problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$? | \frac{33}{7} | math | 34 |
A circle passing through the vertex \( P \) of triangle \( PQR \) touches side \( QR \) at point \( F \) and intersects sides \( PQ \) and \( PR \) at points \( M \) and \( N \), respectively, different from vertex \( P \). Find the ratio \( QF : FR \) if it is known that the length of side \( PQ \) is 1.5 times the l... | 1/2 | math | 115 |
Given the sets $M=\{x|x^2 = 2\}$, $N=\{x|ax=1\}$, determine the value of $a$ such that $N \subseteq M$. | 0, -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} | math | 44 |
What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | 10 | math | 44 |
Given vector $\overrightarrow{a} = (2\cos\phi, 2\sin\phi)$, where $\phi \in \left( \frac{\pi}{2}, \pi \right)$, and vector $\overrightarrow{b} = (0, -1)$, find the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{2}-\phi | math | 80 |
Consider a matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \in \{0, 1\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$. | 10 | math | 94 |
The binomial coefficient in the expansion of the binomial \\((2+x)^{n}(n∈N^{*)}\\) is the largest in the 4th and 5th terms. Find the value of $n$. | 7 | math | 47 |
An underground pipeline can be laid by Team A alone in 12 days, and by Team B alone in 24 days. <br/> $(1)$ If these two teams work simultaneously from both ends, how many days will it take to lay the pipeline?<br/> $(2)$ If Team A and Team B cooperate for the first $(a+3)$ days, and the remaining work is completed by ... | 10 \text{ days} | math | 109 |
In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's line passes through (... | 30 | math | 122 |
Let \( a \) be a positive number. Consider the system of equations:
\[
\begin{cases}
y = ax^2 \\
y^2 + 3 = x^2 + 4y
\end{cases}
\]
Determine the number of different real solutions to the system. | 4 | math | 63 |
Mr. Reader has five different Batman comic books, four different X-Men comic books, and three different Calvin & Hobbes comic books. When stacked, all of the Batman comic books are grouped together, all of the X-Men comic books are grouped together, and all of the Calvin & Hobbes comic books are grouped together. In ho... | 103680 | math | 96 |
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set \[
\{-7,-5,-3,-2,2,4,6,13\}.
\]What is the minimum possible value of \[
(a+b+c+d)^{2} + (e+f+g+h)^{2}?
\] | 34 | math | 75 |
There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as ... | 13:10 | math | 121 |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its left focus $F(- \frac{\sqrt{10}}{2}, 0)$, a circle with equation $(x - \frac{\sqrt{10}}{2})^2 + y^2 = 1$ is drawn. A tangent line to the circle passes through the focus and intersects the hyperbola. Determine the eccentricity of... | \frac{\sqrt{10}}{2} | math | 170 |
Given $\omega$ and $t > 0$, the function $$f(x)= \begin{vmatrix} \left.\begin{matrix}\sqrt {3} & \sin\omega x \\ 1 & \cos\omega x\end{matrix}\right.\end{vmatrix}$$ has a minimum positive period of $2\pi$. If the graph of $f(x)$ is shifted to the left by $t$ units, and the resulting graph corresponds to an even function... | \frac {5\pi}{6} | math | 118 |
Let event $A$ be "The line $ax - by = 0$ intersects the circle $(x - 2\sqrt{2})^2 + y^2 = 6$".
(1) If $a$ and $b$ are the numbers obtained by rolling a dice twice, find the probability of event $A$.
(2) If the real numbers $a$ and $b$ satisfy $(a - \sqrt{3})^2 + (b - 1)^2 \leq 4$, find the probability of event $A$. | \frac{1}{2} | math | 118 |
A boy presses his thumb along a vertical rod that rests on a rough horizontal surface. Then he gradually tilts the rod, keeping the component of the force along the rod constant, which is applied to its end. When the tilt angle of the rod to the horizontal is $\alpha=80^{\circ}$, the rod begins to slide on the surface.... | 0.17 | math | 115 |
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no... | 52 | math | 74 |
If two non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $|\overrightarrow{a}+ \overrightarrow{b}|=|\overrightarrow{a}- \overrightarrow{b}|=2|\overrightarrow{a}|$, then the angle between $\overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}- \overrightarrow{b}$ is ______. | 120^{\circ} | math | 87 |
Xiaoming's living room needs to be tiled. If choosing square tiles with a side length of 0.3 meters requires 500 tiles; how many tiles are needed if choosing square tiles with a side length of 0.5 meters? | 180 | math | 52 |
Given that line $l$ is the tangent line of the curve $y=x^{2}+x-2$ at point $(1,0)$, and $m$ is another tangent line of the curve, with $l \perp m$.
(1) Find the equation of line $m$.
(2) Find the area of the triangle formed by lines $l$, $m$, and the $x$-axis. | \frac{125}{12} | math | 91 |
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on side $AB$ with $AE=3\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$ | \frac{1}{8} | math | 109 |
The route from point A to point B consists only of uphill and downhill segments, with a total distance of 21 kilometers. If the uphill speed is 4 km/h and the downhill speed is 6 km/h, and the journey from point A to point B takes 4.25 hours, how long will it take to travel from point B to point A? | 4.5 \, \text{hours} | math | 76 |
Given a square with its four vertices and center, determine the probability that the distance between any two of these 5 points is not less than the length of the square's side. | \frac{3}{5} | math | 36 |
Find the geometric mean of the two numbers \\(\sqrt {2}+1\\) and \\(\sqrt {2}-1\\). | 1 | math | 28 |
Let the complex number z = 1 - 3i, and its conjugate complex number be \overline{z}. Calculate the absolute value of the quotient of z and \overline{z}. | 1 | math | 42 |
In a triangle, two heights are not less than the sides onto which they are dropped. Find the angles of the triangle. | 90^{\circ}, 45^{\circ}, 45^{\circ} | math | 25 |
1. Write the angle -1120° in the form of 2kπ+α (k∈Z), where 0≤α<2π.
2. Write the set of angles β that have the same terminal side as the angle α in part 1, and write the angle β in the interval [-4π, 0]. | -\frac{20π}{9} | math | 73 |
Expand $(3t^2 - 4t + 3)(-4t^2 + 2t - 6)$. | -12t^4 + 22t^3 - 38t^2 + 30t - 18 | math | 28 |
Three sides \(OAB, OAC\) and \(OBC\) of a tetrahedron \(OABC\) are right-angled triangles, i.e. \(\angle AOB = \angle AOC = \angle BOC = 90^\circ\). Given that \(OA = 7\), \(OB = 2\), and \(OC = 6\), find the value of
\[
(\text{Area of }\triangle OAB)^2 + (\text{Area of }\triangle OAC)^2 + (\text{Area of }\triangle OBC... | 1052 | math | 136 |
In a high school senior class, there are 4 boys and 3 girls in the extracurricular speech group. They need to select 3 boys and 2 girls, who will then give speeches one by one in the class. Calculate the number of ways for the 2 girls not to give speeches consecutively. | 864 | math | 67 |
The negation of the proposition "There exists an $x \in \mathbb{R}$, such that $x^2 - 2x + 1 < 0$" is. | \forall x \in \mathbb{R}, x^2 - 2x + 1 \geq 0 | math | 40 |
Three candles were lit simultaneously. When the first candle burned out, $\frac{2}{5}$ of the second candle and $\frac{3}{7}$ of the third candle remained. What fraction of the third candle will remain when the second candle burns out? | \frac{1}{21} | math | 53 |
Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be unit vectors such that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\theta,\) and the angle between \(\mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}\) is also \(2\theta.\) If \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{3},\) find the sm... | 30^\circ | math | 139 |
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\[
f(x f(y))+f(f(x)+f(y))=y f(x)+f(x+f(y))
\]
for all \( x, y \in \mathbb{R} \). | f(x) = 0 \text{ or } f(x) = x | math | 69 |
A team of mowers needed to mow two meadows - one twice the size of the other. They spent half a day mowing the larger meadow. After noon, the team split in half: the first half remained to finish mowing the larger meadow by evening, and the second half mowed the smaller meadow, where by the evening a section still rema... | 8 \text{ mowers in the team} | math | 100 |
Points \(M\) and \(N\) are the midpoints of two sides of the rectangle. What fraction of the rectangle's area is shaded?
A \(\frac{1}{6}\)
B \(\frac{1}{5}\)
C \(\frac{1}{4}\)
D \(\frac{1}{3}\)
E \(\frac{1}{2}\) | \frac{1}{4} | math | 79 |
Consider this histogram of scores for 100 students taking a test:
- Each interval represents a distinct score range, and the students' scores are distributed over these intervals with provided frequencies. Determine the score interval that contains the median.
- The intervals and student counts are as follows:
- $85... | 70-74 | math | 200 |
The interval that contains the root of the function $f(x) = x + \lg(x - 2)$ is. | (2.001, 2.01) | math | 25 |
How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | 432 | math | 47 |
Let $P$ be a plane passing through the origin. When $\begin{pmatrix} 5 \\ 3 \\ 5 \end{pmatrix}$ is projected onto plane $P,$ the result is $\begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}.$ When $\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix}$ is projected onto plane $P,$ what is the result? | \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix} | math | 99 |
Given the function $f(x) = x^{-3} + \sin x + 1$. If $f(a) = 3$, then $f(-a) =$ | -1 | math | 36 |
Among the numbers from 1000 to 9999, how many four-digit numbers have all distinct digits and the absolute difference between the thousandth digit and the unit digit is 2? | 840 | math | 42 |
Find the number of 8-tuples $(x_1, x_2, \dots, x_8)$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_7 - x_8)^2 + x_8^2 = \frac{1}{8}.\] | 1 | math | 91 |
Let $O$ be the coordinate origin and $P$ be the point with coordinates $(2,1)$. If $A$ and $B$ are points on the positive semi-axes of the $x$-axis and $y$-axis respectively such that $PA \perp PB$, find the maximum area of $\triangle OAB$. | \frac{25}{16} | math | 72 |
Given Ted's favorite number is equal to $\sum_{k=1}^{2007} k \cdot \binom{2007}{k}$, find the remainder when Ted's favorite number is divided by $25$. | 23 | math | 51 |
When making canned boxes with white iron sheets, each sheet can be used to make $14$ box bodies or $32$ box bottoms. One box body and two box bottoms form a set of canned boxes. There are $300$ white iron sheets available. Assuming $x$ sheets are used to make box bodies and $(300-x)$ sheets are used to make box bottoms... | 2 \times 14x = 32(300-x) | math | 100 |
If the side of one square is twice the side of a second square, calculate the ratio of the area of the first square to the area of the second square. | 4 | math | 33 |
Consider the rectangular hyperbola defined by the equation $xy = c^2$, where $c$ is a constant. Find the distance between the foci of this hyperbola. | 2\sqrt{2}c | math | 38 |
Find the equation of the tangent line to the function $y=x^{2}+\frac{1}{x}+1$ at $x=1$. | y = x + 2 | math | 32 |
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bot... | 52 | math | 114 |
A modified deck of 54 cards includes 13 ranks in each of 4 suits and 2 jokers that are suit and rank-less. What is the probability that one of the top two cards is the King of $\diamondsuit$? | \frac{1}{27} | math | 51 |
Let $\{a_n\}$ be a geometric sequence, and $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. For any positive integer $n$, it holds that $a_n + 2a_{n+1} + a_{n+2} = 0$. Given that $a_1 = 2$, calculate the value of $S_{101}$. | 2 | math | 90 |
Given \\(\alpha \in \left( \frac{\pi}{2}, \pi \right)\\) and \\(\sin \alpha = \frac{4}{5}\\), calculate the value of \\(\tan \alpha\\. | -\frac{4}{3} | math | 48 |
Given vectors $\overrightarrow{a}=(m,-1)$ and $\overrightarrow{b}=(1,m+2)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then solve for $m$. | -1 | math | 49 |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | math | 56 |
a) In how many different ways can a convex octagon be divided into triangles by diagonals that do not intersect inside the octagon?
b) Euler's problem. In how many ways can a convex $n$-gon be divided into triangles by diagonals that do not intersect inside the $n$-gon? | \frac{(2n-4)!}{(n-1)!(n-2)!} | math | 65 |
Given that the parametric equation of line $(1)$ is $ \begin{cases} x=3-t \\ y= \sqrt {5}+t \end{cases} (t $ is the parameter$)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ=2 \sqrt... | 3 | math | 160 |
In the equation
$$
x^{3}+a x^{2}+b x+6=0,
$$
determine $a$ and $b$ such that one of the roots is 2 and another root is 3. Additionally, calculate the third root. | a = -4, b = 1, x_3 = -1 | math | 58 |
Given $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+\frac{3}{2}$.
$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\pi]$.
$(2)$ If $f(\alpha)=\frac{2}{5}$ and $\alpha\in(\frac{\pi}{3},\frac{5\pi}{6})$, find the value of $\sin 2\alpha$. | \frac{-3\sqrt{3} - 4}{10} | math | 103 |
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 | math | 54 |
Express $0.6\overline{333}$ as a common fraction. | \frac{19}{30} | math | 18 |
For what value of $n$ is the four-digit number $514n$, with units digit $n$, divisible by 12? | 8 | math | 30 |
What fraction of the Earth's volume lies above the $45$ degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane. | \frac{8-5\sqrt{2}}{16} | math | 54 |
A sign painter paints individual numerals for a row of 100 houses. The houses are numbered with consecutive integers from 1 to 100. How many 9s are painted, and what is the total count of all digits used in painting? | 192 | math | 54 |
Let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Then, for the function $y = x - \lfloor x \rfloor$ ($x \in \mathbb{R}$), the following statements are made: (1) It has a maximum and a minimum value; (2) It is a periodic function; (3) It is an increasing function; (4) It is an even function... | 2 | math | 110 |
A sequence ${a_{n}}$ satisfies $a_{1}=1$ and $(a_{1}+a_{2})+(a_{2}+a_{3})+(a_{3}+a_{4})+…+(a_{n}+a_{n+1})=2^{n+1}-2$. Determine the value of $a_{8}$. | 85 | math | 80 |
A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inche... | 41 | math | 100 |
Given $f(x) = 3^x + 3x - 8$, using the bisection method to find an approximate solution of the equation $3^x + 3x - 8 = 0$ within the interval $x \in (1, 3)$, and taking the midpoint of the interval $x = 2$, determine the next interval containing a root. | (1, 2) | math | 82 |
Given the function $f(x)=x(m+e^{-x})$ where $e$ is the base of the natural logarithm, there exist two different points on the curve $y=f(x)$ such that the tangents to the curve at these two points are both perpendicular to the $y$-axis. The range of the real number $m$ is \_\_\_\_\_\_\_\_. | (0,e^{-2}) | math | 83 |
Let \( p \) be an odd prime number. Each of the numbers \( \binom{p-2}{0}, \binom{p-2}{1}, \ldots, \binom{p-2}{p-2} \) is divided by \( p \) and the remainders are considered. How many different remainders are obtained? | p-1 | math | 76 |
Let $p(x)$ be a monic, quartic polynomial, such that $p(1) = 3,$ $p(3) = 11,$ and $p(5) = 27.$ Find
\[p(-2) + 7p(6).\] | 1112 | math | 62 |
Six seats are arranged in a circular table. Each seat is to be painted in red, blue, or green such that any two adjacent seats have different colors. How many ways are there to paint the seats? | 66 | math | 42 |
Find the value of the constant $a$ such that the coefficient of $x$ in the expansion of $\left(x-\frac{a}{x}\right)\left(1-\sqrt{x}\right)^{6}$ is $31$. | a = -2 | math | 50 |
If the complex number $z$ satisfies $(1+i)z=2$, solve for $z$. | 1-i | math | 21 |
In a diagram, there is an equilateral triangle with a side length of $10$ m. Calculate both the perimeter and the height of the triangle. | 5\sqrt{3} | math | 32 |
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct? | 5040 | math | 59 |
A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals. | 1139.71 | math | 84 |
Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$ If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$ | x^2 | math | 66 |
Given $510^4$, determine the number of its positive integer divisors that are perfect squares or perfect cubes (or both). | 96 | math | 28 |
In the interval $(0,2\pi)$, determine the range of $x$ that satisfies $\sin x > \cos x$. | ( \frac{\pi}{4}, \frac{5\pi}{4}) | math | 28 |
Given that $|2a+b|$ and $\sqrt{3b+12}$ are opposite in sign.<br/>$(1)$ Find the square root of $2a-3b$;<br/>$(2)$ Solve the equation $ax^{2}+4b-2=0$ in terms of $x$. | \pm 3 | math | 68 |
Given two non-zero vectors $m$ and $n$ with an angle of $\frac{\pi}{3}$ between them, and $|n| = λ|m| (λ > 0)$. The vector group $x_1$, $x_2$, $x_3$ is composed of one $m$ and two $n$'s, while the vector group $y_1$, $y_2$, $y_3$ is composed of two $m$'s and one $n$. If the minimum possible value of $x_1 \cdot y_1 + x_... | λ = \frac{8}{3} | math | 156 |
Given the complex number $z$ that satisfies $z= \frac {1}{2}+ \frac {5}{2}i$, find $|z|$. | \frac {\sqrt {26}}{2} | math | 35 |
Given a circle with the equation \\(x^{2}+2x+y^{2}=0\\), find the equation of the circle with the same center and a radius of \\(2\\). | (x+1)^2 + y^2 = 4 | math | 41 |
Given the set $A=\{-1,-\frac{1}{2},-\frac{1}{3},0,\frac{2}{3},2,3\}$, $B=\{y=f\left(x\right)|f\left(x\right)=x^{k}$, $k\in A$ and the graph of $y=f\left(x\right)$ is symmetric about the $y$-axis$\}$, then the number of elements in set $B$ is ______. | 3 | math | 105 |
Given a student throws a die $5$ times, recording the number of points each time, with an average of $3$ and a variance of $0.4$, determine the number of times the number $2$ appears. | 1 | math | 47 |
A unit has 800 employees, among which 160 have senior titles, 320 have intermediate titles, 200 have junior titles, and the remaining 120 people. Determine the number of people to be drawn from each stratum when a sample of 40 people is drawn using stratified sampling. | 8, 16, 10, 6 | math | 71 |
In a geometric sequence with positive terms $\{a_n\}$, it is given that $a_1 a_3 = 1$ and $a_2 + a_3 = \frac{4}{3}$. Determine the limit $\lim_{n\to \infty} (a_1 + a_2 + \ldots + a_n)$. | \frac{9}{2} | math | 77 |
Given the function $f(x)=|2x+1|+|2x-3|$,
$(1)$ Find the solution set of the inequality $f(x) \leqslant 6$;
$(2)$ Given $a > 0$, if the solution set of the inequality $f(x) < |a-2|$ with respect to $x$ is non-empty, find the range of the real number $a$. | (6, +\infty) | math | 91 |
Given that in $\triangle ABC$, the sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and $a\cos B + \sqrt{3}b\sin A = c$.
(I) Find the measure of angle $A$;
(II) If $a=1$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 3$, find the value of $b + c$. | \sqrt{3} + 2 | math | 102 |
For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1 + 2 + 3 + \cdots + n$. What is the greatest possible value of the greatest common divisor of $6T_n$ and $n-1$? | 3 | math | 64 |
Let \( b = 1^{2} - 2^{2} + 3^{2} - 4^{2} + 5^{2} - \ldots - 2012^{2} + 2013^{2} \). Determine the remainder of \( b \) divided by 2015. | 1 | math | 73 |
A tree grows in a rather peculiar manner. Lateral cross-sections of the trunk, leaves, branches, twigs, and so forth are circles. The trunk is 1 meter in diameter to a height of 1 meter, at which point it splits into two sections, each with diameter 0.5 meter. These sections are each one meter long, at which point they... | \frac{\pi}{2} | math | 131 |
Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \le k \le 150$, how many $a_k$ are divisible by both 3 and 5? | 10 | math | 92 |
Given an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. If the line segment $F\_1F\_2$ is divided by the parabola $y^{2}=2bx$'s focus into two segments with a ratio of $5:1$, find the eccentricity of this ellipse. | \frac{3}{5} | math | 106 |
If the pattern of circles ○●○○●○○○●○○○○●○○○○○● continues, forming a series of circles, calculate the number of ● in the first 120 circles. | 14 | math | 47 |
Let \(x\) and \(y\) be real numbers such that \(2(x^3 + y^3) = x + y\). Find the maximum value of \(x - y\). | \frac{\sqrt{2}}{2} | math | 40 |
Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$ | (1, 1, 1) | math | 91 |
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line? | 1000000 | math | 26 |
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