepScaleR-Preview-Dataset | In a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion? | 255 |
|
agentica-org/DeepScaleR-Preview-Dataset | Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? | -10 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$. | 820 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \). | 3/4 |
|
agentica-org/DeepScaleR-Preview-Dataset | 100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}
Criteria: To pass, one must answer at least 6 questions correctly. Based on the table, calculate the minimum number of people who passed. | 62 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps. | 91 |
|
agentica-org/DeepScaleR-Preview-Dataset | Maria bakes a $24$-inch by $30$-inch pan of brownies, and the brownies are cut into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies the pan contains. | 60 |
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