task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | A hot pot restaurant launched two double meal sets to attract customers. The table below shows the income statistics of the two sets in the past two days:
| | Quantity of Set A | Quantity of Set B | Income |
|----------|-------------------|-------------------|--------|
| Day 1 | 20 times | 10 time... | 5 | 333 | 1 |
math | Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the sm... | 14 | 214 | 2 |
math | If each interior angle of a $n$-sided regular polygon is $140^{\circ}$, find $n$.
If the solution of the inequality $2 x^{2}-n x+9<0$ is $k<x<b$, find $b$.
If $c x^{3}-b x+x-1$ is divided by $x+1$, the remainder is -7, find $c$.
If $x+\frac{1}{x}=c$ and $x^{2}+\frac{1}{x^{2}}=d$, find $d$. | d = 62 | 125 | 5 |
math | A regular hexagon $ABCDEF$ has sides of length 3. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{27\sqrt{3}}{4} | 33 | 13 |
math | Let $k$ be a real number such that the product of real roots of the equation $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$ is $-2013$ . Find the sum of the squares of these real roots. | 4027 | 82 | 4 |
math | If two lines $ax+2y-1=0$ and $x+(a-1)y+a^{2}=0$ are parallel, determine the distance between the two lines. | \dfrac {9 \sqrt {2}}{4} | 38 | 12 |
math | The lengths of the sides of a triangle are \( n-1 \), \( n \), and \( n+1 \), and the second largest angle is twice the smallest angle. Find the cosine of the smallest angle. | \frac{3}{4} | 45 | 7 |
math | Suppose that \( n \) is a positive integer and that the set \( S \) contains exactly \( n \) distinct positive integers. If the mean of the elements of \( S \) is equal to \( \frac{2}{5} \) of the largest element of \( S \) and is also equal to \( \frac{7}{4} \) of the smallest element of \( S \), determine the minimum... | 5 | 96 | 1 |
math | Let \( B_1 \) be the midpoint of segment \( BT \). On the extension of segment \( AT \) beyond point \( T \), mark such a point \( H \) that \( TH = T B_1 \). Then
\[
\angle THB_1 = \angle TB_1H = \frac{\angle ATB_1}{2} = 60^\circ
\]
Therefore, \( HB_1 = TB_1 = B_1B \). Then
\[
\angle BHB_1 = \angle B_1B H = \frac{\ang... | 2AB + 2BC + 2CA > 4AT + 3BT + 2CT | 332 | 22 |
math | Given real numbers \( x \) and \( y \) such that \( x^2 + y^2 = 25 \), find the maximum value of the function:
\[
f(x, y) = \sqrt{8y - 6x + 50} + \sqrt{8y + 6x + 50}
\] | 6\sqrt{10} | 76 | 7 |
math | Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ . | 35 | 32 | 2 |
math | Given $0 < \beta < \frac{\pi}{4} < \alpha < \frac{\pi}{2}$, $\cos(2\alpha - \beta) = -\frac{11}{14}$, $\sin(\alpha - 2\beta) = \frac{4\sqrt{3}}{7}$, find the value of $\sin\frac{\alpha + \beta}{2}$. | \frac{1}{2} | 89 | 7 |
math | Medians $\overline{AF}$ and $\overline{BD}$ of $\triangle ABC$ are perpendicular, and they intersect the third median $\overline{CE}$. If $AF = 12$ and $BD = 16$, compute the area of $\triangle ABC$. | 128 | 60 | 3 |
math | (2013•Hubei) Assume the number of passengers traveling from place A to place B per day, $X$, follows a normal distribution $N(800, 50^2)$. Let $p_0$ denote the probability that the number of passengers traveling from place A to place B in a day does not exceed 900.
(1) Find the value of $p_0$.
(Reference data: If $X ... | 12 | 374 | 2 |
math | Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$. | 504 | 72 | 3 |
math | Let \[f(x) = \left\{
\begin{array}{cl}
x^2 - 4 &\text{ if }x>6, \\
3x + 2 &\text{ if } -6 \le x \le 6, \\
5 &\text{ if } x < -6.
\end{array}
\right.\] If \( x \) is a multiple of 3, add 5 to \( f(x) \). Find \( f(-8) + f(0) + f(9) \). | 94 | 118 | 2 |
math | Given vectors $\overrightarrow {m}$=（cosα，sinα）, $\overrightarrow {n}$=（-1，2）
(1) If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, find the value of $\frac {sinα-2cosα}{sin\alpha +cos\alpha }$;
(2) If $| \overrightarrow {m} - \overrightarrow {n}|= \sqrt {2}$, α∈$( \frac {π}{2}，π)$, find the value of tanα. | -\frac{3}{4} | 121 | 7 |
math | Suppose that $a,$ $b,$ and $c$ are three positive numbers that satisfy the equations $abc = 1,$ $a + \frac {1}{c} = 8,$ and $b + \frac {1}{a} = 20.$ Find $c + \frac {1}{b}.$ | \frac{10}{53} | 69 | 9 |
math | Two circles touch each other internally. It is known that two radii of the larger circle, which form an angle of $60^\circ$ between them, are tangent to the smaller circle. Find the ratio of the radii of the circles. | 3 | 51 | 1 |
math | Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together. | 34560 | 47 | 5 |
math | In an isosceles triangle, if two sides have lengths of $8$ and $9$, then the perimeter of the triangle is $\_\_\_\_\_\_$. | 25 \text{ or } 26 | 36 | 10 |
math | In the function $y=\frac{{\sqrt{x+3}}}{x-1}$, the range of the independent variable $x$ is ____. | x \geqslant -3 \quad \text{and} \quad x \neq 1 | 32 | 23 |
math | The number represented by point A on the number line is 2. Moving 7 units to the left reaches point B. The number represented by point B is __________. Moving 1 and $$\frac{2}{3}$$ units to the right from point B reaches point C. The number represented by point C is __________. | -3\frac{1}{3} | 69 | 9 |
math | In the Cartesian coordinate plane $(xOy)$, point $P$ is a moving point on the graph of the function $f(x) = \ln x (x \geqslant 1)$. The tangent line $l$ to the graph at point $P$ intersects the $x$-axis at point $M$. The line perpendicular to $l$ passing through point $P$ intersects the $x$-axis at point $N$. Let $t$ b... | t_{\text{max}} = \frac{1}{2}\left(2e + \frac{1}{e} - e\right) = \frac{e^2 + 1}{2e} | 122 | 46 |
math | It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$ . Find the value of $f(\cos x)$ . | \cos(2011x) | 56 | 9 |
math | The minimum value of the distance $|AB|$ is to be found, where points $A$ and $B$ are the intersections of the line $y=m$ with the curves $y = 2(x+1)$ and $y = x + \ln x$ respectively. | \frac{3}{2} | 58 | 7 |
math | The square quilt block shown is made from sixteen unit squares, four of which have been divided in half diagonally to form triangles. What fraction of the square quilt is shaded? Express your answer as a common fraction.
[asy]size(100);
fill(scale(4)*unitsquare,gray(.6));
path[] interior = (1,0)--(0,1)--(1,1)--cycle^^... | \frac{1}{8} | 211 | 7 |
math | The Rotokas of Papua New Guinea have twelve letters in their alphabet: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters use only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either A or E, end with V, cannot contain I, and have no... | 1440 | 88 | 4 |
math | Find the derivative.
\[ y = -\frac{1}{4} \arcsin \frac{5+3 \cosh x}{3+5 \cosh x} \] | \frac{1}{3 + 5 \cosh x} | 40 | 14 |
math | Let
\[g(x) = \left\{
\begin{array}{cl}
2x + 4 & \text{if $x < 10$}, \\
x - 3 & \text{if $x \ge 10$}.
\end{array}
\right.\]
Find $g^{-1}(8) + g^{-1}(16).$ | 21 | 81 | 2 |
math | Given $m$ and $n$ are two non-coincident lines, and $\alpha$ and $\beta$ are two non-coincident planes. The following four propositions are given:
① If $m \subset \alpha$, $n \subset \alpha$, $m \parallel \beta$, $n \parallel \beta$, then $\alpha \parallel \beta$;
② If $m \perp \alpha$, $n \perp \beta$, $m \paralle... | 2 | 207 | 1 |
math | Given a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$, where $S_n=2a_n-2$,
(Ⅰ) Find the general term formula of the sequence $\{a_n\}$;
(Ⅱ) Let $b_n=\log_2 a_n$ and $c_n=\frac{1}{b_n b_{n+1}}$, with the sum of the first $n$ terms of the sequence $\{c_n\}$ denoted by $T_n$, find $T_n$;
(Ⅲ... | (n-1)2^{n+1} + 2 | 167 | 13 |
math | If for any \(\theta \in \mathbf{R}\), the modulus of the complex number \(z=(a+\cos \theta)+(2a-\sin \theta) \mathrm{i}\) does not exceed 2, then the range of the real number \(a\) is \(\qquad\). | \left[-\frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5}\right] | 65 | 25 |
math | Given the complex number $z$ that satisfies the equation $i \cdot z = 1 + i$, find the imaginary part of the conjugate of $z$. | 1 | 34 | 1 |
math | Given that $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda\overrightarrow{BC}$, find the value of the real number $\lambda$. | -\frac{4}{3} | 59 | 7 |
math | Given $|a|=3$, $|b|=4$, and $a \lt b$, find the value of $\frac{a-b}{a+b}$. | -7 \text{ or } -\frac{1}{7} | 34 | 15 |
math | Triangle $ABC$ has $AB=24$, $AC=26$, and $BC=30$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the centroid of triangle $ABC$. Find $DE$ expressed as a fraction in lowest terms, and calculate $m+n$ where $DE=\f... | 11 | 119 | 2 |
math | Given that \(a, b, c\) are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120 \text {, }
$$
determine the maximum possible value of \(a\). | 240 | 70 | 3 |
math | Find the measure of $\angle FYD$, given that $\overline{AB} \parallel \overline{CD}$ and $\angle AXF = 130^\circ$, and $\triangle XFG$ is an isosceles triangle with $\angle FXG = 36^\circ$. | 50^\circ | 63 | 4 |
math | Convert the binary number $10101_{(2)}$ into a quaternary number. The result is \_\_\_\_\_\_. The greatest common divisor of 918 and 714 is \_\_\_\_\_\_. | 111_{(4)}, 102 | 54 | 11 |
math | Let \( Q \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{QA} + 3\overrightarrow{QB} + 4\overrightarrow{QC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( AQC \). | 8 | 73 | 1 |
math | The sequence \(a_{n}\) is defined as follows:
\[ a_{1} = 1, \quad a_{n+1} = a_{n} + \frac{2a_{n}}{n} \text{ for } n \geq 1 \]
Find \(a_{100}\). | 5151 | 69 | 4 |
math | Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification. | \log \left( \frac{3}{2} \right) | 45 | 15 |
math | Given a dihedral angle with points A and B on its edge, and lines AC and BD are each in the two half-planes of the dihedral angle, with both perpendicular to AB. If AB=4, AC=6, BD=8, and CD = 2√17, find the measurement of the dihedral angle. | 60^\circ | 73 | 4 |
math | Given five positive consecutive integers starting from $a$, their average is $b$. If $b$ is used as the starting point, find the average of the first three integers that are each 10 units apart starting from $b$. | a+12 | 48 | 4 |
math | Compute $\frac{2468_{10}}{121_{3}} + 3456_{7} - 9876_{9}$. Express your answer in base 10. | -5857.75 | 47 | 8 |
math | In triangle \( \triangle ABC \), \(\angle ABC = 50^\circ\), \(\angle ACB = 30^\circ\), \(M\) is a point inside the triangle such that \(\angle MCB = 20^\circ\), \(\angle MAC = 40^\circ\). Find the measure of \(\angle MBC\).
(Problem 1208 from Mathematical Bulletin) | 30 | 92 | 2 |
math | A game begins with 7 coins aligned on a table, all showing heads up. To win the game, you need to flip some coins so that in the end, each pair of adjacent coins has different faces up. The rule of the game is: in each move, you must flip two adjacent coins. What is the minimum number of moves required to win the game? | 4 \text{ jogadas} | 75 | 7 |
math | The number of ordered pairs \((a, b)\) that satisfy the equation \((a+b \mathrm{i})^{6}=a-b \mathrm{i}\) where \(a, b \in \mathbf{R}\) and \(\mathrm{i}^{2}=-1\) is ___. | 8 | 63 | 1 |
math | Over the next 5 days, 3500 people have plans to move to Oregon. What is the average number of people moving to Oregon per hour? | 29 | 33 | 2 |
math | Given a sequence $\{a\_n\}$ with the first term $a\_1=2$, the sum of the first $n$ terms is $S\_n$. The sequence satisfies the equation $2a_{n+1}+S\_n=3 (n∈N^*)$. Find the sum of all $n$ that satisfy $\frac{34}{33} < \frac{S_{2n}}{S\_n} < \frac{16}{15}$. | 9 | 106 | 1 |
math | Given triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, satisfying $\frac{a}{{2\cos A}}=\frac{b}{{3\cos B}}=\frac{c}{{6\cos C}}$, then $\sin 2A=$____. | \frac{3\sqrt{11}}{10} | 75 | 14 |
math | Let \(a_{1}, a_{2}, \ldots, a_{n}\) be real numbers. Consider the \(2^{n}-1\) non-empty sums that can be formed from these numbers. How many of these sums can be positive? | 2^{n-1} | 52 | 6 |
math | Given a sequence of _m_ terms (\(m \in \mathbb{N}^*\), \(m \geq 3\)) denoted as \(\{a_n\}\), where \(a_i \in \{0, 1\}\) (\(i = 1, 2, 3, \ldots, m\)), such a sequence is called a "0-1 sequence". If there exists a positive integer _k_ (\(2 \leq k \leq m - 1\)) such that there are some _k_ consecutive terms in the sequenc... | 5 | 384 | 1 |
math | If the equation $mx^2+2x+1=0$ has at least one negative root, then the range of the real number $m$ is \_\_\_\_\_\_. | (-\infty, 1] | 40 | 8 |
math | Let set $A=\{a\in\mathbb{R}|2^a=4\}$, $B=\{x\in\mathbb{R}|x^2-2(m+1)x+m^2<0\}$.
(1) If $m=4$, find $A\cup B$;
(2) If $A\cap B=B$, find the range of the real number $m$. | (-\infty, -\frac{1}{2}] | 93 | 13 |
math | Given the real number \( a \geqslant -2 \), with the sets defined as
\[
\begin{array}{l}
A=\{x \mid-2 \leqslant x \leqslant a\}, \\
B=\{y \mid y=2 x+3, x \in A\}, \\
C=\left\{z \mid z=x^{2}, x \in A\right\},
\end{array}
\]
if \( C \subseteq B \), then the range of values for \( a \) is | \left[ \frac{1}{2}, 3 \right] | 120 | 15 |
math | Given the inequality $|x+3|+|x+m| \geq 2m$ with respect to $x$ and its solution set is $\mathbb{R}$.
1. Find the range of values for the real number $m$.
2. Given $a > 0$, $b > 0$, $c > 0$ and $a + b + c = m$, find the minimum value of $a^2 + 2b^2 + 3c^2$ and the corresponding values of $a$, $b$, and $c$ when $m$ is a... | \frac{6}{11} | 131 | 8 |
math | The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimite... | 4 \text{ hours} | 144 | 6 |
math | Find the value of \( a \) such that the equation \( \sin 4x \cdot \sin 2x - \sin x \cdot \sin 3x = a \) has a unique solution in the interval \([0, \pi)\). | 1 | 55 | 1 |
math | A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | 986 | 107 | 3 |
math | Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 15$ and $SR = 8$, find the distance from $S$ to the plane $PQR$. | 8 | 80 | 1 |
math | In triangle \(ABC\) with a \(120^\circ\) angle at vertex \(A\), the bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. Find the angle \(C_1 A_1 B_1\). | 90^\circ | 60 | 4 |
math | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $(\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{a}$, and $(2\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 93 | 9 |
math | Find the angle in the range (-360°, 0°) whose terminal side is the same as that of the angle 1250°. | -190° | 34 | 5 |
math | Calculate the sum of the series $2 + 4 + 8 + 16 + 32 + \cdots + 512 + 1000$. | 2022 | 38 | 4 |
math | Let $a_1,a_2,\dots,a_{2021}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2021}=2021^{2021}.\] Find the remainder when $a_1^3+a_2^3+\cdots+a_{2021}^3$ is divided by $6$. | 5 | 91 | 1 |
math | Given two complex numbers $z_1$ and $z_2$ which satisfy $z_1 = m + (4 - m^2)i$ and $z_2 = 2\cos \theta+( \lambda + 3\sin \theta)i$ where $m$, $\lambda$, $\theta \in \mathbb{R}$, and $z_1 = z_2$, determine the range of possible values for $\lambda$. | \left[- \frac{9}{16}, 7\right] | 94 | 16 |
math | Each Kinder Surprise contains exactly 3 different gnomes, and there are 12 types of gnomes in total. The box contains a sufficient number of Kinder Surprises, and no two of them contain the same set of three gnomes. What is the minimum number of Kinder Surprises that need to be purchased to ensure that, after opening t... | 166 | 92 | 3 |
math | Given $$\sin(\pi + \alpha) = \frac{3}{5}$$, and $\alpha$ is an angle in the third quadrant, then $\tan(\alpha) = \_\_\_\_\_\_$, and $$\frac{\sin \frac{\pi + \alpha}{2} - \cos \frac{\pi + \alpha}{2}}{\sin \frac{\pi - \alpha}{2} - \cos \frac{\pi - \alpha}{2}} = \_\_\_\_\_\_.$$ | -\frac{1}{2} | 110 | 7 |
math | Given the set $\{ -3, -2, -1, 0, 0, 2, 4, 5 \}$, what is the probability that the product of two different numbers selected from this set is $0$. | \frac{3}{14} | 50 | 8 |
math | Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A cylinder passes through $(a,b,c)$ with its axis parallel to the z-axis. It intersects the $x$-axis and $y$-axis at points $A$ and $B$, respectively. If $(p,q)$ is the center of the circular base of the cylinder, find:
\[\frac{a}{p} + \frac{b}{q}.\] | 4 | 99 | 1 |
math | Given the inequality $|2x + 1| - |x - 1| \leqslant \log_2 a$ (where $a > 0$).
(1) Find the solution set when $a = 4$;
(2) Find the range of real number $a$ if the inequality has a solution. | a \in [\frac{\sqrt{2}}{4}, +\infty) | 72 | 18 |
math | Find the minimum value of
\[
(\sin x + \tan x)^2 + (\cos x + \cot x)^2
\]
for $0 < x < \frac{\pi}{2}$. | 5 | 44 | 1 |
math | Given sets $A=\{x|x^{2}+\left(m-2\right)x-2m=0\}$ and $B=\{x\left|\right.mx=2x+1\}$, if $B\subseteq A$, find the value of real number $m$. | \frac{5}{2} | 61 | 7 |
math | The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. Determine the number of such pairs. | 3 | 29 | 1 |
math | A number is called flippy if its digits alternate between two distinct digits. How many five-digit flippy numbers are divisible by 11? | 0 | 29 | 1 |
math | Given the parabola $C$: $y^{2}=4x$ with focus $F$, and a line $MN$ passing through the focus $F$ and intersecting the parabola $C$ at points $M$ and $N$, where $D$ is a point on the line segment $MF$ and $|MD|=2|NF|$. If $|DF|=1$, then $|MF|=$ ______. | 2+ \sqrt {3} | 92 | 7 |
math | Let \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0) = p(3) = \cdots = p(3n) = 2 \), \( p(1) = p(4) = \cdots = p(3n-2) = 1 \), \( p(2) = p(5) = \cdots = p(3n-1) = 0 \), and \( p(3n+1) = 730 \). Find \( n \). | 4 | 120 | 1 |
math | Consider the $4\times4$ array of $16$ dots, shown below.
[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
Counting the number of squares whose vert... | 4 | 249 | 1 |
math | Given that each digit of a 5-digit positive integer is an even number, calculate how many such integers are divisible by 4. | 1875 | 27 | 4 |
math | A circle is inscribed in a square and passes through the midpoint of one of the diagonals of the square. The radius of the circle is $r$. Calculate the area of the square in terms of $r$.
A) $r^2$
B) $\sqrt{2}r^2$
C) $2r^2$
D) $4r^2$ | 2r^2 | 79 | 4 |
math | Let $a,$ $b,$ $c,$ $d,$ and $e$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} + \frac{1}{e + \omega} = \frac{3}{\omega^2},
\]
find
\[
\fr... | 3 | 190 | 1 |
math | A square is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1,\]
such that its sides are rotated 45 degrees relative to the coordinate axes. Find the area of this square. | 32 | 56 | 2 |
math | Circle \(\omega\) is inscribed in rhombus \(H M_{1} M_{2} T\) so that \(\omega\) is tangent to \(\overline{H M_{1}}\) at \(A\), \(\overline{M_{1} M_{2}}\) at \(I\), \(\overline{M_{2} T}\) at \(M\), and \(\overline{T H}\) at \(E\). Given that the area of \(H M_{1} M_{2} T\) is 1440 and the area of \(E M T\) is 405, find... | 540 | 148 | 3 |
math | For which integers \( n \in \{1, 2, \ldots, 15\} \) is \( n^n + 1 \) a prime number? | 1, 2, 4 | 38 | 7 |
math | Triangle ABC is an isosceles triangle with sides AB = AC, and O is the center of its inscribed circle. If the area of the circle is $9\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form. | 36 \text{ cm}^2 | 63 | 9 |
math | If regular triangles and dodecagons are used for plane tiling, determine the possible number of kinds of tilings. | 1 | 26 | 1 |
math | For which values of the real parameters \( a \) and \( b \) will the maximum of the function \( \left|x^{2}+a x+b\right| \) on the interval \([-1, 1]\) be the smallest? | a = 0 \quad \text{and} \quad b = -\frac{1}{2} | 53 | 23 |
math | In acute $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $(a^{2}+b^{2}-c^{2})\tan C= \sqrt {3}ab$.
$(1)$ Find angle $C$;
$(2)$ If $c= \sqrt {7}$ and $b=2$, find the value of side $a$ and the area of $\triangle ABC$. | \dfrac {3 \sqrt {3}}{2} | 105 | 12 |
math | A math interest group at a school is composed of $m$ students, and the school has specifically assigned $n$ teachers as advisors. During one of the group's activities, each pair of students posed a question to each other, each student posed a question to each teacher, and each teacher posed a question to the entire gro... | m=6, n=3 | 91 | 7 |
math | Given \(\vec{O}P = (2,1)\), \(\vec{O}A = (1,7)\), and \(\vec{O}B = (5,1)\), let \(X\) be a point on the line \(OP\) (with \(O\) as the coordinate origin). Find the measure of the angle \(\angle AXB\) when the dot product \(\vec{X}A \cdot \vec{X}B\) is minimized. | \arccos\left(-\frac{4 \sqrt{17}}{17}\right) | 102 | 23 |
math | Given that vector $\overrightarrow{a}\cdot(\overrightarrow{a}+2\overrightarrow{b})=0$, $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 66 | 9 |
math | Given Cindy leaves school at the same time every day, and if she cycles at $20 \mathrm{~km} / \mathrm{h}$ she arrives home at $4:30$ in the afternoon, and if she cycles at $10 \mathrm{~km} / \mathrm{h}$ she arrives home at 5:15 in the afternoon, find the speed, in $\mathrm{km} / \mathrm{h}$, at which she must cycle to ... | 12 | 113 | 2 |
math | Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+1$ cents, $101$ cents is the greatest postage that cannot be formed. | 18 | 50 | 2 |
math | Three distinct vertices of a regular tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron. | 0 | 35 | 1 |
math | In triangle $PQR,$ where $p = 8,$ $q = 10,$ and $r = 6.$ Let $J$ be the incenter.
Then
\[\overrightarrow{J} = x \overrightarrow{P} + y \overrightarrow{Q} + z \overrightarrow{R},\] where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$ Enter the ordered triple $(x,y,z).$ | \left(\frac{1}{3}, \frac{5}{12}, \frac{1}{4}\right) | 108 | 26 |
math | The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? | 432 | 54 | 3 |
math | Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following:
(a) $A \not= B$
(b) $A \cup B = X$ | \frac{3^n - 1}{2} | 53 | 11 |
math | The Price 'n' Shine Store, originally increases all items' prices by $30\%$. Subsequently, the store announces a sale, offering $25\%$ off these new prices. Additionally, a tax of $10\%$ is applied to the sale prices. What percentage of the original price does a customer end up paying after these adjustments?
A) $107.2... | 107.25\% | 123 | 8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.